: A researcher theorized that people can hear better when they have just eaten a large meal. Six individuals were randomly assigned to eat either a large meal or a small meal. After eating the meal, their hearing was tested. The hearing ability scores (high numbers indicate greater ability) are given in the following table. Using the .05 level, do the results support the researcher’s theory? (a) Use the steps of hypothesis testing, (b) sketch the distributions involved. (c) Does not need to be completed in this assignment. To complete this question review the chart located on #16 pg. 307.

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To determine whether the results support the researcher's theory, we need to conduct a hypothesis test. Here are the steps you can follow to conduct the hypothesis test:

Step 1: State the hypotheses

The null hypothesis (H₀) states that there is no difference in hearing ability between those who ate a large meal and those who ate a small meal. The alternative hypothesis (H₁) states that people who have eaten a large meal have better hearing ability than those who ate a small meal.

In symbols:
H₀: μ_large = μ_small (no difference in hearing ability)
H₁: μ_large > μ_small (better hearing ability for those who ate a large meal)

Step 2: Set the significance level

The significance level (α) is given as 0.05 in the question. This means that we want to have 5% chance of making a Type I error (rejecting the null hypothesis when it is actually true).

Step 3: Conduct the test

We can use an independent samples t-test to compare the means of two independent groups. In this case, we have six individuals in each group, so we can calculate the mean and standard deviation for each group and perform the t-test.

Step 4: Calculate the test statistic

The t-test formula is:

t = (mean_large - mean_small) / (s / sqrt(n))

Where mean_large is the mean of the large meal group, mean_small is the mean of the small meal group, s is the pooled standard deviation, and n is the sample size.

Step 5: Determine the critical value or p-value

To determine if the results are statistically significant, we need to compare the calculated test statistic to the critical value or calculate the p-value.

Step 6: Make a decision

If the test statistic falls in the rejection region (beyond the critical value) or if the p-value is less than the significance level (α), we reject the null hypothesis. If not, we fail to reject the null hypothesis.

Regarding the sketching of distributions involved, it is necessary to refer to the chart on page 307 which contains the hearing ability scores for both the large meal and small meal groups. Unfortunately, as an AI text-based bot, I cannot access external sources or view charts. However, you can refer to the chart on page 307 to sketch the distributions and have a visual representation of the data.

By following these steps and using the provided data, you can conduct the hypothesis test and determine whether the results support the researcher's theory.