Probability & Statistics 2 — Variant 1

In a certain country, $20\ 540$ adults out of a population of $6\ 012\ 300$ have a degree in medicine.

Use an approximating distribution to calculate the probability that, in a random sample of $1000$ adults in this country, there will be fewer than $4$ adults who have a degree in medicine.

Justify the approximating distribution used in part (a).

In the past, the annual amount of wheat produced per farm by a large number of similar sized farms in a certain region had mean $24.0$ tonnes and standard deviation $5.2$ tonnes. Last summer a new fertiliser was used by all the farms, and it was expected that the mean amount of wheat produced per farm would be greater than $24.0$ tonnes. In order to test whether this was true, a scientist recorded the amounts of wheat produced by a random sample of $50$ farms last summer. He found that the value of the sample mean was $25.8$ tonnes. Stating a necessary assumption, carry out the test at the $1\%$ significance level.

A certain train journey takes place every day throughout the year. The time taken, in minutes, for the journey is normally distributed with variance $11.2$.

The mean time for a random sample of $n$ of these journeys was found. A $94\%$ confidence interval for the population mean time was calculated and was found to have a width of $1.4076$ minutes, correct to 4 decimal places. Find the value of $n$.

A passenger noted the times for $50$ randomly chosen journeys in January, February and March. Give a reason why this sample is unsuitable for use in finding a confidence interval for the population mean time.

A researcher took $4$ random samples and a $94\%$ confidence interval for the population mean was found from each sample. Find the probability that exactly $3$ of these confidence intervals contain the true value of the population mean.

Large packets of rice are packed in cartons, each containing $20$ randomly chosen packets. The masses of these packets are normally distributed with mean $1010$ g and standard deviation $3.4$ g. The masses of the cartons, when empty, are independently normally distributed with mean $50$ g and standard deviation $2.0$ g.

Find the variance of the masses of full cartons. Small packets of rice are packed in boxes. The total masses of full boxes are normally distributed with mean $6730$ g and standard deviation $15.0$ g. The masses of the boxes and cartons are distributed independently of each other.

Find the probability that the mass of a randomly chosen full carton is more than three times the mass of a randomly chosen full box.

A sample of $5$ randomly selected values of a variable $X$ is as follows: $1$ $2$ $6$ $1$ $a$ where $a > 0$. Given that an unbiased estimate of the variance of $X$ calculated from this sample is $\frac{11}{2}$, find the value of $a$.

The number of accidents per week at a certain factory has a Poisson distribution. In the past the mean has been $1.9$ accidents per week. Last year, the manager gave all his employees a new booklet on safety. He decides to test, at the $5\%$ significance level, whether the mean number of accidents has been reduced. He notes the number of accidents during $4$ randomly chosen weeks this year.

State suitable null and alternative hypotheses for the test.

Find the critical region for the test and state the probability of a Type I error.

State what is meant by a Type I error in this context.

During the $4$ randomly chosen weeks there are a total of $3$ accidents. State the conclusion that the manager should reach. Give a reason for your answer.

Assuming that the mean remains $1.9$ accidents per week, use a suitable approximation to calculate the probability that there will be more than $100$ accidents during a $52$-week period.