Questions: The average final exam score for the statistics course is 78. A professor wants to see if the average final exam score for students who are given colored pens on the first day of class is higher. The final exam scores for the 13 randomly selected students who were given the colored pens are shown below. Assume that the distribution of the population is normal. 100,100,74,78,69,65,70,71,95,96,99,84,76 What can be concluded at the α=0.10 level of significance? a. For this study, we should use t-test for a population mean b. The alternative hypothesis would be: H1: μ Select an answer c. The test statistic ?V= (please show your answer to 3 decimal places.) d. Based on this, we should Select an answer the null hypothesis.

The average final exam score for the statistics course is 78. A professor wants to see if the average final exam score for students who are given colored pens on the first day of class is higher. The final exam scores for the 13 randomly selected students who were given the colored pens are shown below. Assume that the distribution of the population is normal.

100,100,74,78,69,65,70,71,95,96,99,84,76

What can be concluded at the α=0.10 level of significance?
a. For this study, we should use t-test for a population mean
b. The alternative hypothesis would be:
H1: μ Select an answer
c. The test statistic ?V= (please show your answer to 3 decimal places.)
d. Based on this, we should Select an answer the null hypothesis.

Transcript text: The average final exam score for the statistics course is 78. A professor wants to see if the average final exam score for students who are given colored pens on the first day of class is higher. The final exam scores for the 13 randomly selected students who were given the colored pens are shown below. Assume that the distribution of the population is normal. \[ 100,100,74,78,69,65,70,71,95,96,99,84,76 \] What can be concluded at the $\alpha=0.10$ level of significance? a. For this study, we should use $\square$ t-test for a population mean $\checkmark$ b. The alternative hypothesis would be: $H_{1}: \mu \vee \checkmark$ Select an answer $\checkmark$ $\square$ c. The test statistic ?V= $\square$ (please show your answer to 3 decimal places.) d. Based on this, we should $\square$ Select an answer $\checkmark$ the null hypothesis.

Solution

Solution Steps

Step 1: Calculate the Sample Mean

The sample mean $ \bar{x} $ is calculated as follows: \[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} = \frac{1077}{13} = 82.846 \]

Step 2: Calculate the Sample Standard Deviation

The variance $ \sigma^2 $ is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = 177.974 \] The standard deviation $ \sigma $ is then: \[ \sigma = \sqrt{177.974} = 13.341 \]

Step 3: Calculate the Standard Error

The standard error $ SE $ is calculated as: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{13.341}{\sqrt{13}} \approx 3.7 \]

Step 4: Perform the t-Test

The test statistic $ t $ is calculated using the formula: \[ t = \frac{\bar{x} - \mu_0}{SE} = \frac{82.846 - 78}{3.7} \approx 1.31 \]

Step 5: Calculate the P-value

For a right-tailed test, the P-value is given by: \[ P = 1 - T(z) \approx 0.107 \]

Step 6: Conclusion

At the significance level $ \alpha = 0.10 $, since the P-value $ 0.107 $ is greater than $ \alpha $, we fail to reject the null hypothesis.

Final Answer

a. We should use $ \checkmark $ t-test for a population mean.
b. The alternative hypothesis would be: $ H_{1}: \mu > 78 $ $ \checkmark $.
c. The test statistic $ t $ = $ 1.31 $.
d. Based on this, we should $ \checkmark $ fail to reject the null hypothesis.

Thus, the final answer is: \[ \boxed{H_{1}: \mu > 78, \, t = 1.31, \, \text{fail to reject the null hypothesis}} \]

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