Edexcel S3 (Statistics 3) 2017 June

1. The ages, in years, of a random sample of 8 parrots are shown in the table below.

A parrot breeder does not know the ages of these 8 parrots. She examines each of these 8 parrots and is asked to put them in order of decreasing age. She puts them in the order $$\begin{array} { l l l l l l l l } D & G & H & C & A & B & F & E \end{array}$$

1. Find, to 3 decimal places, Spearman's rank correlation coefficient between the breeder's order and the actual order.
   (5)
2. Use your value of Spearman's rank correlation coefficient to test for evidence of the breeder's ability to order parrots correctly, by their age, after examining them. Use a \(1 \%\) significance level and state your hypotheses clearly.

2. A school uses online report cards to promote both hard work and good behaviour of its pupils. Each card details a pupil's recent achievement and contains exactly one of three inspirational messages \(A , B\) or \(C\), chosen by the pupil's teacher. The headteacher believes that there is an association between the pupil's gender and the inspirational message chosen. He takes a random sample of 225 pupils and examines the card for each pupil. His results are shown in Table 1. \begin{table}[h] \end{table} Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not there is evidence to support the headteacher's belief. Show your working clearly. You should state your expected frequencies correct to 2 decimal places.

3. The manager of a gym claimed that the mean age of its customers is 30 years. A random sample of 75 customers is taken and their ages have a mean of 28.2 years and a standard deviation, \(s\), of 8.5 years.

1. Stating your hypotheses clearly and using a 10\% level of significance, test whether or not the manager's claim is supported by the data.
2. Explain the relevance of the Central Limit Theorem to your calculation in part (a).
3. State an additional assumption needed to carry out the test in part (a).

4. The number of emergency plumbing calls received per day by a local council was recorded over a period of 80 days. The results are summarised in the table below.

1. Show that the mean number of emergency plumbing calls received per day is 3.5 A council officer suggests that a Poisson distribution can be used to model the number of emergency plumbing calls received per day. He uses the mean from the sample above and calculates the expected frequencies shown in the table below.

   \(\boldsymbol { x }\)	0	1	2	3	4	5	6	7	8 or more
   Expected frequency	2.42	8.46	14.80	\(r\)	15.10	10.57	6.17	3.08	\(s\)

2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
3. Test, at the \(5 \%\) level of significance, whether or not the Poisson distribution is a suitable model for the number of emergency plumbing calls received per day. State your hypotheses clearly.

5. A dance studio has 800 dancers of which \begin{displayquote} 452 are beginners
251 are intermediates
97 are professionals

1. Explain in detail how a stratified sample of size 50 could be taken.
2. State an advantage of stratified sampling rather than simple random sampling in this situation. \end{displayquote} Independent random samples of 80 beginners and 60 intermediates are chosen. Each of these dancers is given an assessment score, \(x\), based on the quality of their dancing. The results are summarised in the table below.

   	\(\bar { x }\)	\(s ^ { 2 }\)	\(n\)
   Beginners	31.7	57.3	80
   Intermediates	36.9	38.1	60

   The studio manager believes that the mean score of intermediates is more than 3 points greater than the mean score of beginners.
3. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not these data support the studio manager's belief.

6. A company produces a certain type of mug. The masses of these mugs are normally distributed with mean \(\mu\) and standard deviation 1.2 grams. A random sample of 5 mugs is taken and the mass, in grams, of each mug is measured. The results are given below. \section*{\(\begin{array} { l l l l l } 229.1 & 229.6 & 230.9 & 231.2 & 231.7 \end{array}\)}

1. Find a \(95 \%\) confidence interval for \(\mu\), giving your limits correct to 1 decimal place. Sonia plans to take 20 random samples, each of 5 mugs. A 95\% confidence interval for \(\mu\) is to be determined for each sample.
2. Find the probability that more than 3 of these intervals will not contain \(\mu\).

7. The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} \left( 30,4.5 ^ { 2 } \right) \text { and } Y \sim \mathrm {~N} \left( 20,3.5 ^ { 2 } \right)$$ The random variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent and each has the same distribution as \(X\). The random variables \(Y _ { 1 }\) and \(Y _ { 2 }\) are independent and each has the same distribution as \(Y\). Given that the random variable \(A\) is defined as $$A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 5 }$$

1. find \(\mathrm { P } ( A < 24 )\) The random variable \(W\) is such that \(W \sim \mathrm {~N} \left( \mu , 2.8 ^ { 2 } \right)\) Given that \(\mathrm { P } ( W - X < 4 ) = 0.1\) and that \(W\) and \(X\) are independent,
2. find the value of \(\mu\), giving your answer to 3 significant figures.

8. The random variable \(X\) has a continuous uniform distribution over the interval \([ \alpha + 3,2 \alpha + 9 ]\) where \(\alpha\) is a constant. The mean of a random sample of size \(n\), taken from this distribution, is denoted by \(\bar { X }\)

1. Show that \(\bar { X }\) is a biased estimator of \(\alpha\)
2. Hence find the bias, in terms of \(\alpha\), when \(\bar { X }\) is used as an estimator of \(\alpha\) Given that \(Y = \frac { 2 \bar { X } } { 3 } + k\) is an unbiased estimator of \(\alpha\)
3. find the value of the constant \(k\) A random sample of 8 values of \(X\) is taken and the results are as follows
   4.8
   5.8
   6.5
   7.1
   8.2
   9.5
   9.9
   10.6
4. Use the sample to estimate the maximum value that \(X\) can take.
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