An electronics engineer is evaluating the operational life of two drone battery models, Model Alpha and Model Beta. Under maximum load, random samples of these batteries are tested, and their flight times, ttt minutes, are recorded. The results are summarized in the table below.
| Sample size nnn | Sample mean tˉ\bar{t}tˉ | ∑t2\sum t^2∑t2 | |
|---|---|---|---|
| Model Alpha | 12 | 45.5 | 25010.0 |
| Model Beta | 15 | 52.8 | 42100.0 |
You may assume that the flight times for both models follow independent normal distributions with the same variance.
The engineer claims that the mean flight time of Model Beta batteries is 8 minutes longer than the mean flight time of Model Alpha batteries.
Test the engineer's claim at the 5% level of significance.
Given that the true population variance for both battery models is actually 15 min215 \text{ min}^215 min2,
show that when samples of size 12 and 15 are used with a 5% level of significance, the engineer's claim is accepted if 5.06<TˉB−TˉA<10.945.06 < \bar{T}_B - \bar{T}_A < 10.945.06<TˉB−TˉA<10.94.
Hence find the probability of a Type II error for this test if, in fact, the true mean flight time of Model Beta batteries is 12 minutes longer than the mean flight time of Model Alpha batteries.
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.