CAIE S2 (Statistics 2) 2020 March

1 The booklets produced by a certain publisher contain, on average, 1 incorrect letter per 30000 letters, and these errors occur randomly. A randomly chosen booklet from this publisher contains 12500 letters. Use a suitable approximating distribution to find the probability that this booklet contains at least 2 errors.

2 Lengths of a certain species of lizard are known to be normally distributed with standard deviation 3.2 cm . A naturalist measures the lengths of a random sample of 100 lizards of this species and obtains an \(\alpha \%\) confidence interval for the population mean. He finds that the total width of this interval is 1.25 cm . Find \(\alpha\).

3 In the past, the mean time taken by Freda for a particular daily journey was 39.2 minutes. Following the introduction of a one-way system, Freda wishes to test whether the mean time for the journey has decreased. She notes the times, \(t\) minutes, for 40 randomly chosen journeys and summarises the results as follows. $$n = 40 \quad \Sigma t = 1504 \quad \Sigma t ^ { 2 } = 57760$$

1. Calculate unbiased estimates of the population mean and variance of the new journey time.
2. Test, at the \(5 \%\) significance level, whether the population mean time has decreased.

4 The number of accidents on a certain road has a Poisson distribution with mean 0.4 per 50-day period.

1. Find the probability that there will be fewer than 3 accidents during a year (365 days).
2. The probability that there will be no accidents during a period of \(n\) days is greater than 0.95 . Find the largest possible value of \(n\).

5 Bottles of Lanta contain approximately 300 ml of juice. The volume of juice, in millilitres, in a bottle is \(300 + X\), where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4000 } \left( 100 - x ^ { 2 } \right) & - 10 \leqslant x \leqslant 10
0 & \text { otherwise } \end{cases}$$

1. Find the probability that a randomly chosen bottle of Lanta contains more than 305 ml of juice.
2. Given that \(25 \%\) of bottles of Lanta contain more than \(( 300 + p ) \mathrm { ml }\) of juice, show that $$p ^ { 3 } - 300 p + 1000 = 0 .$$
3. Given that \(p = 3.47\), and that \(50 \%\) of bottles of Lanta contain between ( \(300 - q\) ) and ( \(300 + q\) ) ml of juice, find \(q\). Justify your answer.

6 The volumes, in millilitres, of large and small cups of tea are modelled by the distributions \(\mathrm { N } ( 200,30 )\) and \(\mathrm { N } ( 110,20 )\) respectively.

1. Find the probability that the total volume of a randomly chosen large cup of tea and a randomly chosen small cup of tea is less than 300 ml .
2. Find the probability that the volume of a randomly chosen large cup of tea is more than twice the volume of a randomly chosen small cup of tea.

7 A national survey shows that \(95 \%\) of year 12 students use social media. Arvin suspects that the percentage of year 12 students at his college who use social media is less than the national percentage. He chooses a random sample of 20 students at his college and notes the number who use social media. He then carries out a test at the \(2 \%\) significance level.

1. Find the rejection region for the test.
2. Find the probability of a Type I error.
3. Jimmy believes that the true percentage at Arvin's college is \(70 \%\). Assuming that Jimmy is correct, find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.