CAIE S2 (Statistics 2) 2022 June

1 The number of characters in emails sent by a particular company is modelled by the distribution \(\mathrm { N } \left( 1250,480 ^ { 2 } \right)\). Find the probability that the mean number of characters in a random sample of 100 emails sent by the company is more than 1300 .

2 Anton believes that \(10 \%\) of students at his college are left-handed. Aliya believes that this is an underestimate. She plans to carry out a hypothesis test of the null hypothesis \(p = 0.1\) against the alternative hypothesis \(p > 0.1\), where \(p\) is the actual proportion of students at the college that are left-handed. She chooses a random sample of 20 students from the college. She will reject the null hypothesis if at least 5 of these students are left-handed.

1. Explain what is meant by a Type I error in this context.
2. Find the probability of a Type I error in the test.
3. Given that the true value of \(p\) is 0.3 , find the probability of a Type II error in the test.

3 Batteries of type \(A\) are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type \(B\), has a shorter mean life than type \(A\) batteries.

1. Give a reason for using a sample rather than the whole population in carrying out this test.
   A random sample of 120 type \(B\) batteries are tested and it is found that their mean life is 147 hours, and an unbiased estimate of the population variance is 225 hours \(^ { 2 }\).
2. Test, at the \(2 \%\) significance level, whether type \(B\) batteries have a shorter mean life than type \(A\) batteries.
3. Calculate a \(94 \%\) confidence interval for the population mean life of type \(B\) batteries.

4 Each box of Seeds \& Raisins contains \(S\) grams of seeds and \(R\) grams of raisins. The weight of a box, when empty, is \(B\) grams. \(S , R\) and \(B\) are independent random variables, where \(S \sim \mathrm {~N} ( 300,45 )\), \(R \sim \mathrm {~N} ( 200,25 )\) and \(\mathrm { B } \sim \mathrm { N } ( 15,4 )\). A full box of Seeds \& Raisins is chosen at random.
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1. Find the probability that the total weight of the box and its contents is more than 500 grams. [5]
2. Find the probability that the weight of seeds in the box is less than 1.4 times the weight of raisins in the box.

5 The number of clients who arrive at an information desk has a Poisson distribution with mean 2.2 per 5-minute period.

1. Find the probability that, in a randomly chosen 15 -minute period, exactly 6 clients arrive at the desk.
2. If more than 4 clients arrive during a 5 -minute period, they cannot all be served. Find the probability that, during a randomly chosen 5 -minute period, not all the clients who arrive at the desk can be served.
3. Use a suitable approximating distribution to find the probability that, during a randomly chosen 1-hour period, fewer than 20 clients arrive at the desk.

6 A random sample of 5 values of a variable \(X\) is given below. $$\begin{array} { l l l l l } 2 & 3 & 3 & 5 & a \end{array}$$

1. Find an expression, in terms of \(a\), for the mean of these values.
   It is given that an unbiased estimate of the population variance of \(X\), using these values, is 4 . It is also given that \(a\) is positive.
2. Find and simplify a quadratic equation in terms of \(a\) and hence find the value of \(a\).

7 The random variables \(X\) and \(W\) have probability density functions f and g defined as follows: $$\begin{gathered} \mathrm { f } ( x ) = \begin{cases} p \left( a ^ { 2 } - x ^ { 2 } \right) & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}
\mathrm { g } ( w ) = \begin{cases} q \left( a ^ { 2 } - w ^ { 2 } \right) & - a \leqslant w \leqslant a
0 & \text { otherwise } \end{cases} \end{gathered}$$ where \(a , p\) and \(q\) are constants.

1. 1. Write down the value of \(\mathrm { P } ( X \geqslant 0 )\).
   2. Write down the value of \(\mathrm { P } ( W \geqslant 0 )\).
   3. Write down an expression for \(q\) in terms of \(p\) only.
2. Given that \(\mathrm { E } ( X ) = 3\), find the value of \(a\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.