Question

to test ( h_{0}: mu = 37 ) versus ( h_{1}: mu
eq 37 ), a simple random sample of size ( n = 35 ) is obtained. complete parts (a) through (f) below. click the icon to view the table of critical t-values. (a) does the population have to be normally distributed to test this hypothesis by using t-distribution methods? why? a. no - there are no constraints in order to perform a hypothesis test. b. yes - the population must be normally distributed in all cases in order to perform a hypothesis test. c. no - since the sample size is at least 30, the underlying population does not need to be normally distributed. d. yes - since the sample size is at not least 50, the underlying population does not need to be normally distributed. (b) if ( \bar{x} = 40.3 ) and ( s = 7.3 ), compute the test statistic. ( t_{0}=square ) (round to two decimal places as needed.) (c) if the researcher decides to test this hypothesis at the ( alpha = 0.01 ) level of significance, will the researcher reject the null hypothesis? why? because the p - value is than ( alpha ), the researcher will the null hypothesis. (d) construct a 99% confidence interval to test the hypothesis.

Explanation:

Step1: Determine if population must be normal

For t - distribution methods, when the sample size \(n\geq30\), by the Central Limit Theorem, the sampling distribution of the sample mean \(\bar{x}\) is approximately normal. Here \(n = 35\geq30\).

Step2: Compute the test statistic

The formula for the t - test statistic is \(t_{0}=\frac{\bar{x}-\mu_{0}}{s/\sqrt{n}}\). Given \(\bar{x}=40.3\), \(\mu_{0} = 37\), \(s = 7.3\), and \(n = 35\).

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Step3: Analyze hypothesis rejection

The degrees of freedom \(df=n - 1=35-1 = 34\). For a two - tailed test with \(\alpha=0.01\), the critical values are \(t_{\alpha/2,df}=t_{0.005,34}\approx\pm 2.728\). The test statistic \(t_{0}=2.67\). Since \(|t_{0}|=2.67<2.728\), the P - value is greater than \(\alpha\).

Answer:

(a) C. No—since the sample size is at least 30, the underlying population does not need to be normally distributed.
(b) \(t_{0}\approx2.67\)
(c) Because the P - value is greater than \(\alpha\), the researcher will not reject the null hypothesis.