Edexcel S3 (Statistics 3) 2013 June

1. A doctor takes a random sample of 100 patients and measures their intake of saturated fats in their food and the level of cholesterol in their blood. The results are summarised in the table below.

Using a \(5 \%\) level of significance, test whether or not there is an association between cholesterol level and intake of saturated fats. State your hypotheses and show your working clearly.

2. The table below shows the number of students per member of staff and the student satisfaction scores for 7 universities.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence of a correlation between the number of students per member of staff and the student satisfaction score.

3. A college manager wants to survey students' opinions of enrichment activities. She decides to survey the students on the courses summarised in the table below. Each student takes only one course.
The manager has access to the college's information system that holds full details of each of the enrolled students including name, address, telephone number and their course of study. She wants to compare the opinions of students on each course and has a generous budget to pay for the cost of the survey.

1. Give one advantage and one disadvantage of carrying out this survey using
   1. quota sampling,
   2. stratified sampling. The manager decides to take a stratified sample of 100 students.
2. Calculate the number of students to be sampled from each course.
3. Describe how to choose students for the stratified sample.

4. Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, \(x\) minutes, are summarised in Table 1. \begin{table}[h] \end{table}

1. Show that an estimate of \(\bar { x } = 5.49\) and an estimate of \(s _ { x } ^ { 2 } = 6.88\) The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
   Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]

   Waiting Time	\(x < 3\)	\(3 - 5\)	\(5 - 6\)	\(6 - 8\)	\(x > 8\)
   Expected Frequency	8.56	12.73	7.56	\(a\)	\(b\)

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Find the value of \(a\) and the value of \(b\).
3. Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.

1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .

One large and 3 small bottles of Blumen are chosen at random.

1. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
2. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.

6. Fruit-n-Veg4U Market Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is \(0.5 \mathrm {~kg} ^ { 2 }\) and the variance of the yield for the new variety of plant is \(0.75 \mathrm {~kg} ^ { 2 }\). A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .

1. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
2. Explain the relevance of the Central Limit Theorem to the test in part (a).

1. Lambs are born in a shed on Mill Farm. The birth weights, \(x \mathrm {~kg}\), of a random sample of 8 newborn lambs are given below.

$$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$

1. Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
2. Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of Mill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
3. Calculate a \(95 \%\) confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.