Edexcel S3 (Statistics 3) 2020 October

1. The random variable \(X\) has the discrete uniform distribution

$$\mathrm { P } ( X = x ) = \frac { 1 } { \alpha } \quad \text { for } x = 1,2 , \ldots , \alpha$$ The mean of a random sample of size \(n\), taken from this distribution, is denoted by \(\bar { X }\)

1. Show that \(2 \bar { X }\) is a biased estimator of \(\alpha\) A random sample of 6 observations of \(X\) is taken and the results are given below. $$\begin{array} { l l l l l l } 8 & 7 & 3 & 7 & 2 & 9 \end{array}$$
2. Use the sample mean to estimate the value of \(\alpha\)
   

2. A university awards its graduates a degree in one of three categories, Distinction, Merit or Pass. Table 1 shows information about a random sample of 200 graduates from three departments, Arts, Humanities and Sciences. \begin{table}[h] \end{table} Xiu wants to carry out a test of independence between the category of degree and the department. Table 2 shows some of the values of \(\frac { ( O - E ) ^ { 2 } } { E }\) for this test. \begin{table}[h] \end{table}

1. Complete Table 2
2. Hence, complete Xiu’s hypothesis test using a \(5 \%\) level of significance. You should state the hypotheses, the degrees of freedom and the critical value used for this test.

3. Each of 7 athletes competed in a 200 metre race and a 400 metre race. The table shows the time, in seconds, taken by each athlete to complete the 200 metre race. The finishing order in the 400 metre race is shown below, with athlete \(A\) finishing in the fastest time.
\(\begin{array} { l l l l l l l } A & B & G & C & D & F & E \end{array}\)

1. Calculate the Spearman's rank correlation coefficient between the finishing order in the 200 metre race and the finishing order in the 400 metre race.
2. Stating your hypotheses clearly, test whether or not there is evidence of a positive correlation between the finishing order in the 200 metre race and the finishing order in the 400 metre race. Use a \(5 \%\) level of significance. The 7 athletes also competed in a long jump competition with the following results.

   Athlete	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
   Long jump (metres)	6.50	6.47	6.12	6.12	6.48	6.38	6.47

   Yuliya wants to calculate the Spearman's rank correlation coefficient between the finishing order in the 200 metre race and the finishing order in the long jump for these athletes.
3. Without carrying out any further calculations, explain how Yuliya should do this.

4. Luka wants to carry out a survey of students at his school. He obtains a list of all 280 students.

1. Explain how he can use this list to select a systematic sample of 40 students. Luka is trying to make his own random number table. He generates 400 digits to put in his table. Figure 1 shows the frequency of each digit in his table. \begin{table}[h]

   Digit generated	0	1	2	3	4	5	6	7	8	9
   Frequency	36	42	33	41	44	43	48	38	32	43

   \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{table} A test is carried out at the \(10 \%\) level of significance to see if the digits Luka generates follow a uniform distribution. For this test \(\sum \frac { ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } } = 5.9\)
2. Determine the conclusion of this test.
   (3) The digits generated by Luka are taken two at a time to form two-digit numbers. Figure 2 shows the frequency of two-digit numbers in his table. \begin{table}[h]

   Two-digit numbers generated	\(00 - 19\)	\(20 - 39\)	\(40 - 59\)	\(60 - 79\)	\(80 - 99\)
   Frequency	31	49	30	42	48

   \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{table}
3. Test, at the \(10 \%\) level of significance, whether the two-digit numbers generated by Luka follow a uniform distribution. You should state the hypotheses, the degrees of freedom and the critical value used for this test. There are 70 students in Year 12 at his school.
4. State, giving a reason, the advice you would give to Luka regarding the use of his table of numbers for generating a simple random sample of 10 of the Year 12 students.

5. A greengrocer is investigating the weights of two types of orange, type \(A\) and type \(B\). She believes that on average type \(A\) oranges weigh greater than 5 grams more than type \(B\) oranges. She collects a random sample of 40 type \(A\) oranges and 32 type \(B\) oranges and records the weight, \(x\) grams, of each orange. The table shows a summary of her data.

1. Calculate unbiased estimates for the variance of the weights of the population of type \(A\) oranges and the variance of the weights of the population of type \(B\) oranges.
2. Test, at the \(5 \%\) level of significance, the greengrocer's belief. You should state the hypotheses and the critical value used for this test.
3. Explain how you have used the fact that the sample sizes are large in your answer to part (b).

6. The number of toasters sold by a shop each week may be modelled by a Poisson distribution with mean 4 A random sample of 35 weeks is taken and the mean number of toasters sold per week is found.

1. Write down the approximate distribution for the mean number of toasters sold per week from a random sample of 35 weeks. The number of kettles sold by the shop each week may be modelled by a Poisson distribution with mean \(\lambda\) A random sample of 40 weeks is taken and the mean number of kettles sold per week is found. The width of the \(99 \%\) confidence interval for \(\lambda\) is 2.6
2. Find an estimate for \(\lambda\) A second, independent random sample of 40 weeks is taken and a second \(99 \%\) confidence interval for \(\lambda\) is found.
3. Find the probability that only one of these two confidence intervals contains \(\lambda\)
   

7. A company makes cricket balls and tennis balls. The weights of cricket balls, \(C\) grams, follow a normal distribution $$C \sim \mathrm {~N} \left( 160,1.25 ^ { 2 } \right)$$ Three cricket balls are selected at random.

1. Find the probability that their total weight is more than 475.8 grams. The weights of tennis balls, \(T\) grams, follow a normal distribution $$T \sim \mathrm {~N} \left( 60,2 ^ { 2 } \right)$$ Five tennis balls and two cricket balls are selected at random.
2. Find the probability that the total weight of the five tennis balls and the two cricket balls is more than 625 grams. A random sample of \(n\) tennis balls \(T _ { 1 } , T _ { 2 } , \ldots , T _ { n }\) is taken.
   The random variable \(Y = ( n - 1 ) T _ { 1 } - \sum _ { r = 2 } ^ { n } T _ { r }\)
   Given that \(\mathrm { P } ( Y > 40 ) = 0.0838\) correct to 4 decimal places,
3. find \(n\).
   

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   Q7

   
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