Questions: The sample of six measurements shown below was randomly selected from a normally distributed population. Complete parts a through c -2,4,2,3,4,3 a. Test the null hypothesis that the mean of the population is 3 against the alternative hypothesis, μ<3. Use α=0.10. If α=0.10, find the rejection region for the test Choose the correct answer below. A. 1<-1.476 B. t<-1.476 or t>1.476 C. t>2015 D. t>1.476 E. t<-2.015 or t>2.015 F. t<-2.015 Calculate the value of the test statistic. t= (Round to two decimal places as needed.) Make the appropriate conclusion. Choose the correct answer below. A. Do not reject H0. There is insufficient evidence at the α=0.10 level of significance to conclude that the true mean of the population is less than 3 . B. Do not reject H0. There is sufficient evidence at the α=0.10 level of significance to conclude that the true mean of the population is less than 3 C. Reject H0. There is insufficient evidence at the α=0.10 level of significance to conclude that the true mean of the population is less than 3 . D. Reject H0. There is sufficient evidence at the α=0.10 level of significance to conclude that the true mean of the population is less than 3.

The sample of six measurements shown below was randomly selected from a normally distributed population. Complete parts a through c -2,4,2,3,4,3 a. Test the null hypothesis that the mean of the population is 3 against the alternative hypothesis, μ<3. Use α=0.10. If α=0.10, find the rejection region for the test Choose the correct answer below. A. 1<-1.476 B. t<-1.476 or t>1.476 C. t>2015 D. t>1.476 E. t<-2.015 or t>2.015 F. t<-2.015 Calculate the value of the test statistic. t= (Round to two decimal places as needed.) Make the appropriate conclusion. Choose the correct answer below. A. Do not reject H0. There is insufficient evidence at the α=0.10 level of significance to conclude that the true mean of the population is less than 3 . B. Do not reject H0. There is sufficient evidence at the α=0.10 level of significance to conclude that the true mean of the population is less than 3 C. Reject H0. There is insufficient evidence at the α=0.10 level of significance to conclude that the true mean of the population is less than 3 . D. Reject H0. There is sufficient evidence at the α=0.10 level of significance to conclude that the true mean of the population is less than 3.
Transcript text: The sample of six measurements shown below was randomly selected from a normally distributed population. Complete parts a through $\mathbf{c}$ \[ -2,4,2,3,4,3 \] a. Test the null hypothesis that the mean of the population is 3 against the alternative hypothesis, $\mu<3$. Use $\boldsymbol{\alpha}=0.10$. If $\alpha=0.10$, find the rejection region for the test Choose the correct answer below. A. $1<-1.476$ B. $\mathrm{t}<-1.476$ or $\mathrm{t}>1.476$ C. $t>2015$ D. $\mathrm{t}>1.476$ E. $\mathrm{t}<-2.015$ or $\mathrm{t}>2.015$ F. $t<-2.015$ Calculate the value of the test statistic. $\mathrm{t}=\square$ $\square$ (Round to two decimal places as needed.) Make the appropriate conclusion. Choose the correct answer below. A. Do not reject $\mathrm{H}_{0}$. There is insufficient evidence at the $\boldsymbol{\alpha}=0.10$ level of significance to conclude that the true mean of the population is less than 3 . B. Do not reject $\mathrm{H}_{0}$. There is sufficient evidence at the $\boldsymbol{\alpha}=0.10$ level of significance to conclude that the true mean of the population is less than 3 C. Reject $\mathrm{H}_{0}$. There is insufficient evidence at the $\alpha=0.10$ level of significance to conclude that the true mean of the population is less than 3 . D. Reject $\mathrm{H}_{0}$. There is sufficient evidence at the $\boldsymbol{\alpha}=0.10$ level of significance to conclude that the true mean of the population is less than 3.
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Solution

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Solution Steps

Step 1: Calculate the Sample Mean

The sample mean \( \bar{x} \) is calculated as follows: \[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} = \frac{14}{6} = 2.33 \]

Step 2: Calculate the Sample Variance and Standard Deviation

The sample variance \( s^2 \) is given by: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = 5.07 \] The sample standard deviation \( s \) is: \[ s = \sqrt{5.07} = 2.25 \]

Step 3: Calculate the Standard Error

The standard error \( SE \) is calculated as: \[ SE = \frac{s}{\sqrt{n}} = \frac{2.25}{\sqrt{6}} \approx 0.9186 \]

Step 4: Calculate the Test Statistic

The test statistic \( t \) is computed using the formula: \[ t = \frac{\bar{x} - \mu_0}{SE} = \frac{2.33 - 3}{0.9186} \approx -0.7294 \]

Step 5: Calculate the P-value

For a left-tailed test, the P-value is: \[ P = T(z) \approx 0.2492 \]

Step 6: Determine the Rejection Region

For a left-tailed test with \( \alpha = 0.10 \), the rejection region is: \[ t < -1.476 \]

Step 7: Conclusion

Since the test statistic \( t \approx -0.7294 \) does not fall within the rejection region \( t < -1.476 \), we do not reject the null hypothesis \( H_0 \). Therefore, there is insufficient evidence at the \( \alpha = 0.10 \) level of significance to conclude that the true mean of the population is less than 3.

Final Answer

The conclusion is: \\(\boxed{\text{Do not reject } H_0}\\)

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