Question

In order to find the 95% confidence interval of the true mean, how large of a sample size would be necessary to be accurate within 0.1 g. given a standard deviation of 0.3 g? (8 points) 9 10, A T V network claims that 20% of the viewing public watches a certain one of its shows. A survey of 100 TV viewers found that 13 watch the show. If a 0.05, is there enough evidence to reject the network's claim? Use the P-value method. (9 points)

Answer 1

(9)

Sample Size (n) is given by:

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

Given:

$\alpha$ = 0.05

From Table, critical values of Z =$\pm$ 1.96

$\sigma$ = 0.3

e = 0.1

Substituting, we get:

$n=\frac{1.96^{2}\times 0.3^{2}}{0.1^{2}}=35$

So,

Answer is:

35

(10)

H0: Null Hypothesis: P = 0.20

HA: Alternative Hypothesis: P $\neq$ 0.20

n = Sample Size = 100

P = Population Proportion = 0.20

Q = 1 - P =0.80

SE = $\sqrt{PQ/n}=\sqrt{0.20\times 0.80/100}=0.04$

p = Sample Proportion = 13/100 = 0.13

Test statistic is given by:
Z = (0.13 - 0.20)/0.04

=- 1.75

Table of Area Under Standard Normal Curve gives area = 0.4599

So,

P value = (0.5 - 0.4599) X 2 = 0.0802

Since P - value = 0.0802 is greater than $\alpha$ = 0.05, the difference is not significant. Fail to reject null hypothesis.

Conclusion:
The data support the claim that 20% of the viewing public watches a certain one of its shows.