A robotic engineering firm is evaluating the battery performance of its latest autonomous vacuum, the 'DustBot-9000'. A random sample of 75 units was tested on hardwood flooring to determine the discharge time before recharging is required. The results, xxx minutes, are summarised as follows:
∑x=2550and∑x2=87400 \sum x = 2550 \quad \text{and} \quad \sum x^2 = 87400 ∑x=2550and∑x2=87400Find unbiased estimates for the mean and variance of the discharge time for units operating on hardwood floors.
An independent random sample of 75 DustBot-9000 units was then tested on thick carpeting. The discharge times, yyy minutes, for this sample are summarised by:
yˉ=35.2andsy2=10.8 \bar{y} = 35.2 \quad \text{and} \quad s_y^2 = 10.8 yˉ=35.2andsy2=10.8Test, at the 5% level of significance, whether there is evidence that the mean discharge time on hardwood floors is different from the mean discharge time on thick carpeting. State your hypotheses clearly.
Explain why the Central Limit Theorem is necessary to justify the test used in part (b).
State an assumption made regarding the population variances when conducting 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.