Pooled variance estimate calculation

6 A company with a large fleet of cars compared two types of tyres, \(A\) and \(B\). They measured the stopping distances of cars when travelling at a fixed speed on a dry road. They selected 20 cars at random from the fleet and divided them randomly into two groups of 10 , one group being fitted with tyres of type \(A\) and the other group with tyres of type \(B\). One of the cars fitted with tyres of type \(A\) broke down so these tyres were tested on only 9 cars. The stopping distances, \(x\) metres, for the two samples are summarised by $$n _ { A } = 9 , \quad \bar { x } _ { A } = 17.30 , \quad s _ { A } ^ { 2 } = 0.7400 , \quad n _ { B } = 10 , \quad \bar { x } _ { B } = 14.74 , \quad s _ { B } ^ { 2 } = 0.8160 ,$$ where \(s _ { A } ^ { 2 }\) and \(s _ { B } ^ { 2 }\) are unbiased estimates of the two population variances.
It is given that the two populations have the same variance.

1. Show that an unbiased estimate of this variance is 0.780 , correct to 3 decimal places. The population mean stopping distances for cars with tyres of types \(A\) and \(B\) are denoted by \(\mu _ { A }\) metres and \(\mu _ { B }\) metres respectively.
2. Stating any further assumption you need to make, calculate a \(98 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\). The manufacturers of Type \(B\) tyres assert that \(\mu _ { B } < \mu _ { A } - 2\).
3. Carry out a significance test of this assertion at the \(5 \%\) significance level. \section*{[Question 7 is printed overleaf.]}

6 The independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of 5 observations of \(X\) and \(n\) observations of \(Y\) are made and the results are summarised by $$\Sigma x = 5.5 , \quad \Sigma x ^ { 2 } = 15.05 , \quad \Sigma y = 8.0 , \quad \Sigma y ^ { 2 } = 36.4$$ Given that the pooled estimate of \(\sigma ^ { 2 }\) is 3 , find the value of \(n\).

1. As part of their research two sports science students, Ali and Bea, select a random sample of 10 adult male swimmers and a random sample of 13 adult male athletes from local sports clubs. They measure the arm span, \(x \mathrm {~cm}\), of each person selected.
   The data are summarised in the table below

The students know that the arm spans of adult male swimmers and of adult male athletes may each be assumed to be normally distributed.
They decide to share out the data analysis, with Ali investigating the means of the two distributions and Bea investigating the variances of the two distributions. Ali assumes that the variances of the two distributions are equal. She calculates the pooled estimate of variance, \(s _ { p } { } ^ { 2 }\)

1. Show that \(s _ { p } { } ^ { 2 } = 112.6\) to 1 decimal place. Ali claims that there is no difference in the mean arm spans of adult male swimmers and of adult male athletes.
2. Stating your hypotheses clearly, test this claim at the \(10 \%\) level of significance.
   (5) Bea believes that the variances of the arm spans of adult male swimmers and adult male athletes are not equal.
3. Show that, at the \(10 \%\) level of significance, the data support Bea's belief. State your hypotheses and show your working clearly. Ali and Bea combine their work and present their results to their tutor, Clive.
4. Explain why Clive is not happy with their research and state, with a reason, which of the tests in parts (b) and (c) is not valid.

3 The fuel economy of two similar cars produced by manufacturers \(A\) and \(B\) was compared. A random sample of 15 cars was selected from manufacturer \(A\) and a random sample of 10 cars was selected from manufacturer \(B\). All the selected cars were driven over the same distance and the petrol consumption in miles per gallon (mpg) was calculated for each car. The results, \(x _ { A } \operatorname { mpg }\) and \(x _ { B } \operatorname { mpg }\) for cars from manufacturers \(A\) and \(B\) respectively, are summarised below, where \(\bar { x }\) denotes the sample mean and \(n\) the sample size. $$\begin{array} { l l l } \Sigma x _ { A } = 460.5 & \Sigma \left( x _ { A } - \bar { x } _ { A } \right) ^ { 2 } = 156.88 & n _ { A } = 15 \\ \Sigma x _ { B } = 334 & \Sigma \left( x _ { B } - \bar { x } _ { B } \right) ^ { 2 } = 123.97 & n _ { B } = 10 \end{array}$$

1. (a) Assuming that the populations are normally distributed with a common variance, show that the pooled estimate of this common variance is 12.21 , correct to 4 significant figures. [2]
   (b) Construct a 95\% confidence interval for \(\mu _ { B } - \mu _ { A }\), the difference in the population means for manufacturers \(A\) and \(B\).
2. Comment on a claim that the fuel economy for manufacturer \(B\) 's cars is better than that for manufacturer \(A\) 's cars.
3. A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { \theta } \mathrm { e } ^ { - \frac { x } { \theta } } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ where \(\theta\) is a positive constant. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population with the distribution in part (i). The estimator \(T\) is defined by \(T = k \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }\), where \(k\) is a constant. Find the value of \(k\) such that \(T\) is an unbiased estimator of \(\theta ^ { 2 }\).
5. The discrete random variable \(X\) has distribution \(\operatorname { Geo } ( p )\). Show that the moment generating function of \(X\) is given by \(\mathrm { M } _ { X } ( t ) = \frac { p \mathrm { e } ^ { t } } { 1 - q \mathrm { e } ^ { t } }\), where \(q = 1 - p\).
6. Use the moment generating function to find
   (a) \(\mathrm { E } ( X )\),
   (b) \(\operatorname { Var } ( X )\).
7. An unbiased six-sided die is thrown repeatedly until a five is obtained, and \(Y\) denotes the number of throws up to and including the throw on which the five is obtained. Find \(\mathrm { P } ( | Y - \mu | < \sigma )\), where \(\mu\) and \(\sigma\) are the mean and standard deviation, respectively, of the distribution of \(Y\).
8. The continuous random variable \(X\) has a uniform distribution over the interval \(0 < x < \frac { 1 } { 2 } \pi\). Show that the probability density function of \(Y\), where \(Y = \sin X\), is given by $$\mathrm { f } ( y ) = \begin{cases} \frac { 2 } { \pi \sqrt { 1 - y ^ { 2 } } } & 0 < y < 1 \\ 0 & \text { otherwise. } \end{cases}$$
9. Deduce, using the probability density function, the exact values of
   (a) the median value of \(Y\),
   (b) \(\mathrm { E } ( Y )\).

The independent variables \(X\) and \(Y\) have distributions with the same variance \(\sigma^2\). Random samples of \(N\) observations of \(X\) and \(2N\) observations of \(Y\) are taken, and the results are summarised by $$\Sigma x = 4, \quad \Sigma x^2 = 10, \quad \Sigma y = 8, \quad \Sigma y^2 = 102.$$ These data give a pooled estimate of \(10\) for \(\sigma^2\). Find \(N\). [5]