Edexcel S1 (Statistics 1)

1. The discrete random variable \(X\) has the following probability distribution.

1. Find and simplify an expression in terms of \(k\) for \(\mathrm { E } ( X )\). Given that \(\mathrm { E } ( X ) = 9\),
2. find the value of \(k\).

2. (a) Explain briefly what is meant by a statistical model.
(b) State, with a reason, whether or not the normal distribution might be suitable for modelling each of the following:

1. The number of children in a family;
2. The time taken for a particular employee to cycle to work each day using the same route;
3. The quarterly electricity bills for a particular house.

3. The probability that Ajita gets up before 6.30 am in the morning is 0.7 The probability that she goes for a run in the morning is 0.35
The probability that Ajita gets up after 6.30 am and does not go for a run is 0.22
Let \(A\) represent the event that Ajita gets up before 6.30 am and \(B\) represent the event that she goes for a run in the morning. Find

1. \(\mathrm { P } ( A \cup B )\),
2. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
3. \(\mathrm { P } ( B \mid A )\).
4. State, with a reason, whether or not events \(A\) and \(B\) are independent.

4. A company produces jars of English Honey. The weight of the glass jars used is normally distributed with a mean of 122.3 g and a standard deviation of 2.6 g . Calculate the probability that a randomly chosen jar will weigh

1. less than 127 g ,
2. less than 121.5 g . The weight of honey put into each jar by a machine is normally distributed with a standard deviation of 1.6 g . The machine operator can adjust the mean weight of the honey put into each jar without changing the standard deviation.
3. Find, correct to 4 significant figures, the minimum that the mean weight can be set to such that at most 1 in 20 of the jars will contain less than 454 g .
   (4 marks)

5. The letters of the word DISTRIBUTION are written on separate cards. The cards are then shuffled and the top three are turned over. Let the random variable \(V\) be the number of vowels that are turned over.

1. Show that \(\mathrm { P } ( V = 1 ) = \frac { 21 } { 44 }\).
2. Find the probability distribution of \(V\).
3. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).

6. A cinema recorded the number of people at each showing of each film during a one-week period. The results are summarised in the table below.

1. Draw a histogram on graph paper to illustrate these data.
2. Calculate estimates of the median and quartiles of these data.
3. Use your answers to part (b) to show that the data is positively skewed.

7. A new vaccine is tested over a six-month period in one health authority. The table shows the number of new cases of the disease, \(d\), reported in the \(m\) th month after the trials began. A doctor suggests that a relationship of the form \(d = a + b x\) where \(x = \frac { 1 } { m }\) can be used to model the situation.

1. Tabulate the values of \(x\) corresponding to the given values of \(d\) and plot a scatter diagram of \(d\) against \(x\).
2. Explain how your scatter diagram supports the suggested model. You may use $$\Sigma x = 2.45 , \quad \Sigma d = 390 , \quad \Sigma x ^ { 2 } = 1.491 , \quad \Sigma x d = 189.733$$
3. Find an equation of the regression line \(d\) on \(x\) in the form \(d = a + b x\).
4. Use your regression line to estimate how many new cases of the disease there will be in the 13th month after the trial began.
5. Comment on the reliability of your answer to part (d).