Question

d) the first group returned to sports activity in a mean of 60.7 days with a standard deviation of 28.5 days, and the other in a mean of 51 days with a standard deviation this difference is statistically significant? explain.
a. yes, since 51 < 60.7, the first exercise program is less effective than the second.
b. yes, since 10.7 < 28.5, the results of the first program are less consistent than the second.
c. yes, for both of the above reasons.
d. it is difficult to say whether use of mean and standard deviation as measures of center and spread is justifiable.
e. no, these numbers are roughly what we would expect from randomization.

Explanation:

To determine if the difference is statistically significant, we analyze each option:

• Option A: Comparing means (51 < 60.7) alone doesn’t prove significance, as we need to consider variability (standard deviation) and sample size (not provided here).
• Option B: Comparing standard deviations (10.7 < 28.5, assuming a typo for the second group’s SD, but even so) alone doesn’t prove significance without knowing if the mean difference is meaningful relative to variability.
• Option C: Combining A and B is invalid, as neither A nor B alone (or together) confirm significance without proper statistical testing (e.g., t - test, which requires sample size).
• Option D: Mean and standard deviation are valid for symmetric, non - skewed data, but the key issue is that we lack information (like sample size) to assess significance. However, the critical point is that without knowing the distribution of the data (e.g., if it’s skewed, mean/SD may not be appropriate measures), it’s hard to justify using them to determine significance. But more importantly, to claim statistical significance, we need to test if the mean difference is larger than what random chance would produce. Since we don’t have sample size or a formal test, we can’t conclude significance. But the most accurate reasoning is that we can’t assume the mean and standard deviation are appropriate (or that the difference is significant) without more context. However, the best answer is D because we don’t know if the data is suitable for mean/SD (e.g., if it’s skewed, these measures are misleading), making it hard to justify using them to assess significance. Wait, no—actually, the correct reasoning is that to determine statistical significance, we need to see if the observed difference (in means) is larger than what we’d expect from random variation. Since we don’t have sample size or a test statistic, we can’t conclude significance. But the options: Option E claims “these numbers are roughly what we would expect from randomization”—but we don’t know that. Option D says it’s difficult to say if mean/SD are justifiable. But the key is that without knowing the data’s distribution (e.g., if it’s skewed, mean/SD may not be appropriate), we can’t rely on them to assess significance. However, the most appropriate answer is D because we can’t be sure if mean and standard deviation are valid measures here (e.g., if the data is skewed, median/IQR would be better), so justifying their use to determine significance is difficult.

Answer:

D. It is difficult to say whether use of mean and standard deviation as measures of center and spread is justifiable.