Probability & Statistics 2 — Variant 2

A javelin thrower noted the lengths of a random sample of $50$ of her throws. The sample mean was $72.3$ m and an unbiased estimate of the population variance was $64.3$ $\text{m}^{2}$. Find a $92\%$ confidence interval for the population mean length of throws by this athlete.

A discus thrower wishes to calculate a $92\%$ confidence interval for the population mean length of his throws. He bases his calculation on his first $50$ throws in a week. Comment on this method.

In the past, the mean height of plants of a particular species has been $2.3$ m. A random sample of $60$ plants of this species was treated with fertiliser and the mean height of these $60$ plants was found to be $2.4$ m. Assume that the standard deviation of the heights of plants treated with fertiliser is $0.4$ m. Carry out a test at the $2.5\%$ significance level of whether the mean height of plants treated with fertiliser is greater than $2.3$ m.

It is known that $1.8\%$ of children in a certain country have not been vaccinated against measles. A random sample of $200$ children in this country is chosen.

Use a suitable approximating distribution to find the probability that there are fewer than $3$ children in the sample who have not been vaccinated against measles.

Justify your approximating distribution.

A random variable $X$ has probability density function given by $f(x) = \begin{cases} \frac{3}{16}(4x - x^{2}) & 2 \leqslant x \leqslant 4, \\ 0 & \text{otherwise.} \end{cases}$

Show that $\mathrm{E}(X) = \frac{11}{4}$.

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

Given that the median of $X$ is $m$, find $\mathrm{P}(m < X < 3)$.

The masses, in kilograms, of large and small sacks of grain have the distributions $\mathrm{N}(53, 11)$ and $\mathrm{N}(14, 3)$ respectively.

Find the probability that the mass of a randomly chosen large sack is greater than four times the mass of a randomly chosen small sack.

A lift can safely carry a maximum mass of $1000$ kg. Find the probability that the lift can safely carry $12$ randomly chosen large sacks and $25$ randomly chosen small sacks.

$X$ is a random variable with distribution $\mathrm{Po}(2.90)$. A random sample of $100$ values of $X$ is taken. Find the probability that the sample mean is less than $2.88$.