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MATH 243 Lab 2
Name: _____________________ Date: ______________________
1.
In 1927, Babe Ruth hit a record of 60 home runs. Thirty-four years later in 1961, Roger Maris broke that record with
61 home runs. In 1998, both Mark McGwire (St. Louis Cardinals) and Sammy Sosa (Chicago Cubs) surpassed
Maris’s record. At the time when McGwire broke the record, his batting average was 0.294. His 62 home runs came
in 456 times at bat, thus his likelihood of hitting a home run at that point was 62/456 = 0.136. Assume that this
proportion remained fixed, use the procedures of Example 1 to simulate home runs hitting for McGwire for the next
500 times at bat.
The procedure will be to list a set of integers from 1 to 500 to represent the number of home runs. Next, a set of values
(0’s and 1’s) will be simulated. The 1’s will represent successful hit (i.e. the home run) with a probability of 0.0.136. The
0’s will represent unsuccessful hit (no home run). A running count of the number of home runs will be kept and also a
running proportion of successful home runs will be computed. Lastly, a graph of the running proportion of success
versus the number of hits will be displayed. The graph will be analyzed to explain the concept of the Law of Large
Numbers.
First label column C1 with the variable name of HIT and column C2 with the variable name of VALUES.
Next we will enter the number of hits (1, 2, 3, 4, …, 500 in column C1 by selecting Calc-> Make Patterned Data->
Simple Set of Numbers. This will enable us to generate a sequence of integers from 1 to 500. The dialog box for the
Simple Set of Numbers will be displayed. Fill in the dialog box.
Now to generate the 500 hits, we will use the Bernoulli distribution to generate the successes and failures. As
mentioned earlier, we will generate 0’s and 1’s for the failures and successes respectively. Select Calc-> Random Data> Bernoulli. In the Bernoulli Distribution dialog box that appears, enter the information required.
Click on the OK button and a sequence of 0’s (failures) and 1’s (successes) will be generated into column C2.
Observe that column C2 was labeled as VALUES.
Note: The probability of success is 0.136.
To explore the Law of Large Numbers for this experiment, we next need a running count of the number of successes.
That is, we need to accumulate the number of home runs made as the experiment is conducted. We can achieve this by
using the function that generates the Partial sums. To employ this function, first rename column C3 as PARSUMS. Next
select Calc-> Calculator and the Calculator dialog box will appear. Enter the information sa required. The Partial sums
function can be obtained by scrolling down in the Functions box. To place it in the Expression box, double click on
Partial sums and PARS will appear in the box. Since we need to accumulate the number of successes in column C2, in
the Expression box make sure that you are finding the partial sums for column C2 (VALUES).
Click on the OK button and the partial sums will be generated in column C3 (PARSUMS).
Recall, the Law of Large Numbers says that as the number of trials increases, the empirical probabilities
should
converge to the theoretical probability P(E). Thus, the next thing we need to do is to generate the empirical probabilities
. Rename column C4 as EMPROBS. That is, we will place the empirical probabilities in column C4. These values
are obtained by dividing the cumulative number of successes (C3) by the number of attempts (C1).
To compute these values, select Calc-> Calculator and the resulting calculator dialog box should be completed as as
required (let c4=C3/C1).
Click on the OK button and the values for
will be generated and saved in column C4.
Now, we will use the generated information to graphically display what is happening to
as the number of free throw
attempts is increasing. To display this, we will plot the values in C1 along the horizontal axis and the values of C4 along
the vertical axis.
To achieve this, select Graph-> ScatterPlot with connect line.
Click on the OK button and the graph will be displayed.
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(a) Produce a graph of the running accumulated proportion of home runs made.
(b) Interpret the graph of the running accumulated proportion versus the number of times at bat. Invoke the Law of
Large Numbers in your discussion. Discuss.
(c) Based on your display in (a), does it appear that McGwire’s home run success per at bat is approaching his
"expected" proportion (0.136) of success? Discuss.
d. Based on your simulation, fill in the table.
What was his proportion of
home runs after
Proportion
50 times at bat?
100 times at bat?
150 times at bat?
200 times at bat?
250 times at bat?
300 times at bat?
350 times at bat?
400 times at bat?
450 times at bat?
500 times at bat?
e.
For the 500 times at bat, on which time at bat was his running accumulated proportion of success (hitting a
home run) the lowest?
Note: You can compute descriptive statistics for the running accumulated proportion of successes to observe the
lowest (minimum) and highest (maximum) accumulated proportion of success for the 500 times at bat. You then
need to match the lowest and highest running accumulated proportion with time at bat.
Time at Bat: __________________________
Lowest Proportion of success: __________________________
(f) For the 500 times at bat, on which time at bat was his proportion of success (hitting a home run) the highest?
Time at Bat: __________________________
Highest Proportion of success: __________________________
1.
The charge for automotive servicing of a 4-cylinder, a 6-cylinder, and a 8-cylinder vehicle, together with
the percentages of these vehicle types being serviced, is given in the following table:
Number of Cylinders
a.
468cylinder cylinder cylinder
Cost of Service ($)
32
36
42
Percentage Serviced (%)
25
45
30
List all possible ordered pairs of two costs. (This will be the sample space S as in Example 4).
S=
3
b.
List the set of corresponding averages for the ordered pairs. (This correspond to the set A in Example 4).
c.
A=
Construct, as in Example 4, a theoretical probability distribution for the average costs.
Average Cost
($)
Theoretical
Probability
d.
e.
f.
g.
h.
(Review Example 4). Enter the service cost (32, 36, and 42) in column C1 and the corresponding
probabilities (0.25, 0.45, and 0.30) in column C2. Rename C1 as Cost and C2 as Prob.
