Chapters 3 and 6 Sample Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Suppose you were to collect data for the pair of given variables in order to make a scatterplot. Determine for each variable if it is the explanatory variable or the response variable. Variables: Cloudy days, rainy days 2) Suppose you were to collect data for the pair of given variables in order to make a scatterplot. Determine for each variable if it is the explanatory variable or the response variable. Variables: Minutes of homework, grade on exam 3) Hi-Tech Agi Inc wants to determine if the rainfall in inches can be used to predict the yield per acre on a corn farm. Identify the response variable. Fill in the blank. 4) The ____________________ defines the groups to be compared with respect to values on the response variable. 5) Whenever we distinguish between a response variable and an explanatory variable, it is natural to form conditional proportions for categories of the ____________________. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 6) The ____________________ is the outcome variable on which comparisons are made. A) lurking variable B) response variable C) explanatory variable D) predictor variable E) Both b. and c. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 7) For the following pairs of variables, which more naturally is the response variable and which is the explanatory variable? a. b. c. College grade point average and college entrance exam score Students' interest and ability in studying a foreign language Speed of professional advancement and standard of dress 8) A large manufacturer hires many handicapped workers and keeps track of both their type of handicap and their level of performance. a. b. Identify the two variables. Identify the response variable and the explanatory variable. 1 Complete the conditional proportion table and use it to solve the problem. 9) The partially filled conditional proportion table gives the relative frequencies of the data on age (in years) and sex from the residents of a retirement home. Age (yrs) 60-69 70-79 Over 79 Total Male 0.19 0.1 0.11 Female 0.2 0.1 0.3 Total 1 What percentage of residents are males over 79? 10) The partially filled conditional proportion table gives the relative frequencies of the data on age (in years) and sex from the residents of a retirement home. Age (yrs) 60-69 70-79 Over 79 Total Male 0.19 0.1 0.11 Female 0.2 0.2 0.2 Total 1 What percentage of residents are females in the age group 70-79? 11) Most patients who undergo surgery make routine recoveries and are discharged as planned, but some patients experience complications and their discharge is delayed. Jamestown has a large hospital and a small hospital, each performing major and minor surgeries. Data is collected at each hospital to see how many surgical patients have their discharges delayed by postsurgical complications. The results are shown in the following table. Discharge Delayed Major surgery Minor surgery Large hospital 70 of 700 12 of 200 Small hospital 15 of 60 24 of 300 Overall, for what percent of surgical patients was discharge delayed? Provide an appropriate response. 12) According to the article "Motion Sickness in Public Road Transport: The Effect of Driver, Route and Vehicle", seat position within a bus may have some effect on whether one experiences motion sickness. The table below classifies each person in a random sample of bus riders by the location of his or her seat and whether nausea was reported. Front Middle Rear Nausea 58 166 193 No Nausea 870 1163 806 Source: Ergonomics (1999): 1646-1664. a. b. c. What is the response variable, and what is the explanatory variable? How do the proportions experiencing nausea compare for the 3 seat positions? What proportion of all sampled bus riders experienced nausea? 2 13) The article "The Association Between Smoking and Unhealthy Behaviors Among a National Sample of Mexican-American Adolescents" presents data resulting from a random sample of Mexican-American male adolescents. Each boy, age 10-18, was classified according to smoking status and his response to a question asking whether he liked to do risky things. Smoking Status Smoker Nonsmoker 45 46 36 153 Likes Risky Things Doesn't Like Risky Things Source: Journal of School Health (1998): 376-379 a. How does the proportion of sampled smokers that likes risky things compare to the proportion of sampled nonsmokers that likes risky things? b. What proportion of all sampled Mexican-American male adolescents likes risky things? 14) According to the article "Motion Sickness in Public Road Transport: The Effect of Driver, Route and Vehicle", seat position within a bus may have some effect on whether one experiences motion sickness. The table below classifies each person in a random sample of bus riders by the location of his or her seat and whether nausea was reported. Front Middle Rear Nausea 58 166 193 No Nausea 870 1163 806 Source: Ergonomics (1999): 1646-1664. a. b. c. Create a side-by-side bar graph that compares the three seat positions with respect to nausea status. Summarize the results of the side-by-side bar graph. Is there an association between motion sickness and seat position in a bus? Explain. 15) The article "The Association Between Smoking and Unhealthy Behaviors Among a National Sample of Mexican-American Adolescents" presents data resulting from a random sample of Mexican-American male adolescents. Each boy, age 10-18, was classified according to smoking status and his response to a question asking whether he liked to do risky things. Smoking Status Smoker Nonsmoker 45 46 36 153 Likes Risky Things Doesn't Like Risky Things Source: Journal of School Health (1998): 376-379 a. Create a side-by-side bar graph that compares smoking status with respect to risky things status. b. Summarize the results of the side-by-side bar graph. c. Describe how the graph would look if there was not an association between smoking and unhealthy behaviors among Mexican-American male adolescents. 3 Complete the contingency table and use it to solve the problem. 16) The partially filled contingency table gives the frequencies of the data on age (in years) and sex from the residents of a retirement home. Age (yrs) 60-69 70-79 Over 79 Total Male 11 9 5 Female 9 2 4 Total What is the proportion of male residents in the age group 60-69? 17) The partially filled contingency table gives the frequencies of the data on age (in years) and sex from the residents of a retirement home. Age (yrs) 60-69 70-79 Over 79 Total Male 15 1 5 Female 5 10 4 Total What proportion of the female residents are younger than 80? 18) Just how accurate are the weather forecasts we hear every day? The table below compares the daily forecast with a city's actual weather for a year. Actual Weather Forecast Rain No rain Rain 33 3 No rain 55 274 GIven that rain was forecast, what proportion of the time did it actually rain? 19) Just how accurate are the weather forecasts we hear every day? The table below compares the daily forecast with a city's actual weather for a year. Actual Weather Forecast Rain No rain Rain 25 10 No rain 60 270 Given that rain was forecast, what proportion of the time was there no rain? 4 20) A survey of autos parked in student and staff lots at a large university classified the brands by country of origin, as seen in the table. Driver Brand American European Asian Student 106 30 66 Staff 99 25 25 Given that the driver is a student, what proportion of the cars are Asian? 