CAIE S1 (Statistics 1) 2018 June

1 The masses in kilograms of 50 children having a medical check-up were recorded correct to the nearest kilogram. The results are shown in the table.

1. Find which class interval contains the lower quartile.
2. On the grid, draw a histogram to illustrate the data in the table.
   \includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-02_1397_1397_1187_415}

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( - 3 , \sigma ^ { 2 } \right)\). The probability that a randomly chosen value of \(X\) is positive is 0.25 .

1. Find the value of \(\sigma\).
2. Find the probability that, of 8 random values of \(X\), fewer than 2 will be positive.

3 The members of a swimming club are classified either as 'Advanced swimmers' or 'Beginners'. The proportion of members who are male is \(x\), and the proportion of males who are Beginners is 0.7 . The proportion of females who are Advanced swimmers is 0.55 . This information is shown in the tree diagram.
\includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-04_435_974_482_587} For a randomly chosen member, the probability of being an Advanced swimmer is the same as the probability of being a Beginner.

1. Find \(x\).
2. Given that a randomly chosen member is an Advanced swimmer, find the probability that the member is male.

4 Farfield Travel and Lacket Travel are two travel companies which arrange tours abroad. The numbers of holidays arranged in a certain week are recorded in the table below, together with the means and standard deviations of the prices.

1. Calculate the mean price of all 51 holidays.
2. The prices of individual holidays with Farfield Travel are denoted by \(
   ) x _ { F }\( and the prices of individual holidays with Lacket Travel are denoted by \)\\( x _ { L }\). By first finding \(\Sigma x _ { F } ^ { 2 }\) and \(\Sigma x _ { L } ^ { 2 }\), find the standard deviation of the prices of all 51 holidays.

5 A game is played with 3 coins, \(A , B\) and \(C\). Coins \(A\) and \(B\) are biased so that the probability of obtaining a head is 0.4 for coin \(A\) and 0.75 for coin \(B\). Coin \(C\) is not biased. The 3 coins are thrown once.

1. Draw up the probability distribution table for the number of heads obtained.
2. Hence calculate the mean and variance of the number of heads obtained.

6 The diameters of apples in an orchard have a normal distribution with mean 5.7 cm and standard deviation 0.8 cm . Apples with diameters between 4.1 cm and 5 cm can be used as toffee apples.

1. Find the probability that an apple selected at random can be used as a toffee apple.
2. 250 apples are chosen at random. Use a suitable approximation to find the probability that fewer than 50 can be used as toffee apples.

7 Find the number of ways the 9 letters of the word SEVENTEEN can be arranged in each of the following cases.

1. One of the letter Es is in the centre with 4 letters on either side.
2. No E is next to another E.
   5 letters are chosen from the 9 letters of the word SEVENTEEN.
3. Find the number of possible selections which contain exactly 2 Es and exactly 2 Ns.
4. Find the number of possible selections which contain at least 2 Es.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.