Edexcel S3 (Statistics 3) 2021 October

1. A machine makes screws with a mean length of 30 mm and a standard deviation of 2.5 mm .

A manager claims that, following some repairs, the machine is now making screws with a mean length of less than 30 mm . The manager takes a random sample of 80 screws and finds that they have a mean length of 29.5 mm . Use a suitable test, at the \(5 \%\) level of significance, to determine whether there is evidence to support the manager's claim. State your hypotheses clearly.

2. Andy has some apple trees. Over many years she has graded each apple from her trees as \(A , B , C , D\) or \(E\) according to the quality of the apple, with \(A\) being the highest quality and \(E\) being the lowest quality. She knows that the proportion of apples in each grade produced by her trees is as follows. Raj advises Andy to add potassium to the soil around her apple trees. Andy believes that adding potassium will not affect the distribution of grades for the quality of the apples. To test her belief Andy adds potassium to the soil around her apple trees. The following year she counts the number of apples in each grade. The number of apples in each grade is shown in the table below. Test Andy's belief using a \(5 \%\) level of significance. Show your working clearly, stating your hypotheses, expected frequencies and degrees of freedom. 2 continued

3. A cafe owner wishes to know whether the price of strawberry jam is related to the taste of the jam. He finds a website that lists the price per 100 grams and a mark for the taste, out of 100, awarded by a judge, for 9 different strawberry jams \(A , B , C , D , E , F , G , H\) and \(I\). He then ranks the marks for taste and the prices. The ranks are shown in the table below.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Test, at the \(5 \%\) level of significance, whether or not there is a relationship between the price and the taste of these strawberry jams. State your hypotheses clearly. A friend suggests that it would be better to use the price per 100 grams, \(c\), and the mark for the taste, \(m\), for each strawberry jam rather than rank them. Given that $$\mathrm { S } _ { c c } = 2.0455 \quad \mathrm {~S} _ { m m } = 243.5556 \quad \mathrm {~S} _ { c m } = 16.4943$$
3. calculate the product moment correlation coefficient between the price and the mark for taste of these strawberry jams, giving your answer correct to 3 decimal places.
4. Use your value of the product moment correlation coefficient to test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the price and the mark for taste of these 9 strawberry jams. State your hypotheses clearly.
5. State which of the tests in parts (b) and (d) is more appropriate for the cafe owner to use. Give a reason for your answer.

1. A local village radio station, LSB, decides to survey adults in its broadcasting area about the programmes it produces.
   \(L S B\) broadcasts to 4 villages \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .
   The number of households in each of the villages is given below.

LSB decides to take a stratified sample of 200 households.

1. Explain how to select the households for this stratified sample.
   (3) One of the questions in the survey related to the age group of each member of the household and whether they listen to \(L S B\). The data received are shown below.

   \multirow{2}{*}{}	Age group
   	18-49	50-69	Older than 69
   Listen to LSB	130	162	65
   Do not listen to LSB	78	98	62

   The data are to be used to determine whether or not there is an association between the age group and whether they listen to \(L S B\).
2. Calculate the expected frequencies for the age group 50-69 that
   1. listen to \(L S B\)
   2. do not listen to \(L S B\)
      (2) Given that for the other 4 classes \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 4.657\) to 3 decimal places,
3. test at the \(5 \%\) level of significance, whether or not there is evidence of an association between age and listening to \(L S B\). Show your working clearly, stating the degrees of freedom and the critical value.

1. Assam produces bags of flour. The stated weight printed on the bags of flour is 3 kg . The weights of the bags of flour are normally distributed with standard deviation 0.015 kg .

Assam weighs a random sample of 9 bags of flour and finds their mean weight is 2.977 kg .

1. Calculate the \(99 \%\) confidence interval for the mean weight of a bag of flour. Give your limits to 3 decimal places. Assam decides to increase the amount of flour put into the bags.
2. Explain why the confidence interval has led Assam to take this action. After the increase a random sample of \(n\) bags of flour is taken. The sample mean weight of these \(n\) bags is 2.995 kg . A \(95 \%\) confidence interval for \(\mu\) gave a lower limit of less than 2.991 kg .
3. Find the maximum value of \(n\).
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6. Amala believes that the resting heart rate is lower in men who exercise regularly compared to men who do not exercise regularly. She measures the resting heart rate, \(h\), of a random sample of 50 men who exercise regularly and a random sample of 40 men who do not exercise regularly. Her results are summarised in the table below.

1. Calculate the value of \(\alpha\) and the value of \(\beta\)
2. Test, at the \(5 \%\) level of significance, whether there is evidence to support Amala's belief. State your hypotheses clearly.
3. Explain the significance of the central limit theorem to the test in part (b).
4. State two assumptions you have made in carrying out the test in part (b).

1. A company produces bricks.

The weight of a brick, \(B \mathrm {~kg}\), is such that \(B \sim \mathrm {~N} \left( 1.96 , \sqrt { 0.003 } ^ { 2 } \right)\)
Two bricks are chosen at random.

1. Find the probability that the difference in weight of the 2 bricks is greater than 0.1 kg A random sample of \(n\) bricks is to be taken.
2. Find the minimum sample size such that the probability of the sample mean being greater than 2 is less than 1\% The bricks are randomly selected and stacked on pallets.
   The weight of an empty pallet, \(E \mathrm {~kg}\), is such that \(E \sim \mathrm {~N} \left( 21.8 , \sqrt { 0.6 } ^ { 2 } \right)\)
   The random variable \(M\) represents the total weight of a pallet stacked with 500 bricks. The random variable \(T\) represents the total weight of a container of cement.
   Given that \(T\) is independent of \(M\) and that \(T \sim \mathrm {~N} \left( 774 , \sqrt { 1.8 } ^ { 2 } \right)\)
3. calculate \(\mathrm { P } ( 4 T > 100 + 3 M )\)