Statistics 104 – Laboratory 9
Distribution of the sample mean, y .
This lab looks at the sample mean y by sampling from a population of 250 females who took an
introductory statistics class. The characteristic we are interested in is the average or mean height
of female students. We can look at the variation in a sample statistic by taking many samples
(called repeated sampling) from the population and looking at the distribution of the sample
statistics obtained.
1. Refer to the table titled “Heights of all females in the population.” This table contains a listing
of the heights, in centimeters (cm), of the population members. Rather than list the names of the
population members, this table numbers them by rows numbered (00, 01, . . . , 09, 10, . . . , 24)
and columns numbered (0,1,2,. . . , 9). For example, student 037 (Row 03 and Column 7) is 165
cm tall.
a) Use the random number table to select a simple random sample of 10 students from this
population. Write the student numbers and heights on the answer sheet. Calculate the
sample mean height. Note: You can make more efficient use of 3 digit random numbers
doing the following. For numbers between 000 and 249 go directly to the table of heights.
For numbers between 250 and 499 subtract 250 and then go to the table of heights. For
numbers between 500 and 749 subtract 500 and then go to the table of heights. For
numbers between 750 and 999 subtract 750 and then go to the table heights.
b) Take three other random samples of size 10 from the population. This requires different
random numbers than the ones used in a). Record the student numbers and heights on the
answer sheet and calculate the sample mean for each sample.
c) How different are the four sample means? What causes the differences?
The population of heights has a population mean μ = 166.6 cm and a population standard
deviation σ = 7.6 cm . The distribution of heights is given below.
0.05
0.03
Density
0.04
0.02
0.01
140
150
160
170
180
190
Height (cm)
We are interested in the distribution of the sample mean for random samples of size n = 10 .
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d) Describe the shape of the theoretical distribution of the sample mean, y , for samples of
size n = 10 .
e) What is the mean of the theoretical distribution of the sample mean for samples of size
n = 10 ?
f) What is the standard deviation of the theoretical distribution of the sample mean for
samples of size n = 10 ?
g) Calculate the probability of getting a sample mean, n = 10 , greater than 170 cm.
Confidence Intervals for the population mean, μ .
2. You will use the sample data your group collected to construct confidence intervals for the
population mean height. Recall that your population standard deviation is σ = 7.6 cm .
a) For each of your group’s 4 samples of size n = 10 construct an 80% confidence interval
for the population mean μ .
b) How many of the 4 confidence intervals you constructed actually contain the population
mean of 166.6 cm? Put this information on the blackboard.
c) How many of the confidence intervals for the whole lab section actually contain the
population mean of 166.6 cm?
d) Theoretically, what percentage of 80% confidence intervals, n = 10 , will contain the
population mean of 166.6 cm?
e) Will a 95% confidence interval be narrower than, the same width as or wider than an
80% confidence interval, n = 10 ? Explain briefly.
f) Will a 95% confidence interval based on n = 10 values be narrower than, the same width
as or wider than a 95% confidence interval based on n = 40 values? Explain briefly.
g) Construct a 95% confidence interval for your combined sample of n = 40 .
h) Theoretically, what percentage of 95% confidence intervals n = 40 , will contain the
population mean of 166.6 cm?
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Stat 104 – Laboratory 9
Group Answer Sheet
Names of Group Members:
____________________, ____________________
____________________, ____________________
1.
Sample 1
Number Height
Sample 2
Number Height
Sample 3
Number Height
Sample 4
Number Height
c) How different are the four sample means? What causes the differences?
d) Describe the shape of the theoretical distribution of the sample mean, y , for samples of
size n = 10 .
e) What is the mean of the theoretical distribution of the sample mean for samples of size
n = 10 ?
f) What is the standard deviation of the theoretical distribution of the sample mean for
samples of size n = 10 ?
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g) Calculate the probability of getting a sample mean, n = 10 , greater than 170 cm.
Confidence Intervals for the population mean, μ .
2.
a) For each of your group’s 4 samples of size n = 10 construct an 80% confidence interval
for the population mean μ .
Sample 1:
Sample 2:
Sample 3:
Sample 4:
b) How many of the 4 confidence intervals you constructed actually contain the population
mean of 166.6 cm?
c) How many of the confidence intervals for the whole lab section actually contain the
population mean of 166.6 cm?
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d) Theoretically, what percentage of 80% confidence intervals, n = 10 , will contain the
population mean μ ?
e) Will a 95% confidence interval be narrower than, the same width as or wider than an
80% confidence interval, n = 10 ? Explain briefly.
f) Will a 95% confidence interval based on n = 10 values be narrower than, the same width
as or wider than a 95% confidence interval based on n = 40 values? Explain briefly.
g) Construct a 95% confidence interval for combined sample of n = 40 .
h) Theoretically, what percentage of 95% confidence intervals n = 40 , will contain the
population mean μ ?
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