You wish to test the following claim ($ H $) at a significance level of $ \alpha = 0.01 $.

Null Hypothesis ($ H_0 $): $ \mu = 83.8 $

Alternative Hypothesis ($ H_1 $): $ \mu > 83.8 $

You believe the population is normally distributed, but you do not know the standard deviation. You obtain the following sample data: 105.5, 121.3, 103.3, 101.2

1. What is the test statistic for this sample? (Report your answer accurate to three decimal places.)

Test Statistic =

2. What is the p-value for this sample? (Report your answer accurate to four decimal places.)

p-value =

3. The p-value is:
- less than (or equal to) $ \alpha $
- greater than $ \alpha $

4. This test statistic leads to a decision to:
- reject the null hypothesis
- accept the null hypothesis
- fail to reject the null hypothesis

5. As such, the final conclusion is that:
- There is sufficient evidence to warrant rejection of the claim that the population mean is greater than 83.8.
- There is not sufficient evidence to warrant rejection of the claim that the population mean is greater than 83.8.
- The sample data support the claim that the population mean is greater than 83.8.
- There is not sufficient sample evidence to support the claim that the population mean is greater than 83.8.

To test the given claim, calculate the test statistic and p-value using a t-test. Compare the p-value to the significance level to make a decision.

In order to test the claim at a significance level of 0.01, we need to calculate the test statistic and the p-value for the given data. Since we don't know the standard deviation of the population, but we assume it is normally distributed, we can use a t-test. First, we calculate the sample mean and sample standard deviation. The sample mean is the sum of the data divided by the number of data points, and the sample standard deviation is the square root of the variance.

Next, we calculate the test statistic by subtracting the hypothesized population mean from the sample mean and dividing it by the standard error of the mean. The standard error of the mean is the sample standard deviation divided by the square root of the sample size. The test statistic can be calculated as follows:

Test Statistic = (Sample Mean - Hypothesized Mean) / (Sample Standard Deviation / sqrt(Sample Size))

After calculating the test statistic, we can determine the p-value. The p-value is the probability of observing a test statistic as extreme as the one obtained, assuming the null hypothesis is true. We can compare the test statistic to the critical value of the t-distribution to find the p-value. If the p-value is less than the significance level, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.

Based on the calculated test statistic and p-value, we compare the p-value to the significance level of 0.01 to make a decision. If the p-value is less than or equal to 0.01, we reject the null hypothesis. If the p-value is greater than 0.01, we fail to reject the null hypothesis. In this case, you'll need the actual values of the test statistic and p-value to determine the final decision and conclusion.

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