An environmental agency is testing the lifespan of two different types of industrial bio-degradable filters. A random sample of 80 filters of Type A was tested until failure. The lifespans, a a\,a hours, are summarised below:
∑a=5200and∑a2=340 844 \sum a = 5200 \quad \text{and} \quad \sum a^2 = 340\,844 ∑a=5200and∑a2=340844Calculate unbiased estimates for the mean and variance of the lifespan of Type A filters.
A separate random sample of 80 filters of Type B was tested under identical conditions. The lifespans, b b\,b hours, yielded the following summary statistics:
bˉ=67.8andsb2=45.2 \bar{b} = 67.8 \quad \text{and} \quad s_b^2 = 45.2 bˉ=67.8andsb2=45.2Test, at the 1% level of significance, whether there is a difference between the mean lifespan of Type A filters and Type B filters. State your hypotheses clearly.
Explain why it was necessary to use the Central Limit Theorem in the context of this test.
State an assumption made regarding the population variances to conduct the test in part (b).
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.