OCR S1 (Statistics 1) 2011 January

2 The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.2 )\). Find

1. \(\mathrm { P } ( X = 3 )\),
2. \(\mathrm { P } ( 3 \leqslant X \leqslant 5 )\),
3. \(\mathrm { P } ( X > 4 )\). Two independent values of \(X\) are found.
4. Find the probability that the total of these two values is 3 .

3 A firm wishes to assess whether there is a linear relationship between the annual amount spent on advertising, \(\pounds x\) thousand, and the annual profit, \(\pounds y\) thousand. A summary of the figures for 12 years is as follows. $$n = 12 \quad \Sigma x = 86.6 \quad \Sigma y = 943.8 \quad \Sigma x ^ { 2 } = 658.76 \quad \Sigma y ^ { 2 } = 83663.00 \quad \Sigma x y = 7351.12$$

1. Calculate the product moment correlation coefficient, showing that it is greater than 0.9 .
2. Comment briefly on this value in this context.
3. A manager claims that this result shows that spending more money on advertising in the future will result in greater profits. Make two criticisms of this claim.
4. Calculate the equation of the regression line of \(y\) on \(x\).
5. Estimate the annual profit during a year when \(\pounds 7400\) was spent on advertising.

4 Jenny and Omar are each allowed two attempts at a high jump.

1. The probability that Jenny will succeed on her first attempt is 0.6 . If she fails on her first attempt, the probability that she will succeed on her second attempt is 0.7 . Calculate the probability that Jenny will succeed.
2. The probability that Omar will succeed on his first attempt is \(p\). If he fails on his first attempt, the probability that he will succeed on his second attempt is also \(p\). The probability that he succeeds is 0.51 . Find \(p\).
   \(530 \%\) of packets of Natural Crunch Crisps contain a free gift. Jan buys 5 packets each week.

1. The number of free gifts that Jan receives in a week is denoted by \(X\). Name a suitable probability distribution with which to model \(X\), giving the value(s) of any parameter(s). State any assumption(s) necessary for the distribution to be a valid model. Assume now that your model is valid.
2. Find
   (a) \(\mathrm { P } ( X \leqslant 2 )\),
   (b) \(\mathrm { P } ( X = 2 )\).
3. Find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts. 6
4. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
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6

1. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number.

   1

   3

   3

   3

   5

   5

   9

   How many different 7 -digit numbers can be formed using these cards?
2. The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it.
   \includegraphics[max width=\textwidth, alt={}, center]{98ac515d-fd47-4864-afd6-321e9848d6cb-04_398_801_596_632} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
   (a) How many selections of seven cards are possible?
   (b) Find the probability that the seven cards include exactly one card showing the letter A .

7 The probability distribution of a discrete random variable, \(X\), is shown below.

1. Find \(\mathrm { E } ( X )\) in terms of \(a\).
2. Show that \(\operatorname { Var } ( X ) = 4 a ( 1 - a )\).