Probability & Statistics 2 — Variant 3

A random variable $X$ has the distribution $\mathrm{N}(410, 400)$. Find the probability that the mean of a random sample of $36$ values of $X$ is less than $405$.

In a survey of $300$ randomly chosen adults in Rickton, $134$ said that they exercised regularly. This information was used to calculate an $\alpha\%$ confidence interval for the proportion of adults in Rickton who exercise regularly. The upper bound of the confidence interval was found to be $0.487$, correct to 3 significant figures. Find the value of $\alpha$ correct to the nearest integer.

The masses, in kilograms, of chemicals $A$ and $B$ produced per day by a factory are modelled by the independent random variables $X$ and $Y$ respectively, where $X \sim \mathrm{N}(10.3, 5.76)$ and $Y \sim \mathrm{N}(11.4, 9.61)$. The income generated by the chemicals is $\$2.50$ per kilogram for $A$ and $\$3.25$ per kilogram for $B$.

Find the mean and variance of the daily income generated by chemical $A$.

Find the probability that, on a randomly chosen day, the income generated by chemical $A$ is greater than the income generated by chemical $B$.

In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution $\mathrm{Po}(0.31)$. Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen $5$-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is $5$, carry out the test at the $2.5\%$ significance level.