This document discusses 12 versions of hypothesis testing examples related to analyzing changes in the average height of female college freshmen. Each version provides sample data and tests whether there is sufficient evidence to believe if the average height has changed. Key differences between versions include the alternative hypothesis being tested, sample size, confidence level, and whether population data is known. Later versions also discuss calculating p-values and determining suitable sample sizes.

1. IME 301 March 2013
Hypothesis Testing Examples
By: Dr. Parisay
This handout assists in discussion on how to interpret hypothesis testing examples, as
well as, effect of different parameters in final conclusion. Detailed solutions will be
discussed in class. Dr. Parisay’s comments are in red
Version 1: The average height of females in the freshman class of a certain college has
been 162.5 centimeters. (that is 162.5 0 m = ) Is there a reason to believe that there has
been a change (that is not equal) in the average height if a random sample of 25 females
(that is n=25) in the present freshman class has an average height of 165.2 centimeters
(that is X =165.2 ) and a variance of 49 Cm2 (that is S 2 = 49 )? Assume that the
confidence level is 0.99.
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
(Notice that you do not have the population variance. Therefore, you should
use ‘t’ test statistics. This version will result in accepting the null hypothesis.
Then, reject the alternate hypothesis. That is, there has been no change in
the average height with 99% CL.)
Version 2: The average height of females in the freshman class of a certain college has
been 162.5 centimeters. Is there reason to believe that the average height has decreased if
a random sample of 25 females in the present freshman class has an average height of
165.2 centimeters and a variance of 49 Cm2? Assume that the confidence level is 0.99.
: 162.5 0 0 H m =m = : 162.5 1 0 H m <m =
(This version will result in NOT rejecting the null hypothesis. Therefore,
we REJECT the alternate hypothesis. That is, the average height has
NOT decreased with 99% CL)
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2. Version 3: The average height of females in the freshman class of a certain college has
been 162.5 centimeters. Is there a reason to believe that there has been a change in the
average height if a random sample of 25 females in the present freshman class has an
average height of 165.2 centimeters and a variance of 49 Cm2? Assume that the
confidence level is 0.9.
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
(This version will result in rejecting the null hypothesis. Then, accept the
alternate hypothesis. That is, there has been a change in the average height
with 90% CL.)
Version 4: The average height of females in the freshman class of a certain college has
been 162.5 centimeters. Is there a reason to believe that there has been a change in the
average height if a random sample of 61 females in the present freshman class has an
average height of 165.2 centimeters and a variance of 49 Cm2? Assume that the
confidence level is 0.99.
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
(This version will result in rejecting the null hypothesis. Then, accept the
alternate hypothesis. That is, there has been a change in the average height
with 99% CL.)
Version 5: The average height of females in the freshman class of a certain college has
been 162.5 centimeters. Is there a reason to believe that there has been a change in the
average height if a random sample of 25 females in the present freshman class has an
average height of 165.2 centimeters and a variance of 16 Cm2? Assume that the
confidence level is 0.99.
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
(This version will result in rejecting the null hypothesis. Then, accept the
alternate hypothesis. That is, there has been a change in the average height
with 99% CL.)
Questions:
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3. Compare Version 1 and Version 2. What is the difference in problem
information/statement and conclusion?
Compare Version 1 and Version 3. What is the difference in problem information and
conclusion?
Compare Version 1 and Version 4. What is the difference in problem information and
conclusion?
Compare Version 1 and Version 5. What is the difference in problem information and
conclusion?
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4. Version 6: The average height of females in the freshman class of a certain college has
been 162.5 centimeters with a standard deviation of 6.9 centimeters (that is s = 6.9 ) .
Is there reason to believe that there has been a change in the average height if a random
sample of 61 females in the present freshman class has an average height of 165.2
centimeters? Assume that the confidence level is 0.99.
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
(We have population’s variance, as well as, Central Limit Theorem will hold as
n>30, variance is known, it is two sided hypothesis testing for mean, and should use
‘Z’ test statistics. It will result in rejecting null hypothesis.)
Version 7: The average height of females in the freshman class of a certain college has
been 162.5 centimeters with a standard deviation of 6.9 centimeters. Is there reason to
believe that the average height is increased if a random sample of 61 females in the
present freshman class has an average height of 165.2 centimeters? Assume that the
confidence level is 0.99.
: 162.5 0 0 H m =m = : 162.5 1 0 H m >m =
(We perform ‘Z’ test. This version will result in rejecting the null hypothesis.
Therefore, we accept the alternate hypothesis.)
Questions:
Compare Version 1 and Version 6. What is the difference in problem information and
conclusion?
Compare Version 6 and Version 7. What is the difference in problem information and
conclusion?
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5. Version 9: The average height of females in the freshman class of a certain college has
been 162.5 centimeters. Is there a reason to believe that there has been a change in the
average height if a random sample of 25 females in the present freshman class has an
average height of 165.2 centimeters and a variance of 49 Cm2? Use a P-value for your
conclusion
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
(That is, find P-value and discuss any significance level less than that will result in
not rejecting the null hypothesis. Another approach is to check if P-value is small,
let’s say less than 1%, then reject null hypothesis. The reason is that usually
significance level will be more than 5%. However, if P-value is large, let’s say more
than 10%, do not reject the null hypothesis.).
Questions:
Compare Version 1 and Version 9. What is the difference in problem information and
conclusion?
Version 11: The average height of females in the freshman class of a certain college has
been 162.5 centimeters with a standard deviation of 6.9 centimeters. Is there reason to
believe that there has been a change in the average height if a random sample of 36
females in the present freshman class has an average height of 165.2 centimeters?
Assume that the confidence level is 0.99. What will be the type II error if the mean of
population is in fact 163 centimeters?
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
(We have population’s variance, as well as, Central Limit Theorem will hold as
n>30. It is two sided hypothesis testing for mean. We should use ‘Z’ test statistics.
It will result in accepting null hypothesis.)
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6. Version 12: The average height of females in the freshman class of a certain college has
been 162.5 centimeters with a standard deviation of 6.9 centimeters. We would like to
collect a sample to test if there has been a change in the average height to 163 cm. What
is a suitable sample size for this test? Assume we would like a confidence level of 0.99
with type II error of 20%.
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
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7. Version 12: The average height of females in the freshman class of a certain college has
been 162.5 centimeters with a standard deviation of 6.9 centimeters. We would like to
collect a sample to test if there has been a change in the average height to 163 cm. What
is a suitable sample size for this test? Assume we would like a confidence level of 0.99
with type II error of 20%.
: 162.5 0 0 H m =m = : 162.5 1 0 H m ¹m =
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