21) A survey of autos parked in student and staff lots at a large university classified the brands by country of origin, as seen in the table. Driver American European Asian Student 108 36 59 Staff 92 18 56 Given that the car is European, what proportion of the drivers are staff? Fill in the blank. 22) A ____________________ is a display for two categorical variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 23) Which of the following displays can be used to describe two categorical variables? A) stem-and-leaf plot B) side-by-side box plot C) histogram D) contingency table E) side-by-side dot plot SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 24) The relationship between the number of games won by a minor league baseball team and the average attendance at their home games is analyzed. A regression to predict the average attendance from the number of games won has an r = 0.73. Interpret this statistic. 25) Using advertised prices for used Ford Escorts a linear model for the relationship between a car's age and its price is found. The regression has an R2 = 87.1%. Describe the relationship 26) A random sample of records of electricity usage of homes gives the amount of electricity used in July and size (in square feet) of 135 homes. A regression was done to predict the amount of electricity used (in kilowatt-hours) from size. The residuals plot indicated that a linear model is appropriate. Do you think the slope is positive or negative? Why? 5 Fill in the blank. 27) A ____________________ association between two quantitative variables is present if y tends to go down as x goes up. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 28) In a positive association between two quantitative variables, A) y tends to go down as x goes down. B) the movement of x does not affect the movement of y. C) y tends to go up as x goes down. D) y tends to go down as x goes up. E) none of the above. 29) In a negative association between two quantitative variables, A) y tends to go up as x goes up. B) y tends to go down as x goes down. C) y tends to go up as x goes down. D) the movement of x does not affect the movement of y. E) none of the above. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the scatterplot shows little or no association, a negative association, a linear association, a moderately strong association, or a very strong association (multiple associations are possible). 30) 6 31) 32) 33) 7 34) 35) Determine which plot shows the strongest linear correlation. 36) Construct a scatterplot for the data. 37) Below are the Olympic gold medal performances in the men's high jump from 1960 to 1984. Year High Jump (in.) 1960 85.25 1964 85.75 1968 88.25 1972 87.75 1976 88.50 1980 92.50 1984 92.25 8 Solve the problem. 38) Use a scatter plot to display the data below. All measurements are in milligrams per cigarette. Brand Benson & Hedges Lucky Strike Marlboro Viceroy True Tar Nicotine 16 1.2 13 1.1 16 1.2 18 1.4 6 0.6 39) The data below represent the numbers of absences and the final grades of 15 randomly selected students from a statistics class. Use a scatter plot to display the data. Is there a relationship between the students' absences and their final grades? Student 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of Absences 5 6 2 12 9 5 8 15 0 1 9 3 10 3 11 Final Grade as a Percent 79 78 86 56 75 90 78 48 92 78 81 86 75 89 65 Fill in the blank. 40) A graphical display for two quantitative variables is called a ____________________. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 41) Which of the following displays can be used to describe two quantitative variables? A) all of the above B) contingency table C) scatterplot D) histogram E) side-by-side bar graph 9 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 42) Coal is still used as an energy source at The University of Georgia. The processing of raw coal involves"washing," in which coal ash is removed. Data (see table below) relating the percentage of ash to the density of a coal particle are given in the article "Quantifying Sampling Precision for Coal Ash Using Gy's Discrete Model of the Fundamental Error." Density (g/cc) % Ash 1.25 1.93 1.325 4.63 1.375 8.95 1.45 15.05 1.55 23.31 Source: Journal of Coal Quality, 2989, 33-39. Construct a scatterplot to show how the percentage of ash depends on the density of a coal particle. Is a linear relationship evident from the graph? Explain. 43) Soda is often considered unhealthy because its content is high in both caffeine and refined sugar. But are the two related? Caffeine and refined sugar contents (in milligrams) of 12 ounces of several brands of soda are shown on the following scatterplot. The correlation between caffeine and refined sugar is 0.187. Describe the association. 44) Almost all of the acidity of soda pop comes from the phosphoric acid which is added to give them a sharper flavor. Is there an association between the pH of the soda and the amount of phosphoric acid (in grams)? The correlation between pH and phosphoric acid is -0.991. Describe the association. 45) A Science instructor assigns a group of students to investigate the relationship between their grade point average and their consistency in studying. The correlation between GPA and studying consistency is 0.853. Assume that the association is linear. Describe the association. 46) Given the length of a human's femur, x, and the length of a human's humerus, y, would you expect a positive correlation, a negative correlation, or no correlation? 47) Given the supply of a commodity, x, and the price of a commodity, y, would you expect a positive correlation, a negative correlation, or no correlation? 48) Given the size of a human's brain, x, and their score on an IQ test, y, would you expect a positive correlation, a negative correlation, or no correlation? 10 49) A random sample of 150 yachts sold in the United States last year was taken. A regression to predict the price (in thousands of dollars) from length (in feet) has an R2 = 18.9%. What is the correlation between length and price? Fill in the blank. 50) The ____________________ is a summary measure that describes the strength of the linear association between two quantitative variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 51) The strength of the linear relationship between two quantitative variables may be measured by the A) y-intercept. B) scatterplot. C) residual. D) slope. E) correlation. 52) The correlation A) does not depend on the units of measurement of y or x. B) depends on the units of measurement of y. C) depends on the units of measurement of y and x. D) depends on the units of measurement of x. E) None of the above. 53) If the correlation is approximately zero, then one can conclude A) that there is a linear relationship between x and y. B) that there is no relationship between x and y. C) that there is a relationship between x and y. D) that there is no linear relationship between x and y. E) none of the above. 