Statistics Paper 4 June 2017

X is a discrete random variable with a geometric distribution, $X \sim Geo(0.4)$, counting the number of trials up to and including the first success.

Find $P(X \le 7)$.

X is a discrete random variable with a geometric distribution, $X \sim Geo(0.4)$, counting the number of trials up to and including the first success.

Find $P(X > 8 \mid X > 3)$.

A survey of 2 000 students at a certain university shows that on average one in every 500 students catches a cold in a week.

Using a suitable approximation, find the probability that exactly one student catches a cold in a week.

A survey of 2 000 students at a certain university shows that on average one in every 500 students catches a cold in a week, so the mean number catching a cold in a week is 4.

Using a Poisson model, find the probability that at least three students catch a cold in a month, assuming the month has exactly 28 days.

The marks obtained by 36 advanced level students in a Mathematics test were

59, 53, 74, 55, 90, 57, 88, 68, 59, 67, 82, 62, 61, 77, 74, 86, 60, 83, 92, 58, 60, 72, 57, 96, 56, 67, 73, 78, 66, 79, 51, 60, 54, 67, 80, 63

Find the median mark.

The marks obtained by 36 advanced level students in a Mathematics test were

59, 53, 74, 55, 90, 57, 88, 68, 59, 67, 82, 62, 61, 77, 74, 86, 60, 83, 92, 58, 60, 72, 57, 96, 56, 67, 73, 78, 66, 79, 51, 60, 54, 67, 80, 63

Find the interquartile range of these marks.

The marks obtained by 36 advanced level students in a Mathematics test were

59, 53, 74, 55, 90, 57, 88, 68, 59, 67, 82, 62, 61, 77, 74, 86, 60, 83, 92, 58, 60, 72, 57, 96, 56, 67, 73, 78, 66, 79, 51, 60, 54, 67, 80, 63

The marks are grouped into class intervals of width 5 marks, so one class holds the marks from 65 to 69 inclusive. How many of the 36 marks fall in that class?

The continuous random variable X has probability density function $f(x) = 2e^{-kx}$ for $x \ge 0$ and $f(x) = 0$ for $x < 0$, where k is a positive integer.

In terms of k, what is the value of $\displaystyle\int_{0}^{\infty} 2e^{-kx}\,dx$?

The continuous random variable X has probability density function $f(x) = 2e^{-2x}$ for $x \ge 0$ and $f(x) = 0$ for $x < 0$.

Which of these is the cumulative distribution function of X?

The continuous random variable X has probability density function $f(x) = 2e^{-2x}$ for $x \ge 0$ and $f(x) = 0$ for $x < 0$, so its cumulative distribution function is $F(x) = 1 - e^{-2x}$ for $x \ge 0$.

Find the exact value of the median of X.

The random variable X is normally distributed with mean $\mu$ and standard deviation $\sigma$, and $P(X > 65) = 0.01$.

Write down the value of z for which $\Phi(z) = 0.99$, giving your answer correct to 3 decimal places.

The random variable X is normally distributed with mean $\mu$ and variance $\sigma^{2}$, with $P(X > 65) = 0.01$ and $P(X < 20) = 0.02$. Solving the two standardised equations gives $\sigma = 10.27$.

Find $\mu$.

The discrete random variable X is distributed as shown in the table below.

Calculate the mean value of X.

A set of 120 observations of a discrete random variable X has mean $\dfrac{29}{30}$.

A Poisson model with the same mean is fitted to the data. Find the expected frequency of the value $X = 0$ out of the 120 observations.

A set of 120 observations of a discrete random variable X has mean $\dfrac{29}{30}$.

A Poisson model with the same mean is fitted to the data. Find the expected frequency of the value $X = 2$ out of the 120 observations.

A goodness of fit test compares 120 observations of X with a Poisson model of the same mean. The observed and expected frequencies are

The last two classes are pooled so that no expected frequency is below 5. Calculate the value of the test statistic $\chi^{2}$.

A goodness of fit test at the 5% level of significance compares 120 observations of X with a Poisson model whose mean was estimated from the same data. After pooling there are four classes, the test statistic is $\chi^{2} = 0.2153$ and the critical value is $\chi^{2}_{0.05}(2) = 5.991$.

What is the conclusion of the test?

An agriculture class applied three fertilisers X, Y and Z to 75 beds of beans and classified the yield per bed as high, medium or low.

