#### 17. Confidence Intervals

*Question 1*

As an example, suppose we want the critical value for a 95% confidence interval. This is the *t*_{4} value with an area of 2.5% to the right. Looking through Table 17.1 we find the four values 2.76 up to 2.79 all have an area listed of .025. Table 17.2 shows that the exact value is 2.776.

In general Table 17.2 is great for making confidence intervals. However, if you carry out a *t*-test and find a *t*_{4} statistic of 3.2, for example, then Table 17.2 can only tell you that the one-sided P-value is between .025 and .01. Table 17.1 gives the exact value of .016, but of course you would need a whole new table if your result had a different degrees of freedom.

*Question 2*

The 10 dissolving times have mean 188.5 seconds with standard deviation 12.030 seconds. The standard error of the sample mean is 12.030/√10 = 3.80 seconds. With 9 degrees of freedom the critical *t*_{9} value for 95% confidence is 2.262 (from Table 17.2). The confidence interval is then

188.5 ± 2.262 × 3.80 = 188.5 ± 8.6 seconds,

or (179.9, 197.1) seconds.

*Question 3*

Data Set #18 gives the pulse rates before and after 20 push-ups for 10 subjects. The 10 increases in pulse rate have mean 28.90 bpm with standard deviation 8.478 bpm. The standard error of the sample mean is 8.478/√10 = 2.681 bpm. With 9 degrees of freedom the critical *t*_{9} value for 95% confidence is 2.262 (from Table 17.2). The confidence interval is then

28.90 ± 2.262 × 2.681 = 28.90 ± 6.06 bpm,

or (22.84, 34.96) bpm.

*Question 4*

The 30 pulse rates have mean 69.53 bpm with standard deviation 8.613 bpm. The standard error of the sample mean is 8.613/√30 = 1.573 bpm. With 29 degrees of freedom the critical *t*_{29} value for 95% confidence is 2.045 (from Table 17.2). The confidence interval is then

69.53 ± 2.045 × 1.573 = 69.53 ± 3.22 bpm,

or (66.31, 72.75) bpm.

*Question 5*

The 30 hours of weekend sleep have mean 16.18 hours with standard deviation 3.834 hours. The standard error of the sample mean is 3.834/√30 = 0.700 hours. With 29 degrees of freedom the critical *t*_{29} value for 95% confidence is 2.045 (from Table 17.2). The confidence interval is then

16.18 ± 2.045 × 0.700 = 16.18 ± 1.43 hours,

or (14.75, 17.61) hours.