Questions: Question 7 You wish to test the following claim (Ha) at a significance level of alpha=0.005. Ho: mu=84.4 Ha: mu ≠ 84.4 You believe the population is normally distributed and you know the standard deviation is sigma=16.9. You obtain a sample mean of M=82 for a sample of size n=62. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p -value = The p-value is... - less than (or equal to) alpha - greater than alpha This test statistic leads to a decision to... - reject the null - accept the null - fail to reject the null As such, the final conclusion is that... - There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 84.4. - There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 84.4. - The sample data support the claim that the population mean is not equal to 84.4. - There is not sufficient sample evidence to support the claim that the population mean is not equal to 84.4.

Question 7 You wish to test the following claim (Ha) at a significance level of alpha=0.005. Ho: mu=84.4 Ha: mu ≠ 84.4 You believe the population is normally distributed and you know the standard deviation is sigma=16.9. You obtain a sample mean of M=82 for a sample of size n=62. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p -value = The p-value is... - less than (or equal to) alpha - greater than alpha This test statistic leads to a decision to... - reject the null - accept the null - fail to reject the null As such, the final conclusion is that... - There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 84.4. - There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 84.4. - The sample data support the claim that the population mean is not equal to 84.4. - There is not sufficient sample evidence to support the claim that the population mean is not equal to 84.4.
Transcript text: Question 7 You wish to test the following claim $\left(H_{a}\right)$ at a significance level of $\alpha=0.005$. \[ \begin{array}{l} H_{\circ}: \mu=84.4 \\ H_{a}: \mu \neq 84.4 \end{array} \] You believe the population is normally distributed and you know the standard deviation is $\sigma=16.9$. You obtain a sample mean of $M=82$ for a sample of size $n=62$. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic $=$ $\square$ What is the p-value for this sample? (Report answer accurate to four decimal places.) p -value = $\square$ The p-value is... less than (or equal to) $\alpha$ greater than $\alpha$ This test statistic leads to a decision to... reject the null accept the null fail to reject the null As such, the final conclusion is that... There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 84.4. There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 84.4 . The sample data support the claim that the population mean is not equal to 84.4. There is not sufficient sample evidence to support the claim that the population mean is not equal to 84.4 . Check Answer
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Solution

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Solution Steps

Step 1: Calculate the Standard Error

The standard error \( SE \) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{16.9}{\sqrt{62}} \approx 2.1463 \]

Step 2: Calculate the Test Statistic

The test statistic \( Z \) is calculated using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{82 - 84.4}{2.1463} \approx -1.1182 \]

Step 3: Calculate the P-value

For a two-tailed test, the p-value \( P \) is calculated as: \[ P = 2 \times (1 - T(|z|)) \approx 0.2635 \]

Step 4: Compare P-value with Alpha

The significance level \( \alpha \) is given as \( 0.005 \). Since: \[ P = 0.2635 > \alpha = 0.005 \] we conclude that the p-value is greater than alpha.

Step 5: Decision on the Null Hypothesis

Since the p-value is greater than the significance level, we fail to reject the null hypothesis \( H_0 \).

Final Conclusion

There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to \( 84.4 \).

Final Answer

\(\boxed{\text{The test statistic is } -1.118, \text{ the p-value is } 0.2635, \text{ and we fail to reject the null hypothesis.}}\)

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