54) Which of the following is not a property of r? A) r is always between -1 and 1. B) r measures the strength of any kind of relationship between x and y. C) r does not depend on the units of y or x. D) The closer r is to zero, the weaker the linear relationship between x and y. E) r does not depend on which variable is treated as the response variable. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the best predicted value of y corresponding to the given value of x. ^ 55) Four pairs of data yield r = 0.942 and the regression equation y = 3x. Also, y = 12.75. What is the best predicted value of y for x = 2.9? ^ 56) Eight pairs of data yield r = 0.708 and the regression equation y = 55.8 + 2.79x. Also, y = 71.125. What is the best predicted value of y for x = 4.2? 57) The regression equation relating dexterity scores (x) and productivity scores (y) for the employees of a company ^ is y = 5.50 + 1.91x. Ten pairs of data were used to obtain the equation. The same data yield r = 0.986 and y = 56.3. What is the best predicted productivity score for a person whose dexterity score is 20? 11 58) The regression equation relating attitude rating (x) and job performance rating (y) for the employees of a ^ company is y = 11.7 + 1.02x. Ten pairs of data were used to obtain the equation. The same data yield r = 0.863 and y = 80.1. What is the best predicted job performance rating for a person whose attitude rating is 80? ^ 59) Nine pairs of data yield r = 0.867 and the regression equation y = 19.4 + 0.93x. Also, y = 64.7. What is the best predicted value of y for x = 40? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 60) The y-intercept is the A) change in the predicted value of y per unit increase in x. B) predicted value of y. C) predicted value of y when x = 0. D) point where the regression line crosses the x-axis. E) smallest value for the residual sum of squares. 61) The slope is the A) predicted value of y when x = 0. B) point where the regression line crosses the y-axis. C) predicted value of y. D) change in the predicted value of y per unit increase in x. E) smallest value for the residual sum of squares. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 62) Which statement is true about residuals? 63) A regression line for predicting the selling prices of homes in Chicago is ^ y = 168 + 102x, where x is the square footage of the house. A house with 1500 square feet recently sold for $140,000. What is the residual for this observation? 64) A regression line for predicting Interenet usage (%) for 39 countries is ^ ^ y = -3.61 + 1.55x, where x is the GDP, in thousands of dollars, per capita and y is Internet usage. What is the residual for one of the 39 countries with GDP per capita of $ 28,000 and actual Internet use of 38 percent? 65) A regression line for predicting Interenet usage (%) for 39 countries is ^ ^ y = -3.61 + 1.55x, where x is the GDP, in thousands of dollars, per capita and y is Internet usage. Interpret the residual for one of the 39 countries with GDP per capita of $ 15,000 and actual Internet use of 20 percent. 66) A regression line for predicting the selling prices of homes in Chicago is ^ y = 168 + 102x, where x is the square footage of the house. Interpret the residual for a house with 1800 square feet that recently sold for $200,000. 12 Provide an appropriate response. 67) A random sample of records of electricity usage of homes in the month of July gives the amount of electricity used and size (in square feet) of 135 homes. A regression was done to predict the amount of electricity used (in ^ kilowatt-hours) from size. Assume that a linear model is appropriate. The model is usage = 1229 + 0.02 size. What would a negative residual mean for people living in a house that is 2290 square feet? 68) A random sample of records of electricity usage of homes gives the amount of electricity used and size (in square feet) of 135 homes. A regression to predict the amount of electricity used (in kilowatt-hours) from size has an r2 =0.71. Assume that a linear model is appropriate. Write a sentence summarizing what r2 says about this regression. 69) The relationship between the number of games won by a minor league baseball team and the average attendance at their home games is analyzed. A regression to predict the average attendance from the number of games won has an r2 = 0.255. The residuals plot indicated that a linear model is appropriate. Write a sentence summarizing what r2 says about this regression. 70) The relationship between the number of games won by a minor league baseball team and the average attendance at their home games is analyzed. A regression to predict the average attendance from the number of games won has an r2 = 0.256. Assume that a linear model is appropriate. What is the correlation between the average attendance and the number of games won? 71) A random sample of records of electricity usage of homes gives the amount of electricity used in July and size (in square feet) of 135 homes. A regression to predict the amount of electricity used (in kilowatt-hours) from size was completed. Assume that a linear model is appropriate. What are the variables and units in this regression? 72) A random sample of records of electricity usage of homes gives the amount of electricity used in July and size (in square feet) of 135 homes. A regression was done to predict the amount of electricity used (in kilowatt-hours) from size. Assume that a linear model is appropriate. What units does the slope have? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 73) Among the possible lines that can go through data points in a scatterplot, the regression line results from the least squares method and has the smallest value for the ____________________. A) correlation B) intercept C) slope D) residual sum E) residual sum of squares 74) The prediction error for an observation, which is the difference between the actual value and the predicted value of the response variable, is called ____________________. A) a residual B) an outlier C) a correlation D) an extrapolation E) an intercept 13 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Fill in the missing information. _ _ ^ 75) x sx y sy r y = a + bx 40 20 76) 77) ^ 8 11 8 y = ? sx _ y sy r y = a + bx 2.6 1 ? 110 ? y = -110 + 40x _ x sx _ y sy ? ? 18 _ x ^ ^ r ^ y = a + bx ^ 4 -0.5 y = 30 - 4x Use a graphing calculator to find the regression line from the given data . 78) x 2 4 5 6 y 7 11 13 20 79) x 0 3 4 5 12 y 8 2 6 9 12 80) x 1.2 1.4 1.6 1.8 2.0 y 54 53 55 54 56 81) Ten students in a graduate program were randomly selected. Their grade point averages (GPAs) when they entered the program were between 3.5 and 4.0. The following data were obtained regarding their GPAs on entering the program versus their current GPAs. x (entering GPA) 3.5 3.8 3.6 3.6 3.5 3.9 4.0 3.9 3.5 3.7 y (current GPA) 3.6 3.7 3.9 3.6 3.9 3.8 3.7 3.9 3.8 4.0 82) Two different tests are designed to measure employee productivity and dexterity. Several employees are randomly selected and tested with these results. y (productivity) 23 25 28 21 21 25 26 30 34 36 x (dexterity) 49 53 59 42 47 53 55 63 67 75 14 83) Managers rate employees according to job performance and attitude. The results for several randomly selected employees are given below. y (performance) 59 63 65 69 58 77 76 69 70 64 x (attitude) 72 67 78 82 75 87 92 83 87 78 Provide an appropriate response. 