Download Making Decisions About a Single Population Mean

Making Decisions about a Single Population Mean
Data File: Sleep-time.JMP
Background: These data come from a study comparing the time it takes for smokers and
non-smokers to fall asleep.
Variables: Sleep-time.JMP
> Smoking Status - smoker or non-smoker
> Sleep Time - time to fall asleep
Question of Interest: Is the average sleep-time for everyone in the population longer
than 20 minutes?
Putting this question into statements that will be used for hypothesis testing:
HO: Average Sleep Time for everyone in the population is 20 minutes or less
HA: Average Sleep Time for everyone in the population is greater than 20 minutes
Intuitive Decision
In order to determine whether or not the null or alternative hypothesis is true, you should
first review the summary statistics for the variable you are interested in testing.
Remember, these summary statistics are for the observations you observe. In the end, we
are trying to make decisions about everyone in the greater population, not just the
observations you observe. In order to make decisions about all observations of interest,
we must apply some inferential technique (i.e. a hypothesis test or construction of a
confidence interval).
Recall, to get summary statistics for a numerical variable, select Analyze >
Distribution. The variable to summarize is Sleep Time. The results are shown on the
following page.
Assumptions
1.
The observations should follow a normal distribution.
To check this assumption, we can place a normal and smooth curve over our histogram
(either select the Sleep Time pull-down menu or right click on the Sleep Time header
then choose Fit Distribution > Normal & Smooth Curve). Another plot that is
commonly used to assess normality is the Normal Quantile Plot. To obtain a normal
quantile plot select Normal Quantile Plot from the same menu.
Using these graphs, do you think normality is being satisfied? Explain.
Performing the Actual Hypothesis Test
Recall, the hypothesis of interest
HO: Average Sleep Time for everyone in the population is 20 minutes or less
HA: Average Sleep Time for everyone in the population is greater than 20 minutes
For convenience, let mean for everyone in the population. Also, a subscript can be
used to identify what variable is being tested. This can be particularly helpful in studies
where multiple variables are being examined. Using this notation we have a more
traditional way of writing our hypotheses for the question of interest.
To perform the test, right click on the Sleep Time header and select the Test Mean. Enter
the value (or boundary value) for the mean from the null hypothesis in the box labeled
Specify Hypothesized Mean. Check the box for Wilcoxon Sign-Rank test and click OK.
The results of the test are shown below.
Number
Description
1
2
3
4
5
6
7
8
So what about our test...

What type of test do we have?

What is the appropriate p-value?

What is our decision for the test?

Write a conclusion for your findings.
From the Moments box, we see that the likely range for the average Sleep Time of
everyone in the population (i.e. Sleeptime ) is 19.82 minutes up to 21.15 minutes. That is, a
95% confidence interval for Sleeptime is (19.82, 21.15).
Interpret the meaning of this interval. Does this agree with what you found in above
using the hypothesis test? Explain.
Nonparametric Alternative
If we believe that the distribution of Sleep Time is not normal, but is symmetric, then a
nonparametric test may be appropriate. To obtain a nonparametric test we check the box
labeled Wilcoxon Signed-Rank Test in the Test Mean window. The results of this test
are shown below.
The p-value for signed-rank test suggests that the typical Sleep Time for the entire
population is not greater than 20 minutes. This test agrees with what was discovered
above in the test that assumes normality.
Comment: If normality of the measurements being tested is in doubt, then it is best to
use a nonparametric inferential procedure to draw conclusions about the typical value of
the population.
Example 2 - Normal Human Body Temperature
Data File: Bodytemp.JMP
Background: These data come from a study of the normal human body temperature.
Mackowiak, P. A., Wasserman, S. S., and Levine, M. M. (1992), "A Critical Appraisal of
98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of
Carl Reinhold August Wunderlich," Journal of the American Medical Association, 268,
1578-1580.
Variables: Bodytemp.JMP
> Gender – gender of subject
> Temperature – body temperature in degrees Farenheit.
> Heart Rate – heart rate of subject
It has long been believed that the normal human body temperature is 98.6o F. The data for
this example comes from a study of body temperature and pulse rate for adults. Suppose
we wish to test:
Ho : normal mean body temperature is 98.6o F)
Ha :    normal mean body temperature is NOT 98.6o F)
Select Analyze > Distribution and put Temperature in the right hand box. To assess
normality (required for using a t-test) we again use a normal curve, a smooth density
estimate, and a normal quantile plot which can be obtained by selecting those options
from the Temperature pull-down menu. The results are shown on the following page.
Using these graphs, do you think normality is being satisfied? Explain.
To perform the t-test select the Test Mean ... option from Temperature pull-down menu
located next to the variable name at the top of the window. Next enter the value for the
mean assuming the null hypothesis is true, 98.6 here, in the box labeled Specify
Hypothesized Value for the Mean then click OK. The results of the t-test are shown
below.
There are three p-values reported for each test along with the value of the t-test statistic.
The p-values are for a two-tailed ( H A :   98.6 o F ) , upper-tailed ( H A :   98.6 o F ) and
lower tailed ( H A :   98.6 o F ) tests respectively. Here we are performing a two-tailed
test so our p-value is < .0001.

What is our decision for the test?

Write a conclusion for your findings.
In both the Moments and Fit Normal boxes above we see that the 95% CI for  is given
by the interval (98.122, 98.3765).

Interpret the meaning of this interval. Does this agree with what you found in
above using the hypothesis test? Explain.
Note: When fitting a normal distribution to our data we also obtain an estimate and CI for
the population SD . We can also perform tests regarding the standard deviation by
selecting the Test SD... option from the variable pull-down menu and entering a
hypothesized value for the standard deviation in the box.