Question

Please LABEL the two populations when writing hypothesis statements. 1. There are three main types of exercises: range-of-motion (flexibility), strengthening, and endurance. Researchers would like to estimate the mean workout length for the strengthening with 99% confidence. They have no idea of the population standard deviation so they take a pilot sample of 24 people who exercise regularly and measure how long their strengthening exercise lasts (minutes). The sample mean time was 73.6 minutes and the sample standard deviation was 10.7 minutes. How many more need to be sampled if the researchers want to be within plus or minus 3 minutes. If you have resources to sample only 60 people and you still need to be 99% confident, what is your new confidence interval width? __(3) If you have resources to sample only 60 people and you still need to be within plus or minus 3 minutes, what is your new confidence level?

Answer 1

(1)

Sample Size (n) is given by:

$n=\frac{Z_{\alpha /2}\times \sigma ^{2}}{e^{2}}$

Given:

$\alpha$ = 0.01

From Table, critical values of Z = $\pm$ 2.576

$\sigma$ = 10.7

e = 3

Substituting, we get:

2.5762 x 10.72 85 32

So,

Additional sample size required = 85 - 24 = 61

So,

Answer is:

61

(2)

n = 60

$\bar{x}$ = 73.6

s = 10.7

SE = s

п

= 10.7/$\sqrt{60}$

1.3814

$\alpha$ = 0.01

From Table, critical values of Z = $\pm$ 2.576

Confidence Interval:

73.6 $\pm$ (2.576 X 1.3814)

= 73.6 $\pm$ 3.5584

= ( 70.0416 ,77.1584)

Confidence Interval:

70.0416 <   $\mu$ <   77.1584

(2)

n = 60

$\bar{x}$ = 73.6

s = 10.7

SE = s

п

= 10.7/$\sqrt{60}$

1.3814

e = 3

Confidence Interval:

73.6 $\pm$ 3

= 73.6 $\pm$ 3

= ( 70.6 ,76.6)

Confidence Interval:

70.6 <   $\mu$ <   76.6