Probability & Statistics 2 — Variant 2

In a survey of $200$ randomly chosen students from a certain college, $23\%$ of the students said that they owned a car. Calculate an approximate $93\%$ confidence interval for the proportion of students from the college who own a car.

The random variable $W$ has a Poisson distribution. State the relationship between $\mathrm{E}(W)$ and $\mathrm{Var}(W)$.

The random variable $X$ has the distribution $\mathrm{B}(n, p)$. Jyothi wishes to use a Poisson distribution as an approximate distribution for $X$. Use the formulae for $\mathrm{E}(X)$ and $\mathrm{Var}(X)$ to explain why it is necessary for $p$ to be close to $0$ for this to be a reasonable approximation.

Given that $Y$ has the distribution $\mathrm{B}(20000, 0.00007)$, use a Poisson distribution to calculate an estimate of $\mathrm{P}(Y > 2)$.

The masses, in kilograms, of newborn babies in country $A$ are represented by the random variable $X$, with mean $\mu$ and variance $\sigma^{2}$. The masses of a random sample of $500$ newborn babies in this country were found and the results are summarised below. $n = 500, \Sigma x = 1625, \Sigma x^{2} = 5663.5$

Calculate unbiased estimates of $\mu$ and $\sigma^{2}$. A researcher wishes to test whether the mean mass of newborn babies in a neighbouring country, $B$, is different from that in country $A$. He chooses a random sample of $60$ newborn babies in country $B$ and finds that their sample mean mass is $2.95$ kg. Assume that your unbiased estimates in part (a) are the correct values for $\mu$ and $\sigma^{2}$. Assume also that the variance of the masses of newborn babies in country $B$ is the same as in country $A$.

Carry out the test at the $1\%$ significance level.

The number, $X$, of books received at a charity shop has a constant mean of $5.1$ per day.

State, in context, one condition for $X$ to be modelled by a Poisson distribution. Assume now that $X$ can be modelled by a Poisson distribution.

Find the probability that exactly $10$ books are received in a $3$-day period.

Use a suitable approximating distribution to find the probability that more than $180$ books are received in a $30$-day period. The number of DVDs received at the same shop is modelled by an independent Poisson distribution with mean $2.5$ per day.

Find the probability that the total number of books and DVDs that are received at the shop in $1$ day is more than $3$.

Two random variables $X$ and $Y$ have the independent distributions $\mathrm{N}(7, 3)$ and $\mathrm{N}(6, 2)$ respectively. A random value of each variable is taken. Find the probability that the two values differ by more than $2$.

Each candidate's overall score in a science test is calculated as follows. The mark for theory is denoted by $T$, the mark for practical is denoted by $P$, and the overall score is given by $T + 1.5P$. The variables $T$ and $P$ are assumed to be independent with distributions $\mathrm{N}(62, 158)$ and $\mathrm{N}(42, 108)$ respectively. You should assume that no continuity corrections are needed when using these distributions.

A pass is awarded to candidates whose overall score is at least $90$. Find the proportion of candidates who pass.

Comment on the assumption that the variables $T$ and $P$ are independent.

When a child completes an online exercise called a Mathlit, they might be awarded a medal. The publishers claim that the probability that a randomly chosen child who completes a Mathlit will be awarded a medal is $\frac{1}{3}$. Asha wishes to test this claim. She decides that if she is awarded no medals while completing $10$ Mathlits, she will conclude that the true probability is less than $\frac{1}{3}$.

Use a binomial distribution to find the probability of a Type I error. The true probability of being awarded a medal is denoted by $p$.

Given that the probability of a Type II error is $0.8926$, find the value of $p$.