Exhibit 10-3
A statistics teacher wants to see if there is any difference in the abilities of students enrolled in statistics today and those enrolled five years ago. Final examination scores from a random sample of students enrolled today and from a random sample of students enrolled five years ago were selected. You are given the following information.

Today

Five Years Ago

82

88

σ2

112.5

54

n

45

36

Refer to Exhibit 10-3. The standard error of is _____.

Question

Exhibit 10-3
A statistics teacher wants to see if there is any difference in the abilities of students enrolled in statistics today and those enrolled five years ago. Final examination scores from a random sample of students enrolled today and from a random sample of students enrolled five years ago were selected. You are given the following information.

Today

Five Years Ago

82

88

σ2

112.5

54

n

45

36

Refer to Exhibit 10-3. The standard error of  is _____.

BlackTom AI · Accepted Answer

To assess whether there is a difference in mean scores between today’s statistics students and those from five years ago, we compare the sampling distributions of the two sample means. The standard error of the difference between two independent means is given by SE(x̄_today − x̄_past) = sqrt( (σ_today^2 / n_today) + (σ_past^2 / n_past) ).

First, compute each component:
- For today: σ_today^2 / n_today = 112.5 / 45 = 2.5
- For five years ago: σ_past^2 / n_past = 54 / 36 = 1.5

Next, add the two variances of the sampling distributions: 2.5 + 1.5 = 4.0.

Finally, take the square root to obtain the standard error of the difference in means: sqrt(4.0) = 2.0.

Now evaluate the given options:
- Option A: 4 — This value represents the sum of the two component quotients (2.5 + 1.5) before taking the square root, i.e., the variance of the difference, not the standard error itself.
- Option B: 9.3 — This number does not correspond to any step in the standard error calculation here; it seems unrelated to the given variances and sample sizes.
- Option C: 12.9 — Similar to B, this is not aligned with the SEM calculation for these data and would be far larger than the derived standard error.
- Option D: 2 — This matches the final standard error after applying the square root to the sum of the component terms 2.5 and 1.5.

Understanding common pitfalls: misplacing the square root or summing variances incorrectly leads to the wrong magnitude. Remember that SE uses the square root of the sum of each variance term divided by its sample size, not the sum of the standard errors themselves.

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