84) Coal is still used as an energy source at The University of Georgia. The processing of raw coal involves"washing," in which coal ash is removed. Data (see table below) relating the percentage of ash to the density of a coal particle are given in the article "Quantifying Sampling Precision for Coal Ash Using Gy's Discrete Model of the Fundamental Error." Density (g/cc) % Ash 1.25 1.93 1.325 4.63 1.375 8.95 1.45 15.05 1.55 23.31 Source: Journal of Coal Quality, 2989, 33-39. Find the regression equation. What is the predicted percentage of ash for a coal particle with density 1.35 g/cc? Fill in the blank. 85) ____________________ refers to using a regression line to predict y-values for x-values outside the observed range of data. Extrapolate from the linear model 86) The figure below shows the life expectancy for persons living in the United States. The regression analysis of the data yields the following values: Lifeexp = -353.87 + 0.2157 Year r2 = 0.9539 Use the regression model to predict the life expectancy in 2010. 15 87) The table below shows the gestation (in days) and average longevity (in years) for a number of different mammals: Black Bear Cat (domestic) Monkey (Rhesus) Lion Horse Gorilla Gray Squirrel Gestation (days) 219 63 166 100 330 258 44 Average Longevity (years) 18 12 15 15 20 20 10 The scatter plot and regression equation are shown below: The regression analysis of this data yields the following values: Long = 9.90 + 0.0345 Gestation r2 = 0.9048 Use this model to predict the average longevity of an African elephant whose gestation is 660 days. Explain what is wrong with each interpretation. Assume calculations are done correctly. 88) A psychologist does an experiment to determine whether an outgoing person can be identified by his or her handwriting. She claims that the correlation of 0.89 shows that there is a strong causal relationship between personality type and handwriting. 89) What is a factor which can cause a high correlation between two variables to be misleading? Fill in the blank. 90) Predictions about the future using time series data are called ____________________. 91) When an observation has a large effect on the results of a regression analysis, it is said to be ____________________. Provide an appropriate response. 92) Why should one always construct a scatterplot first before finding a correlation or a regression line? 93) For an observation to be influential, what two conditions must hold? 16 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 94) The correlation and the regression line are A) resistant. B) nonresistant. C) causal. D) unrelated. E) None of the above. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the lurking variable. 95) A reporter studied the causes of a fire to a house, and established, thanks to a scatterplot a strong correlation between the damages (in dollars) and the number of firefighters at the scene. Find a lurking variable, if there is one. 96) A study shows that the amount of chocolate consumed in Canada and the number of automobile accidents is positively related. Find a lurking variable, if there is one. 97) A study of consumer behavior finds a positive correlation between sales of ice cream and sales of soda. What might explain the strong correlation? Fill in the blank. 98) The fact that the direction of an association between two variables can change after we include a third variable and analyze the data at separate levels of that variable is known as ____________________. 99) When two explanatory variables are both associated with a response variable but are also associated with each other, there is said to be ____________________. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 100) A(n) ____________________ is a variable, usually unobserved, that influences the association between the variables of primary interest. A) response variable B) predictor variable C) lurking variable D) independent variable E) explanatory variable 101) The fact that the direction of an association between two variables can change after we include a third variable and analyze the data at separate levels of that variable is known as A) multiple regression. B) causation. C) Simpson's paradox. D) confounding. E) correlation 17 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 102) What is the difference between a confounding variable and a lurking variable? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 103) Which of the following statements is true? A) Association does not imply causation. B) Correlation does not imply causation. C) The direction of the association between two variables can change with the inclusion of a third variable. D) None of the above. E) All of the above. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 104) Describe what scatterplots are, and discuss the importance of creating scatterplots. Find the specified probability distribution of the binomial random variable. 105) A multiple choice test consists of four questions. Each question has five possible answers of which only one is correct. A student guesses on every question. Find the probability distribution of X, the number of questions she answers correctly. 106) In one city, 21% of the population is under 25 years of age. Three people are selected at random from the city. Find the probability distribution of X, the number among the three that are under 25 years of age. 107) In one city, the probability that a person will pass his or her driving test on the first attempt is 0.66. Four people are selected at random from among those taking their driving test for the first time. Determine the probability distribution of X, the number among the four who pass the test. Determine whether a probability model based on Bernoulli trials can be used to investigate the situation. If not, explain. 108) We draw a card from a deck 6 times (replacing the card after each draw) and get 3 kings. How likely is this? 109) We draw a card from a deck 40 times to find the distribution of the suits. After each draw the card is replaced. 110) A pool of possible jurors consists of 15 men and 18 women. A jury of 12 is picked at random from this group. What is the probability that the jury contains all women? 111) A company realizes that 5% of its pens are defective. In a package of 30 pens, is it likely that more than 6 are defective? Assume that pens in a package are independent of each other. 112) A study found that 56% of people working for large companies are dissatisfied with their job. What is the probability that half of the employees at your company are dissatisfied with their job? 18 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 113) Which of the following is not true about the binomial distribution? A) The probability of success plus the probability of failure is one. B) The random variable X is continuous. C) The probability of success is constant from trial to trial. D) Each outcome is independent of the other. E) None of the above. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 114) What are the conditions for a binomial distribution? 115) When is it appropriate to use the binomial distribution to approximate the number of successes in a random sample of n separate subjects from a population? 116) When is the binomial distribution approximately normal? 117) A college basketball player has probability 0.50 of making any given shot. Her successive shots are independent. In overtime of an important game, she misses all three shots that she takes. Explain why this does not mean that she "choked," because it could happen by chance with reasonable probability. 118) Residents of a particular city worry that a waste dump next to the city's water supply may elevate the chance of cancer for its residents. Of the past 80 deaths of residents, 50 were due to cancer. Investigate whether this number is unusually high or if it could it be explained by random variation, if actually the probability of a death being due to cancer was 0.22 in this town. Answer by finding the mean and standard deviation of the probability distribution of the number of deaths due to cancer and the z-score of the observed result if actually the probability equals 0.22. Explain your reasoning, stating all assumptions. Find the mean of the binomial random variable. 119) According to a college survey, 22% of all students work full time. Find the mean for the random variable X, the number of students who work full time in samples of size 16. 120) A die is rolled 10 times and the number of times that two shows on the upper face is counted. If this experiment is repeated many times, find the mean for the random variable X, the number of twos. 121) On a 10-question multiple choice test , each question has four possible answers, one of which is correct. For students who guess at all answers, find the mean for the random variable X, the number of correct answers. 122) The probability that a person has immunity to a particular disease is 0.6. Find the mean for the random variable X, the number who have immunity in samples of size 26. 123) The probability is 0.7 that a person shopping at a certain store will spend less than $20. For random samples of 28 customers, find the mean number who spend less than $20. Find the standard deviation of the binomial random variable. 124) According to a college survey, 22% of all students work full time. Find the standard deviation for the random variable X, the number of students who work full time in samples of size 16. 19 125) A die is rolled 18 times and the number of twos that come up is tallied. If this experiment is repeated many times, find the standard deviation for the random variable X, the number of twos. 126) On a multiple choice test with 16 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the standard deviation for the random variable X, the number of correct answers. 127) The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 11. Find the standard deviation for the random variable X, the number of seeds germinating in each batch. 128) The probability of winning a certain lottery is 1/51,949. For people who play 560 times, find the standard deviation for the random variable X, the number of wins. Find the indicated probability. 129) An archer is able to hit the bull's-eye 55% of the time. If she shoots 8 arrows, what is the probability that she gets exactly 4 bull's-eyes? Assume each shot is independent of the others. 130) A tennis player makes a successful first serve 59% of the time. If she serves 7 times, what is the probability that she gets exactly3 first serves in? Assume that each serve is independent of the others. 131) A multiple choice test has 10 questions each of which has 4 possible answers, only one of which is correct. If Judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer exactly 3 questions correctly? 132) In one city, the probability that a person will pass his or her driving test on the first attempt is 0.68. 11 people are selected at random from among those taking their driving test for the first time. What is the probability that among these 11 people, the number passing the test is between 2 and 4 inclusive? 133) Suppose that11% of people are left handed. If 6 people are selected at random, what is the probability that exactly 2 of them are left handed? 134) A basketball player has made 70% of his foul shots during the season. If he shoots 3 foul shots in tonight's game, what is the probability that he makes all of the shots? 135) Police estimate that 25% of drivers drive without their seat belts. If they stop 6 drivers at random, find the probability that all of them are wearing their seat belts. 136) The on-line access computer service industry is growing at an extraordinary rate. Current estimates suggest that 20% of people with home-based computers have access to on-line services. Suppose that 15 people with home-based computers were randomly and independently sampled. What is the probability that exactly 5 of those sampled have access to on-line services at home? 137) The on-line access computer service industry is growing at an extraordinary rate. Current estimates suggest that 20% of people with home-based computers have access to on-line services. Suppose that 15 people with home-based computers were randomly and independently sampled. What is the probability that at least 1 of those sampled have access to on-line services at home? 20 Obtain the probability distribution of the random variable. 138) When a coin is tossed four times, sixteen equally likely outcomes are possible as shown below: HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH TTTT Let X denote the total number of tails obtained in the four tosses. Find the probability distribution of the random variable X. Leave your probabilities in fraction form. 139) When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below. (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 2) (3, 3) (3, 4) (4, 2) (4, 3) (4, 4) (5, 2) (5, 3) (5, 4) (6, 2) (6, 3) (6, 4) (1, 5) (1, 6) (2, 5) (2, 6) (3, 5) (3, 6) (4, 5) (4, 6) (5, 5) (5, 6) (6, 5) (6, 6) Let X denote the absolute value of the difference of the two numbers. Find the probability distribution of X. Give the probabilities as decimals rounded to three decimal places. 140) When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below. (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 2) (3, 3) (3, 4) (4, 2) (4, 3) (4, 4) (5, 2) (5, 3) (5, 4) (6, 2) (6, 3) (6, 4) (1, 5) (1, 6) (2, 5) (2, 6) (3, 5) (3, 6) (4, 5) (4, 6) (5, 5) (5, 6) (6, 5) (6, 6) Let X denote the smaller of the two numbers. If both dice come up the same number, then X equals that common value. Find the probability distribution of X. Leave your probabilities in fraction form. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 141) Which of the following is a property of a discrete probability distribution? A) Each interval has probability between 0 and 1. This is the area under the curve, above that interval. B) The sum of the probabilities for all the possible x values equals 1. C) The interval containing all possible values has probability equal to 1, so the total area under the curve equals 1. D) Both a. and b. E) None of the above. 21 142) Which of the following is a property of a continuous probability distribution? A) For each x, the probability p(x) falls between 0 and 1. B) The sum of the probabilities for all the possible x values equals 1. C) The interval containing all possible values has probability equal to 1, so the total area under the curve equals 1. D) Both a. and b. E) None of the above. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine the possible values of the random variable. 143) Suppose a coin is tossed four times. Let X denote the total number of tails obtained in the four tosses. What are the possible values of the random variable X? 144) Suppose that two balanced dice are rolled. Let Y denote the product of the two numbers. What are the possible values of the random variable Y? 145) Suppose that two balanced dice, a red die and a green die, are rolled. Let Y denote the value of G - R where G represents the number on the green die and R represents the number on the red die. What are the possible values of the random variable Y? 146) For a randomly selected student in a particular high school, let Y denote the number of living blood related grandparents of the student. What are the possible values of the random variable Y? 147) The following table displays a frequency distribution for the number of siblings for students in one middle school. For a randomly selected student in the school, let X denote the number of siblings of the student. What are the possible values of the random variable X? Number of siblings 0 1 2 3 4 5 6 7 Frequency 189 245 102 42 24 13 5 2 Find the mean of the given probability distribution. 148) The random variable X is the number of houses sold by a realtor in a single month at the Sendsom's Real Estate office. Its probability distribution is given in the table. x P(X = x) 0 0.24 1 0.01 2 0.12 3 0.16 4 0.01 5 0.14 6 0.11 7 0.21 149) The random variable X is the number of tennis balls ordered by customers at Tennis Online. Its probability distribution is given in the table. x 3 6 9 12 15 P(X = x) 0.14 0.22 0.36 0.18 0.10 22 150) The random variable X is the number of people who have an associate's degree in a randomly selected group of four adults from a particular town. Its probability distribution of X, based on 1000 random samples of size 4 adults, is given in the table. x P(X = x) 0 0.026 1 0.153 2 0.346 3 0.345 4 0.130 151) The random variable X is the number that shows up when a loaded die is rolled. Its probability distribution is given in the table. x P(X = x) 1 0.14 2 0.12 3 0.12 4 0.10 5 0.13 6 0.39 152) The random variable X is the number of siblings of a student selected at random from a particular secondary school. Its probability distribution is given in the table. x 0 7 P(X = x) 24 1 13 48 2 3 16 3 7 48 4 1 16 5 1 24 153) The probabilities that a batch of 4 computers will contain 0, 1, 2, 3, and 4 defective computers are 0.6275, 0.3102, 0.0575, 0.0047 and 0.0001, respectively. Round answer to the nearest hundredth. 154) A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day are 0.50, 0.40, 0.09, and 0.01 respectively. Provide an appropriate response. 155) Computer chips often contain surface imperfections. For a certain type of computer chip, 9% contain no imperfections, 22% contain 1 imperfection, 26% contain 2 imperfections, 20% contain 3 imperfections, 12% contain 4 imperfections, and the remaining 11% contain 5 imperfections. Let X represent the number of imperfections in a randomly chosen chip. a. b. Specify the probability distribution for the number of imperfections in a randomly chosen chip. Find the expected number of imperfections in a randomly chosen chip 156) Let X = the number of days in the past week in which a randomly selected person felt anxious and tense. According to a recent General Social Survey, the probabilities for the potential values of X for adult Americans are approximately: x 0 1 P(x) 0.28 0.20 a. b. 2 3 4 5 6 0.15 0.11 0.07 0.07 0.02 Does this refer to a discrete or a continuous random variable? Find the mean of this probability distribution. Interpret. 23 7 0.10 Use the empirical rule to solve the problem. 157) The systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmHg and a standard deviation of 12 mmHg. What percentage of 18-year-old women have a systolic blood pressure that lies within 3 standard deviations of the mean? 158) The systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmHg and a standard deviation of 12 mmHg. What percentage of 18-year-old women have a systolic blood pressure between 96 mmHg and 144 mmHg? 159) The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 360 hours and a standard deviation of 5 hours. What percentage of the bulbs have lifetimes that lie within 1 standard deviation of the mean? 160) The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 360 hours and a standard deviation of 8 hours. What percentage of the bulbs have lifetimes that lie within 2 standard deviations of the mean? 161) At one college, GPA's are normally distributed with a mean of 2.6 and a standard deviation of 0.4. What percentage of students at the college have a GPA between 2.2 and 3? 162) The amount of Jennifer's monthly phone bill is normally distributed with a mean of $79 and a standard deviation of 9. Fill in the blanks. 68% of her phone bills are between $___ and $___. 163) The annual precipitation for one city is normally distributed with a mean of 390 inches and a standard deviation of 3.2 inches. Fill in the blanks. In 95% of the years, the precipitation in this city is between ___ and ___ inches. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 164) Which of the following is not true about the standard normal distribution? A) The area under the standard normal curve to the left of z = 0 is negative. B) About 68% of its observations fall between -1 and 1. C) The standard normal curve is symmetric about 0. D) About 95% of its observations fall between -2 and 2. E) The total area under the standard normal curve is 1. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 165) Serum cholesterol is an important risk factor for coronary disease. The level of serum cholesterol is approximately normally distributed with a mean of 219 mg/dL and a standard deviation of 50 mg/dL. What is the interquartile range of the level of serum cholesterol? Use a table of areas to find the specified area under the standard normal curve. 166) The area that lies between 0 and 3.01 24 167) The area that lies to the left of 1.13 168) The area that lies between -1.10 and -0.36 169) The area that lies to the right of -1.82 170) The area that lies to the right of 0.59 171) The shaded area shown Use a table of areas for the standard normal curve to find the required z-score. 172) Find the z-score for which the area under the standard normal curve to its left is 0.96 173) Find the z-score having area 0.09 to its left under the standard normal curve. 174) Find the z-score for which the area under the standard normal curve to its left is 0.04 175) Find the z-score having area 0.86 to its right under the standard normal curve. Provide an appropriate response. 176) Suppose that property taxes on homes in Columbus, Ohio, are approximately normal in distribution, with a mean of $3000 and a standard deviation of $1000. The property tax for one particular home is $3500. a. b. Find the z-score for that property tax value. What proportion of the property taxes exceeds $3500? 177) Serum cholesterol is an important risk factor for coronary disease. The level of serum cholesterol is approximately normally distributed with a mean of 219 mg/dL and a standard deviation of 50 mg/dL. If the clinically desirable range for serum cholesterol is < 200 mg/dL, what is the probability that a randomly selected person will have a clinically desirable level of serum cholesterol? 178) Serum cholesterol is an important risk factor for coronary disease. The level of serum cholesterol is approximately normally distributed with a mean of 219 mg/dL and a standard deviation of 50 mg/dL. If serum cholesterol levels of over 250 mg/dL indicate a high-enough risk for heart disease to warrant treatment, what is the probability that a randomly selected person will need treatment? 179) Serum cholesterol is an important risk factor for coronary disease. The level of serum cholesterol is approximately normally distributed with a mean of 219 mg/dL and a standard deviation of 50 mg/dL. If the clinically desirable range for serum cholesterol is < 200 mg/dL and serum cholesterol levels of over 250 mg/dL indicate a high-enough risk for heart disease to warrant treatment, what is the probability that a randomly selected person will have a borderline high serum cholesterol level (that is, > 200, but < 250 mg/dL)? 25 Find the indicated probability for the normally distributed variable. 180) The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches? 181) The incomes of trainees at a local mill are normally distributed with a mean of $1,100 and a standard deviation $150. What percentage of trainees earn less than $900 a month? 182) The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that the volume of soda in a randomly selected bottle will be less than 32 oz? 183) A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 200 and 275. 184) A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 170 and 220. 185) The lengths of human pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. What is the probability that a pregnancy lasts at least 300 days? 186) Assume that the weights of quarters are normally distributed with a mean of 5.67 g and a standard deviation 0.070 g. A vending machine will only accept coins weighing between 5.48 g and 5.82 g. What percentage of legal quarters will be rejected? 187) The diameters of pencils produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What is the probability that the diameter of a randomly selected pencil will be less than 0.285 inches? 188) The weekly salaries of teachers in one state are normally distributed with a mean of $490 and a standard deviation of $45. What is the probability that a randomly selected teacher earns more than $525 a week? 189) In one region, the September energy consumption levels for single-family homes are found to be normally distributed with a mean of 1050 kWh and a standard deviation of 218 kWh. For a randomly selected home, find the probability that the September energy consumption level is between 1100 kWh and 1225 kWh. Find the probability involving proportion. 190) Assume that 20% of students at a university wear contact lenses. We randomly pick 200 students. What is the probability that more than 22% of this sample wear contact lenses? 191) A candy company claims that its jelly bean mix contains 15% blue jelly beans. Suppose that the candies are packaged at random in small bags containing about 200 jelly beans. What is the probability that a bag will contain more than 20% blue jelly beans? 192) When a truckload of peaches arrives at a packing plant, a random sample of 125 is selected and examined. The whole truckload will be rejected if more than 8% of the sample is unsatisfactory. Suppose that in fact 11% of the oranges on the truck do not meet the desired standard. What's the probability that the shipment will be accepted anyway? 26 193) Researchers believe that 6% of children have a gene that may be linked to a certain childhood disease. In an effort to track 50 of these children, researchers test 950 newborns for the presence of this gene. What is the probability that they find enough subjects for their study? 194) Researchers believe that 7% of children have a gene that may be linked to a certain childhood disease. In an effort to track 50 of these children, researchers test 950 newborns for the presence of this gene. What is the probability that they do not find enough subjects for their study? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 195) The closer p is to 0 or 1, the larger n must be in order for the sampling distribution of the sample proportion to A) be approximately binomial. B) be centered at p. C) have a standard error equal to [n(p)(1-p)]. D) be approximately normal. E) have a standard error equal to [p(1-p)/n]. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the mean/standard error of the sampling distribution of the proportion. 196) Based on past experience, a bank believes that 8% of the people who receive loans will not make payments on time. The bank has recently approved 600 loans. Describe the sampling distribution model of the proportion of clients in this group who may not make timely payments. 197) Assume that 26% of students at a university wear contact lenses. We randomly pick 300 students. Describe the sampling distribution model of the proportion of students in this group who wear contact lenses. 198) A realtor has been told that 43% of homeowners in a city prefer to have a finished basement. She surveys a group of 300 homeowners randomly chosen from her client list. Describe the sampling distribution model of the proportion of homeowners in this sample who prefer a finished basement. 199) A candy company claims that its jelly bean mix contains 21% blue jelly beans. Suppose that the candies are packaged at random in small bags containing about 400 jelly beans. Describe the sampling distribution model of the proportion of blue jelly beans in a bag. Provide an appropriate response. 200) A recent poll of 1000 British adults asked, "If there were a referendum on the issue, would you favor Britain becoming a republic or remaining a monarchy?" (www.mori.com/polls). Suppose that the population proportion favoring the monarchy equals 0.70. (This was, in fact, the value for the sample proportion.) For a random sample of 1000 residents, let X denote the number in this category. a. Find the mean of the probability distribution of X. b. Find the standard deviation of the probability distribution of X. c. What range of values falls within 3 standard deviations of the mean? Explain why it is unlikely that X will fall outside this interval. 27 201) Eurobarometer recently took a poll to gauge the approval of adults for the European currency. Of the 1000 people sampled in a particular country in the European Union, consider the sample proportion of people who indicate approval of the euro. a. Find the mean and standard error of the sampling distribution for this sample proportion, if the population proportion equals 0.67. b. What shape would you expect this sampling distribution to have? Explain. Within what limits would the sample proportion almost certainly fall? Explain. Solve the problem. 