CAIE S1 (Statistics 1) 2019 November

1 When Shona goes to college she either catches the bus with probability 0.8 or she cycles with probability 0.2 . If she catches the bus, the probability that she is late is 0.4 . If she cycles, the probability that she is late is \(x\). The probability that Shona is not late for college on a randomly chosen day is 0.63 . Find the value of \(x\).

2 Annan has designed a new logo for a sportswear company. A survey of a large number of customers found that \(42 \%\) of customers rated the logo as good.

1. A random sample of 10 customers is chosen. Find the probability that fewer than 8 of them rate the logo as good.
2. On another occasion, a random sample of \(n\) customers of the company is chosen. Find the smallest value of \(n\) for which the probability that at least one person rates the logo as good is greater than 0.995 .

3 The mean and standard deviation of 20 values of \(x\) are 60 and 4 respectively.

1. Find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
   Another 10 values of \(x\) are such that their sum is 550 and the sum of their squares is 40500 .
2. Find the mean and standard deviation of all these 30 values of \(x\).

4 In a probability distribution the random variable \(X\) takes the values \(- 1,0,1,2,4\). The probability distribution table for \(X\) is as follows.

1. Find the value of \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Given that \(X\) is greater than zero, find the probability that \(X\) is equal to 2 .

5 Ransha measured the lengths, in centimetres, of 160 palm leaves. His results are illustrated in the cumulative frequency graph below.
\includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-08_1090_1424_404_356}

1. Estimate how many leaves have a length between 14 and 24 centimetres.
2. \(10 \%\) of the leaves have a length of \(L\) centimetres or more. Estimate the value of \(L\).
3. Estimate the median and the interquartile range of the lengths.
   Sharim measured the lengths, in centimetres, of 160 palm leaves of a different type. He drew a box-and-whisker plot for the data, as shown on the grid below.
   \includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-09_540_1287_1181_424}
4. Compare the central tendency and the spread of the two sets of data.

6

1. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that all four Es are together.
2. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that the Ss are not next to each other.
   Four letters are selected from the 12 letters of the word STEEPLECHASE.
3. Find the number of different selections if the four letters include exactly one \(S\).

7 The shortest time recorded by an athlete in a 400 m race is called their personal best (PB). The PBs of the athletes in a large athletics club are normally distributed with mean 49.2 seconds and standard deviation 2.8 seconds.

1. Find the probability that a randomly chosen athlete from this club has a PB between 46 and 53 seconds.
2. It is found that \(92 \%\) of athletes from this club have PBs of more than \(t\) seconds. Find the value of \(t\).
   Three athletes from the club are chosen at random.
3. Find the probability that exactly 2 have PBs of less than 46 seconds.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.