A company advertises that its rechargeable batteries can run non-stop for an average of $505$ minu

Question

A company advertises that its rechargeable batteries can run non-stop for an average of $505$ minutes after a full charge. $100$ batteries are randomly selected, and their running time after a full charge, $x$ minutes, are measured. The data obtained is summarised by$$\sum (x - 500) = 320, \quad \sum (x - 500)^2 = 8416.$$ Suggest a reason why, in this context, the given data is summarised in terms of $(x - 500)$ rather than $x$.[1] Calculate unbiased estimates of the population mean and variance of the running time of the batteries.[2] A hypothesis test is carried out at the $5$% level of significance. Determine whether the company is overestimating the average running time of the batteries.[4] State, giving a reason, whether it is necessary to assume a normal distribution for the test to be valid.[1] After a change is made in the manufacturing process, another test at the $5$% significance level is carried out using a new random sample of size $50$. The sample standard deviation is found to be $8.5$ minutes. Find the range of values of the sample mean in order to conclude that the average running time of the batteries after a full charge is more than $505$ minutes.[4]

Accepted Answer

(b)$\bar{x}=503.2$, $s^2=74.7$ (e)$\bar{x} \geq 507$

2025 YIJC P2 Q10

2025 YIJC P2 Q10