Edexcel S3 (Statistics 3) 2006 January

1. A school has 15 classes and a sixth form. In each class there are 30 students. In the sixth form there are 150 students. There are equal numbers of boys and girls in each class. There are equal numbers of boys and girls in the sixth form. The head teacher wishes to obtain the opinions of the students about school uniforms.

Explain how the head teacher would take a stratified sample of size 40.
(7)

2. A workshop makes two types of electrical resistor. The resistance, \(X\) ohms, of resistors of Type A is such that \(X \sim \mathrm {~N} ( 20,4 )\).
The resistance, \(Y\) ohms, of resistors of Type B is such that \(Y \sim \mathrm {~N} ( 10,0.84 )\).
When a resistor of each type is connected into a circuit, the resistance \(R\) ohms of the circuit is given by \(R = X + Y\) where \(X\) and \(Y\) are independent. Find

1. \(\mathrm { E } ( R )\),
2. \(\operatorname { Var } ( R )\),
3. \(\mathrm { P } ( 28.9 < R < 32.64 )\)
   (6)

3. The drying times of paint can be assumed to be normally distributed. A paint manufacturer paints 10 test areas with a new paint. The following drying times, to the nearest minute, were recorded. $$82 , \quad 98 , \quad 140 , \quad 110 , \quad 90 , \quad 125 , \quad 150 , \quad 130 , \quad 70 , \quad 110 .$$

1. Calculate unbiased estimates for the mean and the variance of the population of drying times of this paint. Given that the population standard deviation is 25 ,
2. find a 95\% confidence interval for the mean drying time of this paint. Fifteen similar sets of tests are done and the \(95 \%\) confidence interval is determined for each set.
3. Estimate the expected number of these 15 intervals that will enclose the true value of the population mean \(\mu\).

4. People over the age of 65 are offered an annual flu injection. A health official took a random sample from a list of patients who were over 65 . She recorded their gender and whether or not the offer of an annual flu injection was accepted or rejected. The results are summarised below. Using a \(5 \%\) significance level, test whether or not there is an association between gender and acceptance or rejection of an annual flu injection. State your hypotheses clearly.

5. Upon entering a school, a random sample of eight girls and an independent random sample of eighty boys were given the same examination in mathematics. The girls and boys were then taught in separate classes. After one year, they were all given another common examination in mathematics. The means and standard deviations of the boys’ and the girls’ marks are shown in the table. You may assume that the test results are normally distributed.

1. Test, at the \(5 \%\) level of significance, whether or not the difference between the means of the boys’ and girls’ results was significant when they entered school.
2. Test, at the \(5 \%\) level of significance, whether or not the mean mark of the boys is significantly less than the mean mark of the girls in the 'After 1 year' examination.
3. Interpret the results found in part (a) and part (b).

6. An area of grass was sampled by placing a \(1 \mathrm {~m} \times 1 \mathrm {~m}\) square randomly in 100 places. The numbers of daisies in each of the squares were counted. It was decided that the resulting data could be modelled by a Poisson distribution with mean 2. The expected frequencies were calculated using the model. The following table shows the observed and expected frequencies.

1. Find values for \(r , s\) and \(t\).
2. Using a \(5 \%\) significance level, test whether or not this Poisson model is suitable. State your hypotheses clearly. An alternative test might have been to estimate the population mean by using the data given.
3. Explain how this would have affected the test.
   (2)

7. The numbers of deaths from pneumoconiosis and lung cancer in a developing country are given in the table. The correlation between the number of deaths in the different age groups for each disease is to be investigated.

1. Give one reason why Spearman's rank correlation coefficient should be used.
2. Calculate Spearman's rank correlation coefficient for these data.
3. Use a suitable test, at the \(5 \%\) significance level, to interpret your result. State your hypotheses clearly.
   (5)