STT 200 Sample Problems Part 3
1. The margin of error for a 99% confidence interval for the proportion p increases as the
sample size decreases. SAY TRUE OR FALSE
Answer: TRUE
2. The margin of error for a confidence interval for the proportion p, based on a specified
sample size n, increases as the confidence level increases. SAY TRUE OR FALSE
Answer: TRUE
3. The margin of error for a confidence interval for proportion p will definitely increase
as both the confidence level and the sample size decrease. SAY TRUE OR FALSE
Answer: FALSE
4. The sample size required to obtain a confidence interval of specified margin of error
m, decreases as the confidence level increases. SAY TRUE OR FALSE
Answer: FALSE
Use the following to answer questions 5-6: The distribution of the amount of money
undergraduate students spends on books for a term is slightly right skewed, with a mean
of $500 and a standard deviation of $90. It is ok to use normal distributions for this
problem.
5. If an undergraduate student is selected at random, what is the probability that this
student spends less than $480 for a term on books?
Answer: 0.4129
6. If a simple random sample of 400 undergraduate students is selected, what is the
probability that, on average, these students spend more than $520 on books for a term?
Answer: almost 0
7. In the last senatorial election in Michigan, 58% of the adults over the age of 65 voted
Republican. A researcher wishes to determine if the proportion of adults over the age of
65 in the city who plan to vote Republican in the next senatorial election has increased.
Let p represent the proportion of the population of all adults over the age of 65 in the city
that plan to vote Republican in the next senatorial election. In terms of p, the researcher
should test what null and alternative hypotheses?
Answer: Ho: p = .58 versus Ha: p > .58
Use the following for 8 to 11: The tail area above a test statistic value of z = 1.792 is
0.058. Determine whether each of the following statements is true or false.
8. If the alternative hypothesis is of the form Ha: p > po, the data are statistically
significant at significance level α = 0.05.
Answer: FALSE
9. If the alternative hypothesis is of the form Ha: p > po, the data are statistically
significant at significance level α= 0.10.
Answer: TRUE
10. If the alternative hypothesis is of the form Ha: p ≠ po, the data are statistically
significant at significance level α = 0.05. Answer: FALSE
11. If the alternative hypothesis is of the form Ha: p ≠ po, the data are statistically
significant at significance level α = 0.10 Answer: FALSE
Use the following to answer questions 12-16: A noted psychic was tested for
extrasensory perception. The psychic was presented with 200 cards face down and asked
to determine if the card were one of five symbols: a star, a cross, a circle, a square, or
three wavy lines. The psychic was correct in 50 cases. Let p represent the probability
that the psychic correctly identifies the symbol on the card in a random trial. Assume the
200 trials can be treated as a simple random sample from the population of all guesses the
psychic would make in his lifetime.
12. Based on the results of the test, what is a 95% confidence interval for p?
Answer: 0.25 ± 0.060
13. Suppose you wished to see if there were evidence that the psychic is doing better than
just guessing. To do this, you test the hypotheses Ho: p = 0.20 versus Ha: p > 0.20. What
is the value of the z statistic?
Answer: z = 1.77
14. What do we know about the value of the P-value for the hypothesis test?
Answer: z = 0.0384
15. How large a sample n would you need to estimate p with a margin of error 0.01 with
95% confidence? Use the hypothesized value p = 0.20 as the value for p*.
Answer: n = 6147
16. Suppose you wished to conduct a test at a 5% significance level.
a) What would your decision be? Answer: Reject the null hypothesis
b) Based on that decision, what type of mistake could you have made?
Answer: Accept Ho, Type I error
Use the following to answer questions 17-19: An inspector inspects large truckloads of
potatoes to determine the proportion p in the shipment with major defects prior to using
the potatoes to make potato chips. If there is clear evidence that this proportion is less
than 0.10, she will accept the shipment. To reach a decision, she will test the hypotheses
Ho: p = 0.10, Ha: p < 0.10. To do so, she selects a simple random sample of 150 potatoes
from the more than 3000 potatoes on the truck. Only 8 of the potatoes sampled are found
to have major defects.
17. What is the value of the z statistic? Answer: z = –1.91
18. What is the P-value for this hypothesis test? Answer: z = 0.0284
19. At which significance levels is the data statistically significant?
Answer: α=0.05, 0.10. In fact any α > 0.0284 is correct.
20. A simple random sample of 100 of a certain popular model car in 2003 found that
20 had a certain minor defect in the brakes. A simple random sample of 400 of this
model car in 2004 found that 50 had the minor defect in the brakes. Let p1 and p2 be the
proportion of all cars of this model in 2003 and 2004, respectively, that actually have the
defect. What is a 90% confidence interval for p1 – p2?
Answer: 0.075 ± 0.071