An industrial engineer is evaluating the tensile strength of two specific grades of composite fiber braids, Grade X and Grade Y. Random samples are extracted from production and their strengths, s N/mm2s\text{ N/mm}^2s N/mm2, are recorded. The findings are summarized in the table below.
| Number in sample | Sample mean sˉ\bar{s}sˉ | ∑s2\sum s^2∑s2 | |
|---|---|---|---|
| Grade X | 12 | 85.5 | 87834.0 |
| Grade Y | 15 | 92.2 | 127625.4 |
You may assume that the samples are drawn from independent normal distributions with a common population variance.
The engineer claims that the mean tensile strength of Grade Y fiber is 5 N/mm2\text{N/mm}^2N/mm2 greater than the mean tensile strength of Grade X fiber.
Test the engineer's claim at the 5% level of significance.
Given that the true population variance for both grades is actually 10 (N/mm2)210\text{ (N/mm}^2)^210 (N/mm2)2,
(i) show that when samples of size 12 and 15 are used with a 5% level of significance, the engineer's claim is accepted if 2.60<SˉY−SˉX<7.402.60 < \bar{S}_Y - \bar{S}_X < 7.402.60<SˉY−SˉX<7.40.
(ii) Hence find the probability of a Type II error for this test if, in fact, the true mean tensile strength of Grade Y is 8 N/mm2\text{N/mm}^2N/mm2 greater than the mean tensile strength of Grade X.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.