Download Confidence interval

Confidence interval
Pulse rate is an important measure of the fitness of a person's cardiovascular system. The
mean pulse rate for all U.S. adult males is approximately 72 heart beats per minute. A
random sample of 21 U.S. male adults who jog at least 15 miles per week had a mean
pulse rate of 52.6 beats per minute and a standard deviation of 3.22 beats per minute.
What is the 95% confidence interval for the mean pulse rate of all U.S. adult males who
jog at least 15 miles per week.
Solution. Denote by x the sample mean pulse rate , denote by s the sample standard
deviation, and denote by n the sample size. By hypothesis, x  52.6, s  3.22, n  21 .
By looking up the t-value table, we have t 0.025 (20)  . So, the 95% confidence interval for
the mean pulse rate of all U.S. adult males who jog at least 15 miles per week is
( x  sn t 0.025 ( 20))
 (52.6  3.22
* 2.086)
21
 (52.6  1.5)
 (51.1,54.1)
So, a 95% confidence interval for the mean pulse rate of all U.S. adult males who jog at
least 15 miles per week is (51.1, 54.1).
How do I interpret the interval found above? and does it appear that jogging at least 15
miles per week reduces the mean pulse rate for adult males?
By the above, we know that a 95% confidence interval for the mean pulse rate of all U.S.
adult males who jog at least 15 miles per week is (51.1, 54.1), which means that the
mean pulse rate is in (51.1, 54.1) with probability 95%. Yes, it supports the claim that
jogging at least 15 miles per week reduces the mean pulse rate for adult males, since 72 is
not in (51.1, 54.1). We can test it as follows.
Null hypothesis H0:   72
Alternative hypothesis H1:   72 .
We use t-test. Compute the t-statistic
t  x s72 n  523.6.2272 21  27.61
Then we can compute the p-value at http://duke.usask.ca/~rbaker/Tables.html to get
p-value=0.0
So, we should reject the null hypothesis H0.
What assumptions are required for the validity of the CI?
We need to assume that the pulse rate (roughly )follows normal distribution for the
validity of the CI.