CAIE S2 (Statistics 2) 2020 November

1 It is known that, on average, 1 in 300 flowers of a certain kind are white. A random sample of 200 flowers of this kind is selected.

1. Use an appropriate approximating distribution to find the probability that more than 1 flower in the sample is white.
2. Justify the approximating distribution used in part (a).
   The probability that a randomly chosen flower of another kind is white is 0.02 . A random sample of 150 of these flowers is selected.
3. Use an appropriate approximating distribution to find the probability that the total number of white flowers in the two samples is less than 4 .

2 In a survey, a random sample of 250 adults in Fromleigh were asked to fill in a questionnaire about their travel.

1. It was found that 102 adults in the sample travel by bus. Find an approximate \(90 \%\) confidence interval for the proportion of all the adults in Fromleigh who travel by bus.
2. The survey included a question about the amount, \(x\) dollars, spent on travel per year. The results are summarised as follows. $$n = 250 \quad \Sigma x = 50460 \quad \Sigma x ^ { 2 } = 19854200$$ Find unbiased estimates of the population mean and variance of the amount spent per year on travel.
   A councillor wanted to select a random sample of houses in Fromleigh. He planned to select the first house on each of the 143 streets in Fromleigh.
3. Explain why this would not provide a random sample.

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male.
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4
\includegraphics[max width=\textwidth, alt={}, center]{65b50bfb-5fd8-4cf3-ae3b-cffc12e23cd8-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

5 The number of absences per week by workers at a factory has the distribution \(\operatorname { Po } ( 2.1 )\).

1. Find the standard deviation of the number of absences per week.
2. Find the probability that the number of absences in a 2-week period is at least 2 .
3. Find the probability that the number of absences in a 3-week period is more than 4 and less than 8 .
   Following a change in working conditions, the management wished to test whether the mean number of absences has decreased. They found that, in a randomly chosen 3-week period, there were exactly 2 absences.
4. Carry out the test at the \(10 \%\) significance level.
5. State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (d).

6 The time, in minutes, for Anjan's journey to work on Mondays has mean 38.4 and standard deviation 6.9.

1. Find the probability that Anjan's mean journey time for a random sample of 30 Mondays is between 38 and 40 minutes.
   Anjan wishes to test whether his mean journey time is different on Tuesdays. He chooses a random sample of 30 Tuesdays and finds that his mean journey time for these 30 Tuesdays is 40.2 minutes. Assume that the standard deviation for his journey time on Tuesdays is 6.9 minutes.
2. 1. State, with a reason, whether Anjan should use a one-tail or a two-tail test.
   2. Carry out the test at the \(10 \%\) significance level.
   3. Explain whether it was necessary to use the Central Limit theorem in part (b)(ii).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.