Mathematics (AS & A Level) · May / June 2025 · Paper 62 · Question 5

The amount of time, in minutes, spent by a customer on one visit to a certain shop is modelled by the random variable ( ) N X 2 + v , n . In the past, the values of n and v were 10.5 and 3.8 respectively. The shop has recently moved to a new location, and the manager hopes that the new value of n will be greater than 10.5. He takes a random sample of 10 customers and notes the time they each spend in the shop. He then calculates the sample mean x for these 10 times. Using a hypothesis test at the 5% significance level, the manager finds that there is sufficient evidence to conclude that the new value of n is greater than 10.5. Stating a necessary assumption, find the smallest possible value of x. [4]

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