Edexcel S4 (Statistics 4) 2013 June

1. (a) Find the value of the constant \(a\) such that

$$\mathrm { P } \left( 1.690 < \chi _ { 7 } ^ { 2 } < a \right) = 0.95$$ The random variable \(Y\) follows an \(F\)-distribution with 6 and 4 degrees of freedom.
(b) (i) Find the upper \(1 \%\) critical value for \(Y\).
(ii) Find the lower \(1 \%\) critical value for \(Y\).

2. The time, \(t\) hours, that a typist can sit before incurring back pain is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 30 typists gave unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) as shown below. $$\hat { \mu } = 2.5 \quad s ^ { 2 } = 0.36$$

1. Find a 95\% confidence interval for \(\sigma ^ { 2 }\)
2. State with a reason whether or not the confidence interval supports the assertion that \(\sigma ^ { 2 } = 0.495\)

3. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2 . The firm believes that the appointment of a new salesman will increase the number of houses sold. The firm tests its belief by recording the number of houses sold, \(x\), in the week following the appointment. The firm sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 2\) and \(\mathrm { H } _ { 1 } : \lambda > 2\), where \(\lambda\) is the mean number of houses sold per week, and rejects the null hypothesis if \(x \geqslant 3\)

1. Find the size of the test.
2. Show that the power function for this test is $$1 - \frac { 1 } { 2 } e ^ { - \lambda } \left( 2 + 2 \lambda + \lambda ^ { 2 } \right)$$ The table below gives the values of the power function to 2 decimal places. \begin{table}[h]

   \(\lambda\)	2.5	3.0	3.5	4.0	5.0	7.0
   Power	0.46	\(r\)	0.68	\(s\)	0.88	0.97

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
3. Calculate the values of \(r\) and \(s\).
4. Draw a graph of the power function on the graph paper provided on page 6
5. Find the range of values of \(\lambda\) for which the power of this test is greater than 0.6 For your convenience Table 1 is repeated here.

   \(\lambda\)	2.5	3.0	3.5	4.0	5.0	7.0
   Power	0.46	\(r\)	0.68	\(s\)	0.88	0.97

   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Table 1} \includegraphics[alt={},max width=\textwidth]{4f096806-33da-453f-a4c1-12be20d1a96d-06_2125_1603_614_166}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-07_72_47_2615_1886}

4. A company carries out an investigation into the strengths of rods from two different suppliers, Ardo and Bards. Independent random samples of rods were taken from each supplier and the force, \(x \mathrm { kN }\), needed to break each rod was recorded. The company wrote the results on a piece of paper but unfortunately spilt ink on it so some of the results can not be seen.
The paper with the results on is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-08_435_1522_541_244}

1. 1. Use the data from Ardo to calculate an unbiased estimate, \(s _ { A } ^ { 2 }\), of the variance.
   2. Hence find an unbiased estimate, \(s _ { B } ^ { 2 }\), of the variance for the sample of 9 values from Bards.
2. Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not there is a difference in variability of strength between the rods from Ardo and the rods from Bards.
   (You may assume the two samples come from independent normal distributions.)
3. Use a \(5 \%\) level of significance to test whether the mean strength of rods from Bards is more than 0.9 kN greater than the mean strength of rods from Ardo.
   (6)

1. Students studying for their Mathematics GCSE are assessed by two examination papers. A teacher believes that on average the score on paper I is more than 1 mark higher than the score on paper II. To test this belief the scores of 8 randomly selected students are recorded. The results are given in the table below.

Assuming that the scores are normally distributed and stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence to support the teacher's belief.

1. A machine fills bottles with water. The amount of water in each bottle is normally distributed. To check the machine is working properly, a random sample of 12 bottles is selected and the amount of water, in ml, in each bottle is recorded. Unbiased estimates for the mean and variance are

$$\hat { \mu } = 502 \quad s ^ { 2 } = 5.6$$ Stating your hypotheses clearly, test at the 1\% level of significance

1. whether or not the mean amount of water in a bottle is more than 500 ml ,
2. whether or not the standard deviation of the amount of water in a bottle is less than 3 ml .

7. A machine produces bricks. The lengths, \(x \mathrm {~mm}\), of the bricks are distributed \(\mathrm { N } \left( \mu , 2 ^ { 2 } \right)\). At the start of each week a random sample of \(n\) bricks is taken to check the machine is working correctly.
A test is then carried out at the \(1 \%\) level of significance with $$\mathrm { H } _ { 0 } : \mu = 202 \text { and } \mathrm { H } _ { 1 } : \mu < 202$$

1. Find, in terms of \(n\), the critical region of the test. The probability of a type II error, when \(\mu = 200\), is less than 0.05
2. Find the minimum value of \(n\).

8. A random sample \(W _ { 1 } , W _ { 2 } \ldots , W _ { n }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\)

1. Write down \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } \right)\) and show that \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } \right) = n \left( \sigma ^ { 2 } + \mu ^ { 2 } \right)\) An estimator for \(\mu\) is $$\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i }$$
2. Show that \(\bar { X }\) is a consistent estimator for \(\mu\). An estimator of \(\sigma ^ { 2 }\) is $$U = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } - \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } \right) ^ { 2 }$$
3. Find the bias of \(U\).
4. Write down an unbiased estimator of \(\sigma ^ { 2 }\) in the form \(k U\), where \(k\) is in terms of \(n\).