Generate two sets of 1000 cost values from the distribution in part (d) and store in columns C3 and C4.
Rename column C3 as Cost1 and column C4 as Cost2.
Compute the average cost for C3 and C4 and save in column C5. Rename column C5 as AvgCost.
Construct a histogram for the average costs located in column C5. Follow the procedure in Example 4.
Note: you will have to use the appropriate range of values from part (c) for the midpoint/cutpoint positions.
Use the histogram to construct a table of empirical probabilities for the average costs.
Average Cost
Theoretical Probability
Empirical Probability
Absolute Error
i.
Compare these empirical probabilities with the theoretical probabilities by considering the absolute errors.
Discuss.
2
A large jar is filled with nickels, dimes, and quarters. The jar contains 30% nickels, 50% dimes, and 20%
quarters.
a. What probability distribution will represent the information given above. Fill in the values in the following
table.
Value (cents)
b.
Probability
An experiment consists of randomly selecting two coins (with replacement) from the jar and recording the
sum of the two values. Complete the tree diagram for this experiment where D = dime, N = nickel, and Q =
quarter. Label the branches appropriately.
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(c) List all possible ordered pairs for the selection of the two coins. (This will be sample space S, as in Example 4).
S=
(d) List the set of corresponding sum for the ordered pairs. (This correspond to the set A, as in Example 4).
A=
(e) Construct, as in Example 4, a probability distribution for the sum of the values (values in set A).
Note: You will have to use the concept of independence to compute the theoretical probabilities.
Total Value
Theoretical Probability
d.
e.
h.
Present a projection graph for the information from part (e).
Describe the shape of the graph.
Enter the values for the values from part (a) in column C1 and the corresponding theoretical probabilities in
column C2. Rename C1 as Value and C2 as Prob.
i. Generate two sets of 1000 values for the total from the distribution in part (h) and store in columns C3 and
C4. Rename column C3 as Value1 and column C4 as Value2.
j.
Compute the sum for C3 and C4 and save in column C5. Rename column C5 as Total.
Note: To compute the totals for C3 and C4, use the Row Statistics feature of the software and select the
Sum button. See Figure 4.16 to see where the Sum option is located.
k. Construct a histogram for the totals located in column C5. Follow the procedure in Example 4.
Note: you will have to use the appropriate range of values from part (c) for the midpoint/cutpoint positions. Attach a
hard copy of this graph with the work you turn in to your instructor. Label the graph appropriately.
l. Use the histogram to construct a table of empirical probabilities for the average costs.
Average Cost
Theoretical Probability
Empirical Probability
Absolute Error
m. Compare these empirical probabilities with the theoretical probabilities by considering the absolute errors.
Discuss.
3. A couple decided to have a three-child family. Consider this as a probability experiment and assume the
probability of having a boy or a girl is 0.5 respectively. Also, we will assume that the gender of the children is
independent from child to child.
(a) Observe that the first child could be a boy (B) or a girl (G). The outcomes can be represented by the picture (tree
diagram) below.
The outcomes are: B, G.
What is the theoretical probability of a girl (G)?
_________________________
What is the theoretical probability of a boy (B)?
__________________________
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(b) Observe that the second child could also be a boy or a girl. The outcomes can be represented by the tree diagram
below.
The outcomes are: BB, BG, GB, GG.
What is the theoretical probability of the outcome BB?
Answer =
since the gender of the child is independent of each other.
What is the theoretical probability of the outcome BG?
________________________
What is the theoretical probability of the outcome GB?
________________________
What is the theoretical probability of the outcome GG?
________________________
(c) Complete the following tree diagram for the third child.
(d) Fill in the table below with the different possible outcomes for a three-child family and the theoretical
probability of those outcomes occurring. Note that each possible outcome will have an equal probability of
occurring.
Outcomes
BBB
GGG
Theoretical Probability
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(e) From the table in part (d), what is the theoretical probability of 0, 1, 2, or 3 girls in a three-child family? Fill in
the values in the following table. Hint: For example, when the number of girls is 2, consider the appropriate
outcomes in which two girls are present and sum the probabilities to get the theoretical probability for 2 girls.
Number of Girls in a
3-child Family
Theoretical Probability for the
Number of Girls (3-decimal places)
0
1
2
3
(f) What is the theoretical probability that the family name will be carried on in a three-child family? That is, what is
the probability of having at least one boy in a three-child family? Compute from the table in part (c).
Probability: ____________________________.
Discuss the rationale for your answer.
(g) Use MINITAB to simulate the number of girls in a three-child family. We will consider 5000 different families.
To complete the simulation, use the following:
o Use the Bernoulli distribution to simulate 0s and 1s.
o A 0 will represent a boy and a 1 will represent a girl.
o For the Bernoulli distribution, use 0.5 as the probability. Here we are assuming that the probability
of a boy equals the probability of a girl.
o Simulate 3 columns of data.
o Use the row statistic feature of the software to sum the three columns of data and save in a
different column.
o Construct a table for the sum using the tally feature of the software.
o From the tally, fill in the following table.
Note: To obtain the proportion, to three decimal places, for the number of girls you will need to divide the
frequencies in the middle column by 5000.
Number of Girls in a
3-child Family
Observed Frequency for
the Number of Girls
Proportion for the Number of
Girls (3-decimal places)
0
1
2
3
(h) Summarize your results in the following table.
Number of Girls
0
1
2
3
Theoretical Probability
Empirical Probability
Absolute Error
(i) Compare these empirical probabilities with the theoretical probabilities by considering the absolute errors.
Discuss.