Suppose again that two new groups of 50 people have been randomly selected and asked their height in centimetres. The two groups are labelled “Group A” and “Group B”. Further suppose that we wish to test the following hypotheses: H0:μ1=μ2versusH1:μ1≠μ2,H0:μ1=μ2versusH1:μ1≠μ2, where: μ1μ1 denotes the true average height of people in the Group A population μ2μ2 denotes the true average height of people in the Group B population. Using the data collected, an independent samples tt-test was carried out. The associated 95% confidence interval of the difference between μ1μ1 and μ2μ2 is (-1.281, 1.934). Using the confidence interval provided, and with a significance level of α=0.05α=0.05, is there a statistically significant difference in average height between Group A and Group B?

Question

BlackTom AI · Accepted Answer

A 95% confidence interval that includes 0 means the two-sided test at α = 0.05 will not reject H0. Specifically, the reported 95% CI for μ1 − μ2 is (-1.281, 1.934). Because 0 lies between −1.281 and 1.934, a zero difference (μ1 = μ2) is a plausible value given the data, so we do not have sufficient evidence to declare a statistically significant difference at the 5% level.

1) Relevant concepts
- Relationship between two-sided hypothesis test and confidence interval: a 100(1−α)% CI for μ1 − μ2 excludes 0 if and only if the two-sided test at level α rejects H0: μ1 = μ2.
- If 0 is inside the CI, the p-value for the two-sided test is > α, so fail to reject H0.

2) Step-by-step reasoning
- Given CI = (−1.281, 1.934) and α = 0.05 (so use 95% CI).
- Check whether 0 is inside the interval: −1.281 < 0 < 1.934. Yes, 0 is inside.

3) Calculations (intermediate steps, approximate)
- Estimate of difference (center of CI) = (lower + upper)/2 = (−1.281 + 1.934)/2 = 0.653/2 ≈ 0.3265.
- Approximate standard error (using tcrit ≈ 1.984 for df ≈ 98) = (upper − lower) / (2 · tcrit) = 3.215 / (2 · 1.984) = 3.215 / 3.968 ≈ 0.810.
- Approximate t-statistic = estimate / SE ≈ 0.3265 / 0.810 ≈ 0.403 → two‑sided p-value ≈ 0.69 (≫ 0.05).
These computations illustrate the test statistic is small and the p-value is large, consistent with failing to reject H0.

4) Why the correct answer is correct
Because the 95% CI includes 0, the observed difference is not statistically significant at the 5% level. The CI shows a range of plausible differences that includes no difference, so we cannot conclude a significant difference exists.

5) Why the other choices are incorrect
- (a) is wrong: the confidence interval gives enough information to decide significance for a two-sided α = 0.05 test (check whether 0 is inside the CI).
- (b) is wrong: a CI that includes 0 indicates no statistically significant difference at the chosen α.

6) Conclusion
Therefore, there is not a statistically significant difference in average height between Group A and Group B. This matches the provided answer: c. No, there is not a statistically significant difference.

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Determine if a paired t-test or independent two-sample t-test is appropriate: The basal average metabolic rate of seven chimpanzees is compared with the basal average metabolic rate of seven gorillas.

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