Section 10.1 ~ t Distribution for Inferences About a Mean
Objective: After this section you will understand when it is appropriate to use the t
distribution rather than the normal distribution for constructing confidence intervals
or conducting hypothesis tests for population means, and know how to make proper
use of the t distribution.
When dealing with confidence intervals (ch.8) and hypothesis testing (ch.9), we
worked with samples that were large enough to assume a normal distribution which
allowed us to use the standard scores (z-scores) to find probabilities of certain values
occurring.
Recall that in order to find the z-score, the population standard deviation is needed.
In real applications, the population standard deviation is typically not available,
which means that in order to find the confidence interval or conduct the hypothesis
test we would estimate it using the sample standard deviation.
Many statisticians believe that this is not the best approach and they use what is
known as a t distribution (or student t distribution) in place of the normal
distribution.
As long as the sample size is at least 30 or the population assumes a normal
distribution, a t distribution can be used to find a confidence interval and/or conduct
a hypothesis test.
The t distribution is similar in shape and symmetry to the normal distribution, but it
accounts for greater variability that is expected with small samples.
Note ~ when you know the population standard deviation and the sample size is
greater than 30 or the population is normally distributed, the normal distribution is
best to use.
t value:
Formula for margin of error using a t distribution:
A t value is found by:
Example 1
Here are five measures of diastolic blood pressure from randomly selected adult men: 78,
54, 81, 68, 66. These five values result in these sample statistics: n = 5, x  69.4 , and
s = 10.7. Using this sample, construct the 95% confidence interval estimate of the mean
diastolic blood pressure level for the population of all men.
When a t distribution is used to conduct a hypothesis test, the t value plays the role that
the z-score played when we worked with the normal distribution.
Formula to calculate the t value:
Once you calculate t, you can decide whether to reject or not reject the null hypothesis by
using this following criteria:
Right-tailed test: reject the null if the t value that you found is ___________
the t value from the table (that corresponds to the appropriate degrees of freedom)
Use column 2 as a comparison if you want a 97.5% confidence level and
column 3 if you want a 95% confidence level
Left-tailed test: reject the null if the t value that you found is ___________
the negative of the t value from the table (that corresponds to the appropriate
degrees of freedom)
Use column 2 as a comparison if you want a 97.5% confidence level and
column 3 if you want a 95% confidence level
Two-tailed test: reject the null if the absolute value of the t value that you
found is ___________ the t value from the table (that corresponds to the
appropriate degrees of freedom)
Use column 2 as a comparison if you want a 95% confidence level and
column 3 if you want a 90% confidence level
Example 2:
Listed below are ten randomly selected IQ scores of statistics students:
111 115 118 100 106 108 110 105 113 109
Using methods from Chapter 4, you can confirm that these data have the following
sample statistics: n = 10, x  109.5 , and s = 5.2. Using a 0.05 significance level, test the
claim that statistics students have a mean IQ score greater than 100, which is the mean IQ
score of the general population.
Example 3:
Using the same data from example 2 and the same significance level of .05, test the
claim that the mean IQ score is equal to 100.