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Comparing two normal means questions
Two samples, two means, and a question about whether the populations behind them differ. The difference of the sample means is itself normal, and everything follows from its standard error.
6 original questions · 22 marks · the comparing two normal means notes · Further Statistics 2
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Two independent samples of size 25 come from populations with variances 50 and 30. Find the standard error of the difference of the sample means.
Worked answer
Each mean has variance σ²/n, so 50/25 = 2 and 30/25 = 1.2. Adding gives 3.2, so the standard error is √3.2 = 1.789 to three decimal places. M1 for adding the two variances of the means, A1 for the standard error. The variances are added, never subtracted, even though the means are subtracted.Sample 1: n = 36, mean 21.4, population variance 9. Sample 2: n = 49, mean 19.6, population variance 16. Find the standard error of the difference and the test statistic for equal means.
Worked answer
Standard error = √(9/36 + 16/49) = √(0.25 + 0.3265) = √0.5765 = 0.759. The difference of means is 1.8, so z = 1.8/0.759 = 2.37. M1 A1 for the standard error, A1 for the statistic.Sample 1: n = 36, mean 21.4, population variance 9. Sample 2: n = 49, mean 19.6, population variance 16, giving a standard error of 0.759 and a test statistic of 2.37. Test at the 5% level whether the population means differ.
Worked answer
H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂, where μ₁ and μ₂ are the population means. Two-tailed at 5%, so the critical values are ±1.96. The statistic 2.37 exceeds 1.96 and lies in the critical region, so reject H₀. There is evidence at the 5% level that the two population means differ, with the first appearing larger. B1 for the hypotheses in terms of population means, B1 for the critical values, M1 for the comparison, A1 for the conclusion in context. Hypotheses written about x̄₁ and x̄₂ score nothing, since the sample means are known and nothing is being claimed about them.Two samples give means 21.4 and 19.6, with a standard error of 0.759 for their difference. Find a 95% confidence interval for the difference of the population means, and check it agrees with a two-tailed test at the 5% level.
Worked answer
The point estimate is 1.8 and the standard error 0.759, so the interval is 1.8 ± 1.96 × 0.759 = 1.8 ± 1.488, that is (0.31, 3.29). M1 for the form estimate plus or minus z times the standard error, A1 for the half-width, A1 for the interval, B1 for the comparison with the test. Zero lies outside, which is exactly the verdict the two-tailed test at 5% reached. The interval says more: the difference could plausibly be anywhere from a third of a unit to over three.Explain what changes if the population variances are unknown but both samples are large.
Worked answer
Replace each σ² by the sample variance s² from that sample. The Central Limit Theorem keeps the sample means approximately normal whatever the population shape, and with large samples the estimated variances are close enough to the real ones for the normal critical values to remain usable. The conclusion becomes approximate rather than exact, and this should be stated. B1 for replacing each variance by its sample estimate, B1 for the Central Limit Theorem, B1 for saying the result is approximate.Sample 1: n = 36, mean 21.4, population variance 9. Sample 2: n = 49, mean 19.6, population variance 16, giving a standard error of 0.7593 and a test statistic of 2.371. Test at the 1% level, one-tailed, with critical value 2.3263. Then find the smallest difference of sample means that would have been significant at that level, and comment on the pair of results.
Worked answer
H₀: μ₁ = μ₂; H₁: μ₁ > μ₂. The statistic is 2.371, which exceeds 2.3263, so reject H₀ at the 1% level as well.
The test rejects when the difference d satisfies d/0.7593 > 2.3263, that is d > 2.3263 × 0.7593 = 1.766.
The observed 1.8 clears that by only 0.034. A difference of 1.76 would have been significant at 5% but not at 1%, so the two levels would have disagreed. B1 for the one-tailed hypotheses, M1 for the comparison with 2.3263, A1 for the conclusion in context, M1 A1 for the smallest significant difference, B1 for the comment on how marginal it is. Report a statistic this close to its critical value as marginal rather than as a firm verdict, and quote the level with every conclusion.
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