CAIE S1 (Statistics 1) 2010 November

1 Name the distribution and suggest suitable numerical parameters that you could use to model the weights in kilograms of female 18-year-old students.

2 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x\), where \(x\) takes values \(1,2,3,4,5\) only.

1. Draw up a probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\).

3 It was found that \(68 \%\) of the passengers on a train used a cell phone during their train journey. Of those using a cell phone, \(70 \%\) were under 30 years old, \(25 \%\) were between 30 and 65 years old and the rest were over 65 years old. Of those not using a cell phone, \(26 \%\) were under 30 years old and \(64 \%\) were over 65 years old.

1. Draw a tree diagram to represent this information, giving all probabilities as decimals.
2. Given that one of the passengers is 45 years old, find the probability of this passenger using a cell phone during the journey.

4 Delip measured the speeds, \(x \mathrm {~km}\) per hour, of 70 cars on a road where the speed limit is 60 km per hour. His results are summarised by \(\Sigma ( x - 60 ) = 245\).

1. Calculate the mean speed of these 70 cars. His friend Sachim used values of \(( x - 50 )\) to calculate the mean.
2. Find \(\Sigma ( x - 50 )\).
3. The standard deviation of the speeds is 10.6 km per hour. Calculate \(\Sigma ( x - 50 ) ^ { 2 }\).

5 The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower.
\includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-3_1031_1326_372_406}

1. Copy and complete the following frequency table for the data.

   Time \(( t\) minutes \()\)	\(2 < t \leqslant 4\)	\(4 < t \leqslant 6\)	\(6 < t \leqslant 7\)	\(7 < t \leqslant 8\)	\(8 < t \leqslant 10\)	\(10 < t \leqslant 16\)
   Frequency

2. Calculate an estimate of the mean time to take a shower.
3. Two of these students are chosen at random. Find the probability that exactly one takes between 7 and 10 minutes to take a shower.

6
\includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-4_387_899_255_623} A small aeroplane has 14 seats for passengers. The seats are arranged in 4 rows of 3 seats and a back row of 2 seats (see diagram). 12 passengers board the aeroplane.

1. How many possible seating arrangements are there for the 12 passengers? Give your answer correct to 3 significant figures. These 12 passengers consist of 2 married couples (Mr and Mrs Lin and Mr and Mrs Brown), 5 students and 3 business people.
2. The 3 business people sit in the front row. The 5 students each sit at a window seat. Mr and Mrs Lin sit in the same row on the same side of the aisle. Mr and Mrs Brown sit in another row on the same side of the aisle. How many possible seating arrangements are there?
3. If, instead, the 12 passengers are seated randomly, find the probability that Mrs Lin sits directly behind a student and Mrs Brown sits in the front row.

7 The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. \(79 \%\) of people who visit this dentist have visits lasting less than 10 minutes.

1. Find the standard deviation of the times spent by people visiting this dentist.
2. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates from the mean by more than 1 minute.
3. Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than 10 minutes.
4. Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than 8.2 minutes. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }