Edexcel FS1 AS (Further Statistics 1 AS) 2020 June

1. A plumbing company receives call-outs during the working day at an average rate of 2.4 per hour.
   1. Find the probability that the company receives exactly 7 call-outs in a randomly selected 3 -hour period of a working day.
   The company has enough staff to respond to 28 call-outs in an 8 -hour working day.
2. Show that the probability that the company receives more than 28 call-outs in a randomly selected 8 -hour working day is 0.022 to 3 decimal places. In a random sample of 100 working days each of 8 hours,
3. 1. find the expected number of days that the company receives more than 28 call-outs,
   2. find the standard deviation of the number of days that the company receives more than 28 call-outs,
   3. use a Poisson approximation to estimate the probability that the company receives more than 28 call-outs on at least 6 of these days.

1. In an experiment, James flips a coin 3 times and records the number of heads. He carries out the experiment 100 times with his left hand and 100 times with his right hand.

1. Test, at the \(5 \%\) level of significance, whether or not there is an association between the hand he flips the coin with and the number of heads. You should state your hypotheses, the degrees of freedom and the critical value used for this test.
2. Assuming the coin is unbiased, write down the distribution of the number of heads in 3 flips.
3. Carry out a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether or not the distribution you wrote down in part (b) is a suitable model for the number of heads obtained in the 200 trials of James' experiment. You should state your hypotheses, the degrees of freedom and the critical value used for this test.

1. The probability distribution of the discrete random variable \(X\) is

$$P ( X = x ) = \begin{cases} \frac { k } { x } & \text { for } x = 1,2 \text { and } 3
\frac { m } { 2 x } & \text { for } x = 6 \text { and } 9
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(m\) are positive constants.
Given that \(\mathrm { E } ( X ) = 3.8\), find \(\operatorname { Var } ( X )\)

1. During the morning, the number of cyclists passing a particular point on a cycle path in a 10-minute interval travelling eastbound can be modelled by a Poisson distribution with mean 8

The number of cyclists passing the same point in a 10 -minute interval travelling westbound can be modelled by a Poisson distribution with mean 3

1. Suggest a model for the total number of cyclists passing the point on the cycle path in a 10-minute interval, stating a necessary assumption. Given that exactly 12 cyclists pass the point in a 10 -minute interval,
2. find the probability that at least 11 are travelling eastbound. After some roadworks were completed, the total number of cyclists passing the point in a randomly selected 20-minute interval one morning is found to be 14
3. Test, at the \(5 \%\) level of significance, whether there is evidence of a decrease in the rate of cyclists passing the point.
   State your hypotheses clearly.