202) In one region, the September energy consumption levels for single-family homes are found to be normally distributed with a mean of 1050 kWh and a standard deviation of 218 kWh. If 50 different homes are randomly selected, find the probability that their mean energy consumption level for September is greater than 1075 kWh. 203) Assume that women's heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 90 women are randomly selected, find the probability that they have a mean height between 62.9 inches and 64.0 inches. 204) Suppose that replacement times for washing machines are normally distributed with a mean of 9.3 years and a standard deviation of 1.1 years. Find the probability that 70 randomly selected washing machines will have a mean replacement time less than 9.1 years. 205) Human body temperatures are normally distributed with a mean of 98.20В°F and a standard deviation of 0.62В°F. If 19 people are randomly selected, find the probability that their mean body temperature will be less than 98.50В°F. 206) For women aged 18-24, systolic blood pressures (in mm Hg) are normally distributed with a mean of 114.8 and a standard deviation of 13.1. If 23 women aged 18-24 are randomly selected, find the probability that their mean systolic blood pressure is between 119 and 122. 207) A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 8.7 hours. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 208) A population distribution has mean 50 and standard deviation 20. For a random sample of size 100, the sampling distribution of the sample mean has A) mean 50/100 and standard error 20/100. B) mean 50 and standard error 0.20. C) mean 50/10 and standard error 20/10. D) mean and standard error that are unknown unless we know the exact shape of the population distribution. E) mean 50 and standard error 2. 28 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 209) The sampling distribution of sample means is from a highly skewed population with Вµ = 4.47 and repeated random samples of size 100 from this population: = 1.40. For a. Find the mean and standard error of the sampling distribution of the sample mean. b. Explain why the sampling distribution of the sample mean is bell-shaped, even though the population was highly skewed. 210) To estimate the mean acreage of ranches in Alberta, Canada, a researcher plans to obtain the acreage for a random sample of 64 farms. Results from an earlier study suggest that 800 acres is a reasonable guess for the standard deviation of ranch size. a. Find the probability that the sample mean acreage falls within 100 acres of the population mean acreage. b. If the researcher can increase n above 64, will the probability that the sample mean falls within 100 acres of the population mean increase or decrease? Why? 211) .In North America, female adult heights are approximately normal with Вµ = 65 inches and = 3.5 inches. The heights of 50 females were measured at a national collegiate volleyball tournament. The sample mean height was found to be 70 inches. a. Using the population parameters given above, what is the probability of obtaining a sample mean height of 70 inches or higher with a random sample of n = 50? b. Does this probability make you question the population mean stated for female heights? Justify why you believe this sample mean may not be representative of the population of female heights. For samples of the specified size from the population described, find the mean and standard error if the sampling distribution of the sample mean x. 212) The mean and the standard deviation of the sampled population are, respectively, 113.9 and 32.1. n = 64 213) The mean and the standard deviation of the sampled population are, respectively, 77.4and 4.0. n = 225 214) The National Weather Service keeps records of snowfall in mountain ranges. Records indicate that in a certain range, the annual snowfall has a mean of 97 inches and a standard deviation of 16 inches. Suppose the snowfalls are sampled during randomly picked years. For samples of size 64 determine the mean and standard deviation of x. 215) One barge from Inland Waterways, Inc. can carry a load of 5516.8 lb. Records of past trips show that the weight of cans that it carries have a mean of 82 lb and standard deviation of 16 lb. For samples of size 64 find the mean and standard deviation of x. 216) One truck from Pee Dee Trucking, Inc. can carry a load of 4876.8 lb. Records of past trips show that the weights of boxes that it carries have a mean of 72 lb and a standard deviation of 16 lb. For samples of size 64, find the mean and standard deviation of x. 29 Provide an appropriate response. 217) The lengths of pregnancies are normally distributed with a mean of 266 days and a standard deviation of 25 days. If 100 women are randomly selected, find the probability that they have a mean pregnancy between 266 days and 268days. 218) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 100 women are randomly selected, find the probability that they have a mean height greater than 63.0 inches. 219) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 75 women are randomly selected, find the probability that they have a mean height between 63 and65 inches. 220) Assume that the heights of men are normally distributed with a mean of 67.9inches and a standard deviation of 2.8 inches. If 64 men are randomly selected, find the probability that they have a mean height greater than 68.9 inches. 221) The body temperatures of adults are normally distributed with a mean of 98.6В° F and a standard deviation of 0.60В° F. If 36 adults are randomly selected, find the probability that their mean body temperature is greater than 98.4В° F. 222) Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $32,000 and a standard deviation of $3000. If 100 teachers are randomly selected, find the probability that their mean salary is greater than $32,500. 223) Assume that blood pressure readings are normally distributed with a mean of 120 and a standard deviation of 8. If 100 people are randomly selected, find the probability that their mean blood pressure will be greater than 122. 224) The average number of pounds of red meat a person consumes each year is 196 with a standard deviation of 22 pounds (Source: American Dietetic Association). If a sample of 50 individuals is randomly selected, find the probability that the mean of the sample will be less than 200 pounds. 225) A coffee machine dispenses normally distributed amounts of coffee with a mean of 12 ounces and a standard deviation of 0.2 ounce. If a sample of 9 cups is selected, find the probability that the mean of the sample will be greater than 12.1 ounces. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 226) The ____________________ is considered powerful in statistics because it works for any population distribution provided the sample size is sufficiently large and the population mean and standard deviation are known. A) binomial distribution B) central limit theorem C) law of large numbers D) normal distribution E) empirical rule 30