Question

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (a) 95\% confidence interval for \(p: \quad 0.53\) to 0.57 (b) \(95 \%\) confidence interval for \(p: \quad 0.41\) to 0.52 (c) 99\% confidence interval for \(p: \quad 0.35\) to 0.55

Short answer

For part (a), the null hypothesis \(H_{0}: p=0.5\) is rejected at a 95% confidence level because the interval does not include the hypothesis value. For parts (b) and (c), we do not reject the null hypothesis at a 95% and 99% confidence levels respectively because the confidence intervals include the value of the null hypothesis.

Step-by-step solution

Analyzing Confidence Interval (a)

The given 95% confidence interval for \(p\) is from 0.53 to 0.57. The value for \(H_{0}\) which is 0.5 is not lying in this range. Therefore, the null hypothesis of \(p = 0.5\) is rejected, concluding \(H_{a}: p \neq 0.5\).

Analyzing Confidence Interval (b)

The given 95% confidence interval for \(p\) is from 0.41 to 0.52. Here, the hypothesis value \(p = 0.5\) lies within the interval. Therefore, we do not reject the null hypothesis, implying there's not enough evidence to conclude \(H_{a}: p \neq 0.5\).

Analyzing Confidence Interval (c)

The given 99% confidence interval for \(p\) is from 0.35 to 0.55. The value \(p = 0.5\) lies within this interval. Therefore, we do not reject the null hypothesis in this case as well. Again, we cannot conclude \(H_{a}: p \neq 0.5\).