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
(b) if ( \bar{x} = 40.3 ) and ( s = 7.3 ), compute the test statistic
( t_{0} = ) (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 greater than ( alpha ), the researcher will fail to reject the null hypothesis
(d) construct a 99% confidence interval to test the hypothesis.
the lower bound is
the upper bound is
(round to three decimal places as needed.)
because the value lies the confidence interval, we the null hypothesis
(type an integer or decimal. do not round)

Explanation:

Step1: Recall the formula for the confidence interval

The formula for a confidence interval for the population mean \(\mu\) when the population standard deviation \(\sigma\) is unknown is \(\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}\)

Step2: Determine the degrees of freedom and \(t - \)value

The degrees of freedom \(df=n - 1=35-1 = 34\). For a \(99\%\) confidence interval, \(\alpha=1 - 0.99=0.01\), and \(\alpha/2=0.005\). Looking up in the \(t -\)distribution table (or using a calculator), \(t_{0.005,34}\approx 2.728\)

Step3: Calculate the margin of error \(E\)

The margin of error \(E=t_{\alpha/2}\frac{s}{\sqrt{n}}\). Substitute \(t_{\alpha/2} = 2.728\), \(s = 7.3\), and \(n = 35\)

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Step4: Calculate the lower and upper bounds

The lower bound \(L=\bar{x}-E\), and the upper bound \(U=\bar{x}+E\). Substitute \(\bar{x}=40.3\) and \(E = 3.367\)

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Step5: Check the null hypothesis value

The null hypothesis value \(\mu_0 = 37\). Since \(36.933<37<43.667\), the value \(37\) lies within the confidence interval.

Answer:

The lower bound is \(36.933\). The upper bound is \(43.667\). Because the value \(37\) lies within the confidence interval, we fail to reject the null hypothesis.