SPS SPS FM Statistics (SPS FM Statistics) 2022 January

1. A local authority official wishes to conduct a survey of households in the borough. He decides to select a stratified sample of 2000 households using Council Tax property bands as the strata. At the time of the survey there are 79368 households in the borough. The table shows the numbers of households in the different tax bands.

a. Calculate the number of households that the official should choose from each stratum in order to obtain his sample of 2000 households so that each stratum is represented proportionally.
b. State one advantage of stratified sampling over simple random sampling.

2. A survey is carried out into the length of time for which customers wait for a response on a telephone helpline. A statistician who is analysing the results of the survey starts by modelling the waiting time, \(X\) minutes, by an exponential distribution with probability density function $$f ( x ) = \left\{ \begin{array} { c c } \lambda e ^ { - \lambda x } & x \geq 0
0 & x < 0 \end{array} \right.$$ The mean waiting time is found to be 5 minutes.
a. State the value of \(\lambda\).
b. Use the model to calculate the probability that a customer has to wait longer that 20 minutes for a response.

3. A shop sells carrots and broccoli. The weights of carrots can be modelled by a normal distribution with mean 130 grams and variance 25 grams \(^ { 2 }\) and the weights of broccoli can be modelled by a normal distribution with mean 400 grams and variance 80 grams \({ } ^ { 2 }\). Find the probability that the weight of six randomly chosen carrots is more than two times the weight of one randomly chosen broccoli.

4. The strength of beams compared against the moisture content of the beam is indicated in the following table. a. Use your calculator to write down the value of the product moment correlation coefficient for these data.
b. Perform a two-tailed test, at the \(5 \%\) level of significance, to investigate whether there is correlation between strength and moisture content.
c. Use your calculator to write down the equation of the regression line of strength on moisture content.
d. Use the regression line to estimate the strength of a beam that has a moisture content of 9.5.

5. In a large population of hens, the weight of a hen is normally distributed with mean \(\mu \mathrm { kg }\) and standard deviation \(\sigma \mathrm { kg }\). A random sample of 100 hens is taken from the population. The mean weight for the sample is denoted \(\bar { X }\).
a. State the distribution of \(\bar { X }\) giving its mean and variance. The sample values are summarised by \(\sum x = 199.8\) and \(\sum x ^ { 2 } = 407.8\) where \(x \mathrm {~kg}\) is the weight of a hen.
b. Find an unbiased estimate for \(\mu\).
c. Find an unbiased estimate for \(\sigma ^ { 2 }\).
d. Find a \(90 \%\) confidence interval for \(\mu\). It is found that \(\sigma = 0.27\). It is decided to test, at the \(1 \%\) level of significance, the null hypothesis \(\mu = 1.95\) against the alternative hypothesis \(\mu > 1.95\).
e. Find the \(p\)-value for the test.
f. Write down the conclusion reached.
g. Explain whether or not the central limit theorem was required in part e.

6. The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematical examinations each year can be modelled by a Poisson distribution with a mean of 3 .
a. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations.
b. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with mean of 7 . Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A-grades in their Mathematics and English examinations.
c. Lowkey School is given a performance rating, \(P = 2 X + 3 Y\), based on the number of A-grades achieved in Mathematics and English. Find: $$\begin{array} { l l } \text { i. } & \mathrm { E } ( P )
\text { ii. } & \operatorname { Var } ( P ) \end{array}$$ d. What assumption did you make in answering part (b)? Did you need this assumption to answer part (c)? Justify your answers.

7. The continuous random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c l } 0 & x < 1
\frac { 4 } { x ^ { 5 } } & x \geq 1 \end{array} \right.$$ a. Find the cumulative distribution function, \(F ( x )\), of \(X\).
b. Find the interquartile range of \(X\).
c. Show that the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), is given by $$g ( y ) = \left\{ \begin{array} { c l } 2 y & 0 < y \leq 1
0 & \text { otherwise } \end{array} \right.$$ d. Find the value of \(a\) for which \(\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = a \mathrm { E } \left( X ^ { 2 } \right)\).