Answer to Question #174193 in Statistics and Probability for anoni

Question

Of two similar groups of patients, A and B, consisting of 50 and 100 individuals, respectively, the first was

given a new type of sleeping pill and the second was given a conventional type. For patients in group A the

mean number of hours of sleep was 7.82 with a standard deviation of 0.24 hour. For patients in group B the

mean number of hours of sleep was 6.75 with a standard deviation of 0.30 hour. Find (a) 95% and (b) 99%

confidence limits for the difference in the mean number of hours of sleep induced by the two types of

sleeping pills.

Answer 1

Confidence limits for the difference in the mean number of hours of sleep induced by the two types of sleeping pills:

$\overline{x_1}-\overline{x_2}\pm Z_{\alpha}\sqrt{\frac{\sigma^2_1}{n_1}+\frac{\sigma^2_2}{n_2}}$

a) lower limit:

$7.82-6.75-1.96\sqrt{\frac{0.24^2}{50}+\frac{0.3^2}{100}}=1.07-1.96\cdot0.045=0.98$

upper limit:

$1.07+1.96\cdot0.045=1.16$

b) lower limit:

$1.07-2.58\cdot0.045=0.95$

upper limit:

$1.07+2.58\cdot0.045=1.19$