A chi-squared test of association is to be carried out. Calculate the expected frequency for a high yield with fertiliser Z.

An agriculture class applied three fertilisers X, Y and Z to 75 beds of beans and classified the yield per bed as high, medium or low.

A chi-squared test of association is to be carried out. Calculate the expected frequency for a medium yield with fertiliser Y.

An agriculture class applied three fertilisers X, Y and Z to 75 beds of beans and classified the yield per bed as high, medium or low. The observed frequencies are

and the expected frequencies under no association are

Calculate the value of the test statistic $\chi^{2}$.

A chi-squared test of association is carried out at the 1% level of significance on a table with 3 rows and 3 columns.

Write down the critical value $\chi^{2}_{0.01}$ for this test.

A chi-squared test of association between type of fertiliser and yield is carried out at the 1% level of significance on a 3 by 3 table. The test statistic is $\chi^{2} = 7.810$ and the critical value is $\chi^{2}_{0.01}(4) = 13.277$.

What is the conclusion of the test?

The random variable R is normally distributed with $R \sim N(54, 36)$.

Find the value of r such that $P(R \le r) = 0.484$.

The random variable S is normally distributed with $S \sim N(48, 25)$.

Find the value of s such that $P(S \ge s) = 0.484$.

The random variables R and S are independent and normally distributed, with $R \sim N(54, 36)$ and $S \sim N(48, 25)$.

What is the distribution of $R - S$?

The random variables R and S are independent and normally distributed, with $R \sim N(54, 36)$ and $S \sim N(48, 25)$.

Find $P(R \ge S)$.

The random variable R is normally distributed with $R \sim N(54, 36)$. Six independent observations of R are taken.

Find the probability that the sum of the six observations is less than 300.

The marks x obtained by a random sample of n students in a test are summarised by $\sum(x - 25) = 144$, and the sample mean is $\bar{x} = 28.6$.

Find the value of n.

The marks x obtained by a random sample of 40 students in a test have mean $\bar{x} = 28.6$.

Find $\sum x$.

For a sample of 40 marks with $\sum x = 1144$, the coded sum of squares is $\sum(x - 25)^{2} = 3650$.

Find $\sum x^{2}$.

A random sample of 40 marks has $\sum x = 1144$ and $\sum x^{2} = 35\,850$.

Calculate the unbiased estimate of the population variance.

A random sample of 40 marks has mean 28.6 and unbiased estimate of the population variance 80.3.

Which of these is the 99% confidence interval for the population mean?

A random sample of 40 marks from a normal population has mean 28.6 and unbiased estimate of the population variance 80.3.

The hypothesis $\mu = 30$ is tested against the alternative $\mu < 30$. Calculate the value of the test statistic z.

A one-tailed test at the 5% level of significance of $\mu = 30$ against $\mu < 30$ gives a test statistic of $z = -0.988$. The critical value is $z = -1.645$.

What is the conclusion of the test?

Ten candidates obtained the following marks, where X is the paper 1 mark and Y is the paper 2 mark.

Find the mean paper 1 mark, $\bar{x}$.

Ten candidates obtained the following marks, where X is the paper 1 mark and Y is the paper 2 mark.

For these data $\sum x = 810$, $\sum y = 700$, $\sum x^{2} = 66\,984$ and $\sum xy = 58\,103$.

Find the gradient m of the regression line of Y on X, in the form $y = mx + c$.

For ten candidates, the paper 1 marks have mean $\bar{x} = 81$ and the paper 2 marks have mean $\bar{y} = 70$. The regression line of Y on X has gradient $m = 1.021$.

Find the intercept c in $y = mx + c$.

For ten candidates the regression line of the paper 2 mark Y on the paper 1 mark X is $y = 1.021x - 12.7$.

Estimate the paper 2 mark for a candidate who has a paper 1 mark of 75, giving your answer to the nearest whole mark.

For ten candidates' paper 1 marks X and paper 2 marks Y,

$n\sum xy - \sum x \sum y = 14\,030$, $n\sum x^{2} - \left(\sum x\right)^{2} = 13\,740$ and $n\sum y^{2} - \left(\sum y\right)^{2} = 15\,940$.

Calculate the product moment correlation coefficient.

For ten candidates, the product moment correlation coefficient between the paper 1 mark and the paper 2 mark is $r = 0.948$.

Which comment on this value is correct?