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)(t test chart)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.
Sophisticated statistics questions involving t tests, chi-squared tests and null hypotheses etc. are usually not answered here because we are shorthanded in that area.
To determine whether the results support the researcher's theory, we can follow the steps of hypothesis testing:
Step 1: Formulate the null hypothesis (H0) and alternative hypothesis (Ha):
The null hypothesis is 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 is that people who ate a large meal have better hearing ability compared to those who ate a small meal.
Step 2: Determine the significance level:
The significance level, denoted by α, is typically set at 0.05 for this type of test.
Step 3: Gather and analyze the data:
The hearing ability scores are given in the following table:
```
Large Meal: 14, 15, 12
Small Meal: 10, 8, 11
```
Step 4: Conduct the t-test:
To compare the means of two groups, we can use a two-sample t-test. By calculating the t-value and comparing it to a critical value from the t-test chart, we can determine whether the results are statistically significant.
Step 5: Determine the critical region:
The critical region is the area in the tail(s) of the distribution(s) that would lead us to reject the null hypothesis. Since this is a two-tailed test (we are testing for the possibility of either better or worse hearing ability), we need to split the significance level (α) equally between both tails. In this case, we will use α/2 = 0.025 for each tail.
Step 6: Calculate the test statistic (t-value):
The t-value is calculated using the formula:
```
t = (mean1 - mean2) / (sqrt((s1^2 / n1) + (s2^2 / n2)))
```
where:
mean1 and mean2 are the means of the two groups,
s1 and s2 are the standard deviations of the two groups,
n1 and n2 are the sample sizes of the two groups.
Using the given data, we can calculate the t-value.
Step 7: Compare the t-value with the critical value:
Using the t-test chart (with degrees of freedom = n1 + n2 - 2), we find the critical value corresponding to α/2 = 0.025 and the appropriate degrees of freedom.
Step 8: Make a decision and interpret the results:
If the absolute value of the t-value is greater than the critical value, we reject the null hypothesis. If it is not, we fail to reject the null hypothesis.
Next, let's sketch the distributions involved to better visualize the t-test results:
1. Draw the distribution for the large meal group:
a. Calculate the mean and standard deviation of the large meal group.
b. Sketch a normal distribution curve representing the large meal group's hearing ability scores.
2. Draw the distribution for the small meal group:
a. Calculate the mean and standard deviation of the small meal group.
b. Sketch a normal distribution curve representing the small meal group's hearing ability scores.
3. Indicate the critical region(s) on the distributions:
a. Mark the critical region(s) that correspond to the chosen significance level (α) on both distributions.
b. The critical region(s) will be the area(s) that would lead us to reject the null hypothesis.
4. Plot the t-value on the distributions:
a. Locate the t-value on the appropriate distribution(s).
b. Determine whether the t-value falls within or outside the critical region(s) to make a decision.
By following these steps, we can determine whether the results of the hypothesis test support the researcher's theory that people can hear better after eating a large meal.