CAIE S2 (Statistics 2) 2016 November

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 3.5 )\). Find \(\mathrm { P } ( X < 3 )\).

2 Dominic wishes to choose a random sample of five students from the 150 students in his year. He numbers the students from 1 to 150 . Then he uses his calculator to generate five random numbers between 0 and 1 . He multiplies each random number by 150 and rounds up to the next whole number to give a student number.

1. Dominic's first random number is 0.392 . Find the student number that is produced by this random number.
2. Dominic's second student number is 104 . Find a possible random number that would produce this student number.
3. Explain briefly why five random numbers may not be enough to produce a sample of five student numbers.

3 A men's triathlon consists of three parts: swimming, cycling and running. Competitors' times, in minutes, for the three parts can be modelled by three independent normal variables with means 34.0, 87.1 and 56.9, and standard deviations 3.2, 4.1 and 3.8, respectively. For each competitor, the total of his three times is called the race time. Find the probability that the mean race time of a random sample of 15 competitors is less than 175 minutes.

4 The manufacturer of a tablet computer claims that the mean battery life is 11 hours. A consumer organisation wished to test whether the mean is actually greater than 11 hours. They invited a random sample of members to report the battery life of their tablets. They then calculated the sample mean. Unfortunately a fire destroyed the records of this test except for the following partial document.
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-2_467_593_1612_776} Given that the population of battery lives is normally distributed with standard deviation 1.6 hours, find the set of possible values of the sample size, \(n\).

5 It is claimed that \(30 \%\) of packets of Froogum contain a free gift. Andre thinks that the actual proportion is less than \(30 \%\) and he decides to carry out a hypothesis test at the \(5 \%\) significance level. He buys 20 packets of Froogum and notes the number of free gifts he obtains.

1. State null and alternative hypotheses for the test.
2. Use a binomial distribution to find the probability of a Type I error. Andre finds that 3 of the 20 packets contain free gifts.
3. Carry out the test.

6 A variable \(X\) takes values \(1,2,3,4,5\), and these values are generated at random by a machine. Each value is supposed to be equally likely, but it is suspected that the machine is not working properly. A random sample of 100 values of \(X\), generated by the machine, gives the following results. $$n = 100 \quad \Sigma x = 340 \quad \Sigma x ^ { 2 } = 1356$$

1. Find a 95\% confidence interval for the population mean of the values generated by the machine.
2. Use your answer to part (i) to comment on whether the machine may be working properly.

7 Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.

1. Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5 -minute period.
2. Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.

8
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_517_276_427}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_304_508_274_1215}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_305_506_717_431}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_504_717_1217} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between - 3 and 3 only, and their standard deviations are \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) respectively.

1. List \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) in order of size, starting with the largest.
2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3
   0 & \text { otherwise } \end{cases}$$ (a) Show that \(\sigma _ { X } = 2.32\) correct to 3 significant figures.
   (b) Calculate \(\mathrm { P } \left( X > \sigma _ { X } \right)\).
   (c) Write down the value of \(\mathrm { P } \left( X > 2 \sigma _ { X } \right)\).