Edexcel S3 (Statistics 3) 2021 January

1. A journalist is going to interview a sample of 10 players from the 60 players in a local football club. The journalist uses the random numbers on page 27 of the formula booklet and starts at the top of the 10th column, where the first number is 96

The journalist worked down the 10th column to select 10 numbers. The first 3 numbers selected were: 33, 15 and 23

1. Find the other 7 numbers to complete the sample of ten. There are 24 girls and 36 boys who play football for the club.
   The journalist labels the girls from 1 to 24 and the boys from 25 to 60
2. Show how the journalist can use her 10 random numbers to select a stratified sample of 10 players from the club to interview. The club provided the journalist with a list of the players in ascending order of ages, numbered 1 to 60. The journalist uses the 10 random numbers to select a simple random sample of the players.
3. State, giving a reason, a group of players who may not be represented in this sample.

2. A teacher believes that those of her students with strong mathematical ability may also have enhanced short-term memory. She shows a random sample of 11 students a tray of different objects for eight seconds and then asks them to write down as many of the objects as they can remember. The results, along with their percentage score in a recent mathematics test, are given in the table below.

1. Calculate Spearman's rank correlation coefficient for these data. Show your working clearly.
2. Stating your hypotheses clearly, carry out a suitable test to assess the teacher's belief. Use a \(5 \%\) level of significance and state your critical value. The teacher shows these results to her class and argues that spending more time trying to improve their short-term memory would improve their mathematical ability.
3. Explain whether or not you agree with the teacher's argument.

3. The students in a group of schools can choose a club to join. There are 4 clubs available: Music, Art, Sports and Computers. The director collected information about the number of students in each club, using a random sample of 88 students from across the schools. The results are given in Table 1 below. \begin{table}[h] \end{table} The director uses a chi-squared test to determine whether or not the students are uniformly distributed across the 4 clubs.

1. 1. Find the expected frequencies he should use. Given that the test statistic he calculated was 6.09 (to 3 significant figures)
   2. use a \(5 \%\) level of significance to complete the test. You should state the degrees of freedom and the critical value used. The director wishes to examine the situation in more detail and takes a second random sample of 88 students. The director assumes that within each school, students select their clubs independently. The students come from 3 schools and the distribution of the students from each school amongst the clubs is given in Table 2 below. \begin{table}[h]

      School Club	Music	Art	Sports	Computers
      School \(\boldsymbol { A }\)	3	10	9	8
      School \(\boldsymbol { B }\)	1	11	13	5
      School \(\boldsymbol { C }\)	11	6	7	4

      \captionsetup{labelformat=empty} \caption{Table 2}
      \end{table} The director wishes to test for an association between a student's school and the club they choose.
2. State hypotheses suitable for such a test.
3. Calculate the expected frequency for School \(C\) and the Computers club. The director calculates the test statistic to be 7.29 (to 3 significant figures) with 4 degrees of freedom.
4. Explain clearly why his test has 4 degrees of freedom.
5. Complete the test using a \(5 \%\) level of significance and stating clearly your critical value.

4. The scores in a national test of seven-year-old children are normally distributed with a standard deviation of 18
A random sample of 25 seven-year-old children from town \(A\) had a mean score of 52.4

1. Calculate a 98\% confidence interval for the mean score of the seven-year-old children from town \(A\).
   (4) An independent random sample of 30 seven-year-old children from town \(B\) had a mean score of 57.8
   A local newspaper claimed that the mean score of seven-year-old children from town \(B\) was greater than the mean score of seven-year-old children from town \(A\).
2. Stating your hypotheses clearly, use a \(5 \%\) significance level to test the newspaper's claim. You should show your working clearly. The mean score for the national test of seven-year-old children is \(\mu\). Considering the two samples of seven-year-old children separately, at the \(5 \%\) level of significance, there is insufficient evidence that the mean score for town \(A\) is less than \(\mu\), and insufficient evidence that the mean score for town \(B\) is less than \(\mu\).
3. Find the largest possible value for \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-11_2255_50_314_34}
   

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5. Chrystal is studying the lengths of pine cones that have fallen from a tree. She believes that the length, \(X \mathrm {~cm}\), of the pine cones can be modelled by a normal distribution with mean 6 cm and standard deviation 0.75 cm . She collects a random sample of 80 pine cones and their lengths are recorded in the table below.

1. Stating your hypotheses clearly and using a \(10 \%\) level of significance, test Chrystal's belief. Show your working clearly and state the expected frequencies, the test statistic and the critical value used.
   (10) Chrystal's friend David asked for more information about the lengths of the 80 pine cones. Chrystal told him that $$\sum x = 464 \quad \text { and } \quad \sum x ^ { 2 } = 2722.59$$
2. Calculate unbiased estimates of the mean and variance of the lengths of the pine cones. David used the calculations from part (b) to test whether or not the lengths of the pine cones are normally distributed using Chrystal's sample. His test statistic was 3.50 (to 3 significant figures) and he did not pool any classes.
3. Using a \(10 \%\) level of significance, complete David's test stating the critical value and the degrees of freedom used.
4. Estimate, to 2 significant figures, the proportion of pine cones from the tree that are longer than 7 cm . \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-15_2255_50_314_34}

6. A potter makes decorative tiles in two colours, red and yellow. The length, \(R \mathrm {~cm}\), of the red tiles has a normal distribution with mean 15 cm and standard deviation 1.5 cm . The length, \(Y \mathrm {~cm}\), of the yellow tiles has the normal distribution \(\mathrm { N } \left( 12,0.8 ^ { 2 } \right)\). The random variables \(R\) and \(Y\) are independent. A red tile and a yellow tile are chosen at random.

1. Find the probability that the yellow tile is longer than the red tile. Taruni buys 3 red tiles and 1 yellow tile.
2. Find the probability that the total length of the 3 red tiles is less than 4 times the length of the yellow tile. Stefan defines the random variable \(X = a R + b Y\), where \(a\) and \(b\) are constants. He wants to use values of \(a\) and \(b\) such that \(X\) has a mean of 780 and minimum variance.
3. Find the value of \(a\) and the value of \(b\) that Stefan should use.
   \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-19_2255_50_314_34}