Question

Researchers have claimed that the average number of headaches during a semester of Statistics is 14. Statistics professors dispute this claim vehemently. Statistic professors believe the average is much more than this. They sample 13 students and find the sample mean is 16 and the sample standard deviation is 2.0. The statistics professor would like to prove their claim at the 1% level of significance.

What sample size is necessary to estimate the average number of headaches with a 95% confidence interval of width 1?

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 14
Alternative Hypothesis, Ha: μ > 14

Rejection Region
This is right tailed test, for α = 0.01 and df = 12
Critical value of t is 2.681.
Hence reject H0 if t > 2.681

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (16 - 14)/(2/sqrt(13))
t = 3.606

P-value Approach
P-value = 0.0018
As P-value < 0.01, reject the null hypothesis.

b)

The following information is provided,
Significance Level, α = 0.05, Margin or Error, E = 0.5, σ = 2

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 * 2/0.5)^2
n = 61.47

Therefore, the sample size needed to satisfy the condition n >= 61.47 and it must be an integer number, we conclude that the minimum required sample size is n = 62
Ans : Sample size, n = 62 or 61