Researchers are studying whether a high-protein diet reduces mortality in lab mice exposed to a toxin. Among 900 mice on a standard diet, 36 died. Among 950 mice on a high-protein diet, 15 died.

Definitions:
p1 = proportion of mice that died on the standard diet
p2 = proportion of mice that died on the high-protein diet

Hypotheses:
H0: p1 = p2 (The diet does not affect mortality; both groups have the same mortality rate)
H1: p1 p2 (The diet reduces mortality; mice on the standard diet have a higher mortality rate)

What is your decision, and what does it mean in the context of the study?

Question

Researchers are studying whether a high-protein diet reduces mortality in lab mice exposed to a toxin. Among 900 mice on a standard diet, 36 died. Among 950 mice on a high-protein diet, 15 died.

Definitions:
p1 = proportion of mice that died on the standard diet
p2 = proportion of mice that died on the high-protein diet

Hypotheses:
H0: p1 = p2 (The diet does not affect mortality; both groups have the same mortality rate)
H1: p1 > p2 (The diet reduces mortality; mice on the standard diet have a higher mortality rate)

What is your decision, and what does it mean in the context of the study?

BlackTom AI · Accepted Answer

We start by restating the scenario and the choices to ensure clarity of what’s being evaluated. Question setup: We compare mortality between two diet groups in 900 mice on a standard diet (p1 = mortality proportion) and 950 mice on a high-protein diet (p2 = mortality proportion). Observed data: 36 deaths in standard diet group, 15 deaths in high-protein group. H0: p1 = p2 (no difference in mortality between diets). H1: p1 > p2 (standard diet has higher mortality, i.e., the high-protein diet reduces mortality). Now we analyze each answer option one by one, noting what each claims and whether it matches the data and the stated hypotheses. Option A: “Fail to reject H0; this data is inconsistent with the assumption that there is no effect. It seems the treatment reduces mortality. p-value = 0.07 (not significant at alpha = 0.05)” - First, the claim that we fail to reject H0 conflicts with the observed large difference in mortality rates: 4.0% vs about 1.58%, which suggests evidence against equal mortality. The p-value given (0.07) is not aligned with this sizable separation and would typically be considered not significant at 0.05, but it does not reflect the actual data if we compute a proper test for p1 vs p2. Moreover, the statement “it seems the treatment reduces mortality” correctly describes the direction (p2 < p1), but under H0 we would not conclude a reduction. Overall, the numerical p-value and the conclusion do not consistently reflect a significant difference at the 0.05 level, making this option dubious. Option B: “Fail to reject H0; this data is consistent with the assumption that there is no effect. There is not enough evidence to conclude the treatment reduces mortality. p-value = 0.12 (not significant at alpha = 0.05)” - This option asserts no effect and a non-significant p-value, which would be plausible only if the data showed little to no difference. However, the observed death counts yield a substantial difference in proportions (4% vs ~1.58%), which would typically produce a small p-value with a reasonable test. The claim that the data are consistent with no effect contradicts the observed proportions and is unlikely given the actual counts. The p-value of 0.12 also does not reflect the observed disparity, so this option mischaracterizes the data. Option C: “Reject H0; this data is inconsistent with the assumption that there is no effect. It seems the treatment reduces mortality. p-value = 0.0007 (significant at alpha = 0.05)” - This option claims a rejection of H0, which aligns with the large difference in mortality observed (4.0% vs 1.58%). The stated direction, that the high-protein diet reduces mortality, matches the data (p2 < p1). The p-value of 0.0007 is consistent with a highly significant result at the 0.05 level, which correctly supports rejecting H0 in favor of p1 > p2. This option correctly captures both the conclusion and the statistical significance. Option D: “Reject H0; this data is consistent with the assumption that there is no effect. It seems the treatment does not reduce mortality. p-value = 0.08 (not significant at alpha = 0.05)” - This option attempts to combine a rejection of H0 with a claim that the treatment does not reduce mortality, which is internally inconsistent: rejecting H0 would imply a difference between groups, not no effect. The assertion that it is not significant at 0.05 is also incompatible with a rejection. The stated inconsistency and the contradictory phrasing (rejection with “does not reduce mortality”) make this option logically flawed, even before considering the p-value. To summarize the reasoning for each option in a cohesive way: Options A and B present conclusions that either misstate the significance or mischaracterize the data; they rely on p-values and interpretations that do not align with the observed proportions and the stated hypotheses. Option D contains a logical contradiction between rejecting H0 and claiming no effect, plus an incongruent interpretation of the p-value. Option C, in contrast, correctly identifies a real difference in mortality favoring the high-protein diet, with a p-value indicating strong statistical significance and a conclusion aligned with the alternative hypothesis. The directional hypothesis H1: p1 > p2 requests attention to higher mortality in the standard diet group, and the observed data support the opposite direction, leading to rejection of H0 under the given alternative. Note on the practical interpretation: Rejecting H0 here means there is statistically significant evidence at alpha = 0.05 that mortality differs between diets, with the data indicating lower mortality on the high-protein diet side, which is consistent with the stated alternative p1 > p2 being false and p2 < p1 being the observed outcome. The exact conveyed message in a test report would reflect that the diet reduces mortality relative to the standard diet, given the data, and the p-value quantifies the strength of that evidence.

BTRY3010/BTRY5010/STSCI2200/STSCI5200 Lecture #26 Pages ( 126 - 129, 134 )

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