To compare the means of two groups, the researcher can use a statistical test to determine whether there is a significant difference between the average sugar intake of the migraine group (M) and the no-migraine group (O). The research hypotheses are as follows:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\( \mu_M = \mu_O \)[/tex]
This hypothesis asserts that there is no difference in the mean sugar intake between the two groups.
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\( \mu_M > \mu_O \)[/tex]
This hypothesis asserts that the mean sugar intake of the migraine group is greater than that of the no-migraine group.
The researcher has collected data from 50 teenagers in each group and computed the necessary statistics to perform a hypothesis test. The value of the test statistic is then calculated based on the sample data.
The test statistic helps determine whether there is a significant difference between the means of the two groups. Large values of the test statistic indicate that the mean difference observed in the sample data is unlikely to have occurred by chance, suggesting that the alternative hypothesis ([tex]\(H_a: \mu_M > \mu_O\)[/tex]) might be true.
Here are the given test statistic values:
- 1.645
- 1.96
- 10.6
- 0.193
Among the given values, the test statistic relevant to comparing the means in this case is 10.6. This value is significantly larger than typical critical values (e.g., 1.645 or 1.96 for common confidence levels), indicating a strong possibility that the mean sugar intake of the migraine group is indeed greater than that of the no-migraine group.
Thus, the test statistic that supports the conclusion that the mean sugar intake is higher in the migraine group compared to the no-migraine group is:
[tex]\[ \boxed{10.6} \][/tex]
This large test statistic suggests that the data provide strong evidence against the null hypothesis ([tex]\(H_0\)[/tex]) and in favor of the alternative hypothesis ([tex]\(H_a: \mu_M > \mu_O\)[/tex]).
