Question

1.  A study is is taken to find the mean distance commuters drove to work in a large city. We want an error margin of no more than 2.0 km , 95% confidence, and we use an estimate of 5.9 km for the standard deviation. How large would a minimum sample need to be?

2. It is claimed that 20% of all people eat fish every week. We take a sample of 500 people and 85 report that they eat fish every week. At an alpha = .10, do we have evidence to refute this claim? (Note: Either high or low).

a. Null/Alternate Hypotheses;

b. Test Statistic;

c. Critical regions;

d. Reject/Fail to Reject;

e. Find the p-value;

f. Answer the question posed

Answer 1

1)

The following information is provided,
Significance Level, α = 0.05, Margin or Error, E = 2, σ = 5.9

The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.96 * 5.9/2)^2
n = 33.43

Therefore, the sample size needed to satisfy the condition n >= 33.43 and it must be an integer number, we conclude that the minimum required sample size is n = 34
Ans : Sample size, n = 34 or 33

2)
a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.2
Alternative Hypothesis, Ha: p ≠ 0.2

b)

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.17 - 0.2)/sqrt(0.2*(1-0.2)/500)
z = -1.68

c)

Rejection Region
This is two tailed test, for α = 0.1
Critical value of z are -1.64 and 1.64.
Hence reject H0 if z < -1.64 or z > 1.64

d)
REject H0

e)

P-value Approach
P-value = 0.093
As P-value < 0.1, reject the null hypothesis.

f)

There is sufficient evidene to refute this claim