Probability & Statistics 2 — Variant 2

On average, $1$ in $50000$ people have a certain gene. Use a suitable approximating distribution to find the probability that more than $2$ people in a random sample of $150000$ have the gene.

A six-sided die has faces marked $1, 2, 3, 4, 5, 6$. When the die is thrown $300$ times it shows a six on $56$ throws.

Calculate an approximate $96\%$ confidence interval for the probability that the die shows a six on one throw.

Maroulla claims that the die is biased. Use your answer to part (a) to comment on this claim.

The areas, $X$ $\text{cm}^{2}$, of petals of a certain kind of flower have mean $\mu$ $\text{cm}^{2}$. In the past it has been found that $\mu = 8.9$. Following a change in the climate, a botanist claims that the mean is no longer $8.9$. The areas of a random sample of $200$ petals from this kind of flower are measured, and the results are summarized by $\Sigma x = 1850$, $\Sigma x^{2} = 17850$. Test the botanist's claim at the $2.5\%$ significance level.

Customers arrive at a shop at a constant average rate of $2.3$ per minute.

State another condition for the number of customers arriving per minute to have a Poisson distribution. It is now given that the number of customers arriving per minute has the distribution $\mathrm{Po}(2.3)$.

Find the probability that exactly $3$ customers arrive during a $1$-minute period.

Find the probability that more than $3$ customers arrive during a $2$-minute period.

Five $1$-minute periods are chosen at random. Find the probability that no customers arrive during exactly $2$ of these $5$ periods.

A biscuit manufacturer claims that, on average, $1$ in $3$ packets of biscuits contain a prize offer. Gerry suspects that the proportion of packets containing the prize offer is less than $1$ in $3$. In order to test the manufacturer's claim, he buys $20$ randomly selected packets. He finds that exactly $2$ of these packets contain the prize offer.

Carry out the test at the $10\%$ significance level.

Maria also suspects that the proportion of packets containing the prize offer is less than $1$ in $3$. She also carries out a significance test at the $10\%$ level using $20$ randomly selected packets. She will reject the manufacturer's claim if she finds that there are $3$ or fewer packets containing the prize offer. Find the probability of a Type II error in Maria's test if the proportion of packets containing the prize offer is actually $1$ in $7$.

Explain what is meant by a Type II error in this context.

Before a certain type of book is published it is checked for errors, which are then corrected. For costing purposes each error is classified as either minor or major. The numbers of minor and major errors in a book are modelled by the independent distributions $\mathrm{N}(380, 140)$ and $\mathrm{N}(210, 80)$ respectively. You should assume that no continuity corrections are needed when using these models. A book of this type is chosen at random.

Find the probability that the number of minor errors is at least $200$ more than the number of major errors. The costs of correcting a minor error and a major error are $20$ cents and $50$ cents respectively.

Find the probability that the total cost of correcting the errors in the book is less than $\$190$.