OCR S3 (Statistics 3) 2015 June

1 A laminate consists of 4 layers of material \(C\) and 3 layers of material \(D\). The thickness of a layer of material \(C\) has a normal distribution with mean 1 mm and standard deviation 0.1 mm , and the thickness of a layer of material \(D\) has a normal distribution with mean 8 mm and standard deviation 0.2 mm . The layers are independent of one another.

1. Find the mean and variance of the total thickness of the laminate.
2. What total thickness is exceeded by \(1 \%\) of the laminates?

2 In a poll of people aged 18-21, 46 out of 200 randomly chosen university students agreed with a proposition. 51 out of 300 randomly chosen others who were not university students agreed with it. Test, at the \(5 \%\) significance level, whether the proportion of university students who agree with the proposition differs from the proportion of those who are not university students.

3 A tutor gave an assessment to 6 randomly chosen new eleven-year-old students. After each student had received 30 hours of tuition, they were given a second assessment. The scores are shown in the table.

1. Show that, at the \(5 \%\) significance level, there is insufficient evidence that students' scores are higher, on average, after tuition than before tuition. State a necessary assumption.
2. Disappointed by this result, the tutor looked again at the first assessment. She discovered that the first assessment was too easy, in fact being a test for ten-year-olds, not eleven-year-olds. She decided to reduce each score for the first assessment by a constant integer \(k\). Find the least value of \(k\) for which there is evidence at the \(5 \%\) significance level that the students' scores have, on average, improved.

4 A set of bathroom scales is known to operate with an error which is normally distributed. One morning a man weighs himself 4 times. The 4 values for his mass, in kg , which can be considered to be a random sample are as follows. $$\begin{array} { l l l l } 62.6 & 62.8 & 62.1 & 62.5 \end{array}$$

1. Find a \(95 \%\) confidence interval for his mass. Give the end-points of the interval correct to 3 decimal places.
2. Based on these results, a \(y \%\) confidence interval has width 0.482 . Find \(y\).

5 Two guesthouses, the Albion and the Blighty, have 8 and 6 rooms respectively. The demand for rooms at the Albion has a Poisson distribution with mean 6.5 and the demand for rooms at the Blighty has an independent Poisson distribution with mean 5.5. The owners have agreed that if their guesthouse is full, they will re-direct guests to the other.

1. Find the probability that, on any particular night, the two guesthouses together do not have enough rooms to meet demand.
2. The Albion charges \(\pounds 60\) per room per night, and the Blighty \(\pounds 80\). Find the probability, that on a particular night, the total income of the two guesthouses is exactly \(\pounds 400\).
3. If \(A\) is the number of rooms demanded at the Albion each night, and \(B\) the number of rooms demanded at the Blighty each night, find the mean and variance of the variable \(C = 60 A + 80 B\). State whether \(C\) has a Poisson distribution, giving a reason for your answer.

6 In each of 38 randomly selected weeks of the English Premier Football League there were 10 matches. Table 1 summarises the number of home wins in 10 matches, \(X\), and the corresponding number of weeks. \begin{table}[h] \end{table} A researcher investigates whether \(X\) can be modelled by the distribution \(\mathrm { B } ( 10 , p )\). He calculates the expected frequencies using a value of \(p\) obtained from the sample mean.

1. Show that \(p = 0.45\). Table 2 shows the observed and expected number of weeks. \begin{table}[h]

   Number of home wins	0	1	2	3	4	5	6	7	8	9	10	Totals
   Observed number of weeks	0	1	2	8	8	9	7	1	2	0	0	38
   Expected number of weeks	0.096	0.788	2.899	6.326	9.058	8.893	6.064	2.835	0.870	0.158	0.013	38

   \captionsetup{labelformat=empty} \caption{Table 2
2. Show how the value of 2.835 for 7 home wins is obtained.}