Probability & Statistics 2 — Variant 3

It is known that the height $H$, in metres, of trees of a certain kind has the distribution $\mathrm{N}(12.5, 10.24)$. A scientist takes a random sample of 25 trees of this kind and finds the sample mean, $\bar{H}$, of the heights.

State the distribution of $\bar{H}$, giving the values of any parameters.

Find $\mathrm{P}(12 < \bar{H} < 13)$.

The number of enquiries received per day at a customer service desk has a Poisson distribution with mean 45.2. If more than 60 enquiries are received in a day, the customer service desk cannot deal with them all. Use a suitable approximating distribution to find the probability that, on a randomly chosen day, the customer service desk cannot deal with all the enquiries that are received.

A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate $\alpha\%$ confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162. Calculate the value of $\alpha$ correct to 2 significant figures.

The proportion of people having a particular medical condition is 1 in 100 000. A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by $X$.

State, with a justification, a suitable approximating distribution for $X$, giving the values of any parameters.

Use the approximating distribution to calculate $\mathrm{P}(X > 0)$.

The percentage of people having a different medical condition is thought to be $30\%$. A researcher suspects that the true percentage is less than $30\%$. In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the $2\%$ significance level.

The masses, in grams, of apples from a certain farm have mean $\mu$ and standard deviation 5.2. The farmer says that the value of $\mu$ is 64.6. A quality control inspector claims that the value of $\mu$ is actually less than 64.6. In order to test his claim he chooses a random sample of 100 apples from the farm.

The mean mass of the 100 apples is found to be $63.5$ g. Carry out the test at the $2.5\%$ significance level.

Later another test of the same hypotheses at the $2.5\%$ significance level, with another random sample of 100 apples from the same farm, is carried out. Given that the value of $\mu$ is in fact 62.7, calculate the probability of a Type II error.