Probability & Statistics 2 — Variant 3

A random sample of 100 values of a variable $X$ is taken. These values are summarised below. $n = 100$ $\Sigma x = 1556$ $\Sigma x^{2} = 29004$ Calculate unbiased estimates of the population mean and variance of $X$.

Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions $W \sim \mathrm{N}(1520, 450)$, $X \sim \mathrm{N}(2250, 720)$ and $Y \sim \mathrm{N}(3860, 1050)$. Find the probability that, on a particular day, $Y$ is less than the total of $W$ and $X$.

The number of customers who visit a particular shop between 9.00 am and 10.00 am has the distribution $\mathrm{Po}(\lambda)$. In the past the value of $\lambda$ was $5.2$. Following some new advertising, the manager wishes to test whether the value of $\lambda$ has increased. He chooses a random sample of 20 days and finds that the total number of customers who visited the shop between 9.00 am and 10.00 am on those days is 125. Use an approximating distribution to test at the 2.5% significance level whether the value of $\lambda$ has increased.

The random variable $A$ has the distribution $\mathrm{Po}(1.5)$. $A_{1}$ and $A_{2}$ are independent values of $A$.

Find $\mathrm{P}(A_{1} + A_{2} < 2)$.

Given that $A_{1} + A_{2} < 2$, find $\mathrm{P}(A_{1} = 1)$.

Give a reason why $A_{1} - A_{2}$ cannot have a Poisson distribution.

Sunita has a six-sided die with faces marked 1, 2, 3, 4, 5, 6. The probability that the die shows a six on any throw is $p$. Sunita throws the die 500 times and finds that it shows a six 70 times.

Calculate an approximate 99% confidence interval for $p$.

Sunita believes that the die is fair. Use your answer to part (a) to comment on her belief.

Sunita uses the result of her 500 throws to calculate an $\alpha\%$ confidence interval for $p$. This interval has width $0.04$. Find the value of $\alpha$.

The length, $X$ centimetres, of worms of a certain type is modelled by the probability density function $f(x) = \begin{cases} \frac{6}{125}(10 - x)(x - 5) & 5 \leqslant x \leqslant 10, \\ 0 & \text{otherwise.} \end{cases}$

State the value of $\mathrm{E}(X)$.

Find $\mathrm{Var}(X)$.

Two worms of this type are chosen at random. Find the probability that exactly one of them has length less than $6$ cm.