Download One-Sample Inference for the Population Mean and Variance

Making Decisions about a
Single Population Mean (  )
Example 1: Time it takes to fall asleep
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:
H o : Mean Sleep Time for everyone in the population is 20 minutes or less
H a : Mean 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. hypothesis tests or confidence intervals)
Recall, to get summary statistics for a numerical variable, select Analyze >
Distribution. The variable to summarize is Sleep Time.
Assumptions
1. The observations should follow a normal distribution.
To check this assumption, we can place a normal curve over our histogram (right click on
the Sleep Time header > Fit Distribution > Normal). Another plot that is commonly
used to assess normality is the Normal Quantile Plot. To obtain a Normal Quantile Plot
in JMP, you can either right click on the Sleep Time header and select Normal Quantile
Plot from the menu that pops-up or select that option from the pull-down menu to the left
of the variable name.
Using these graphs, do you think normality is being satisfied? Explain.
While the distribution does appear to be symmetric it is quite kurtotic. This is
evidenced by the S-shape in the normal quantile plot and the near uniform appearance
of the histogram. Our sample size is large (n = 84), so in spite of the fact the
distribution is kurtotic we will proceed with t-test and CI.
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 Average for everyone in the population. Also, a subscript may
be used to identify what variable is being tested. As a result, the following is a more
traditional way of writing our hypotheses for the question of interest.
H o :  sleeptime  20 min .
H a :  sleeptime  20 min .
To perform the test, right click on the Sleep Time header and select the Test Mean. Enter
the value (or boundary value) from for the mean from the null hypothesis in the box
labeled Specify Hypothesized Mean. Also check the box for Wilcoxon Sign-Rank test
and click OK.
The results of the test are shown next.
Number
6
Description
The hypothesized mean assuming the null hypothesis is
true, 20 minutes in this case.
The estimated sample mean, X  20.49 minutes in this
case.
The degrees of freedom for the t-statistic,
df = n – 1
= 84 – 1 = 83.
The sample standard deviation, s = 3.05.
The test statistic
X   o 20.49  20
t

 1.46
s
3.05
n
84
p-value for two-tailed test ( H a :   20 )
7
p-value for upper-tail test ( H a :   20 )
8
p-value for lower-tail test ( H a :   20 )
1
2
3
4
5
So what about our test...

What type of test do we have?
Upper-tailed t-test

What is the appropriate p-value?
p-value = .0739, we have weak evidence against the null hypothesis.

What is our decision for the test?
We have weak evidence against the null hypothesis in support of the alternative.
Using a  level of significance we fail to reject.
Write a conclusion for your findings.
There is insufficient evidence to indicate the mean time it takes adults to fall
asleep is greater than 20 minutes.
From the Moments box, we see that the likely range for the mean 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.
We estimate that the mean time it takes adults to fall asleep is between 19.82 and 21.15
minutes. There is a 95% chance this interval will cover the true mean time it takes to
fall asleep.
An 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 (mean or median) 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.
The details of the Wilcoxon Signed-Rank test procedure will be covered
in class.
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.
Inference for the Population Variance and Standard Deviation
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. Inference for the population
variance and standard deviation are covered in your text:
* Section 7.9 for testing and CI’s for  2 (pop. variance) and  (pop. standard deviation)