Probability & Statistics 2 — Variant 2

The lengths, $X$ cm, of a sample of 100 insects of a certain type were summarised as follows. $n = 100$ $\sum x = 36.8$ $\sum x^{2} = 17.34$

Calculate unbiased estimates for the population mean and variance of $X$.

State a necessary condition for the estimates found in part (a) to be reliable.

A random sample of 250 people living in Barapet was chosen. It was found that 78 of these people owned a BETEC phone.

Calculate an approximate 98% confidence interval for the proportion of people living in Barapet who own a BETEC phone.

Manjit claims that more than 40% of the people living in Barapet own a BETEC phone. Use your answer to part (a) to comment on this claim.

In a certain lottery, on average 1 in every 10 000 tickets is a prize-winning ticket. An agent sells 6000 tickets.

Use a suitable approximating distribution to find the probability that at least 3 of the tickets sold by the agent are prize-winning tickets.

Justify the use of your approximating distribution in this context.

Each year a transport firm uses $X$ litres of gasoline and $Y$ litres of diesel fuel, where $X$ and $Y$ have the independent distributions $X \sim \mathrm{N}(10700, 950^{2})$ and $Y \sim \mathrm{N}(13400, 1210^{2})$.

Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel. The costs per litre of gasoline and diesel fuel are $\$0.80$ and $\$0.85$ respectively.

Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is between $\$20000$ and $\$22000$.

A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables $G \sim \mathrm{Po}(0.10)$ and $B \sim \mathrm{Po}(0.15)$ respectively.

Find the probability that during a randomly chosen 2-day period no girls arrive late.

Find the probability that during a randomly chosen 5-day period the total number of students who arrive late is less than 3.

It is given that the values of $\mathrm{P}(G = r)$ and $\mathrm{P}(B = r)$ for $r \geqslant 3$ are very small and can be ignored. Find the probability that on a randomly chosen day more girls arrive late than boys. Following a timetable change the teacher claims that on average more students arrive late than before the change. During a randomly chosen 5-day period a total of 4 students are late.

Test the teacher's claim at the 5% significance level.

The graph of the probability density function f of a random variable $X$ is symmetrical about the line $x = 2$. It is given that $\mathrm{P}(2 < X < 5) = \frac{117}{256}$.

Using only this information show that $\mathrm{P}(X > -1) = \frac{245}{256}$. It is now given that, for $x$ in a suitable domain, $f(x) = k(12 + 4x - x^{2})$, where $k$ is a constant.

Find the value of $k$.

A different random variable $X$ has probability density function $g(x) = \frac{2}{9}(2 + x - x^{2})$. The domain of $X$ is all values of $x$ for which $g(x) \geqslant 0$. Find $\mathrm{Var}(X)$.

The heights, in centimetres, of adult females in Litania have mean $\mu$ and standard deviation $\sigma$. It is known that in 2004 the values of $\mu$ and $\sigma$ were 163.21 and 6.95 respectively. The government claims that the value of $\mu$ this year is greater than it was in 2004. In order to test this claim a researcher plans to carry out a hypothesis test at the 1% significance level. He records the heights of a random sample of 300 adult females in Litania this year and finds the value of the sample mean.

State the probability of a Type I error. You should assume that the value of $\sigma$ after 2004 remains at 6.95.

Given that the value of $\mu$ this year is actually 164.91, find the probability of a Type II error.