5-3 and 5-5 Notes
5-3: Confident Interval for a Population Mean: Student’s t-Statistic
Example 5.3.1
Suppose you have selected a random sample of n = 7 measurements from a normal
distribution. Compare the standard normal z-values with the corresponding t-values
if you were forming the following confidence intervals.
a. 80% confidence interval
b. 95% confidence interval
Example 5.3.2
Let 𝑡0 be a specific value of t. Use Table IV to find 𝑡0 values such that the following
statements are true:
a. 𝑃(𝑡 ≥ 𝑡0 ) = .01, where df = 17
b. 𝑃(𝑡 ≤ 𝑡0 ) = .005, where df = 6
Example 5.3.3.
Consider the pharmaceutical company that desires an estimate of the mean increase
in blood pressure of patients who take a new drug. The blood pressure increases
(measured in points) for the n = 6 patients in the human testing phase are shown
below. Use this information to construct a 95% confidence interval for 𝜇, the mean
increase in blood pressure associated with the new drug for all patients in the
population.
Blood Pressure Increase (Points) for Six Patients
1.7
3.0
.8
3.4
2.7
2.1
5-5: Determining the Sample Size
Estimating a Population Mean
Go back to the hospital example, which we estimated the mean length of stay for
patients in a large hospital. The sample was 100 patients’ records produced the 95%
confidence interval 4.5 ± .78. Our estimate 𝑥̅ was within .78 days of the true mean
length of stay, 𝜇, for all the hospital’s patients at the 95% confidence level for 𝜇 was
2(.78) = 1.56 days wide when 100 accounts were sampled.. Suppose we want to
estimate 𝜇 to within .25 days with 95% confidence is, we want to narrow the width
of the confidence interval from 1.56 days to .50 days. (Recall that the 𝜎 was 4 days.)
How much will the sample size have to be increased to accomplish this?
Example 5.51.
Suppose the manufacturer of official NFL footballs uses a machine to inflate the new
balls to pressure of 13.5 pounds. When the machine is properly calibrated, the mean
inflation pressure is 13.5 pounds, but uncontrollable factors cause the pressures of
individual foots to vary randomly from 13.3 to 13.7 pounds. For quality control
purposes, the manufacturer wished to estimate the mean inflation pressure to with
.025 pounds of its true value with a 99% confidence interval. What sample size
should be specified for the experiment?