Edexcel S3 (Statistics 3) Specimen

1. A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.

Test, at the \(5 \%\) level of significance, the managing director's claim. State your hypotheses clearly.

2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.

1. Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
2. find the probability that Philip beats James in the race by more than 2 minutes.

3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .

1. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
2. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
3. find a 98\% confidence interval for \(w\).

1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at seven positions. The results are shown in the table below.

1. Calculate Spearman's rank correlation coefficient between \(b\) and \(s\).
2. Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a \(1 \%\) level of significance.

5. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below. Test, at the \(5 \%\) level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly.
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6. A total of 228 items are collected from an archaeological site. The distance from the centre of the site is recorded for each item. The results are summarised in the table below. Test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a continuous uniform distribution. State your hypotheses clearly.

1. A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full-time staff and 4000 part-time staff.
   1. Describe how a stratified sample of 200 staff could be taken.
   2. Explain an advantage of using a stratified sample rather than a simple random sample.
   A random sample of 80 full-time staff and an independent random sample of 80 part-time staff were given a test of policy awareness. The results are summarised in the table below.

   	Mean score \(( \bar { x } )\)	Variance of scores \(\left( s ^ { 2 } \right)\)
   Full-time staff	52	21
   Part-time staff	50	19

2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean policy awareness scores for full-time and part-time staff are different.
3. Explain the significance of the Central Limit Theorem to the test in part (c).
4. State an assumption you have made in carrying out the test in part (c). After all the staff had completed a training course the 80 full-time staff and the 80 part-time staff were given another test of policy awareness. The value of the test statistic \(z\) was 2.53
5. Comment on the awareness of company policy for the full-time and part-time staff in light of this result. Use a \(1 \%\) level of significance.
6. Interpret your answers to part (c) and part (f).