Edexcel S1 (Statistics 1) 2018 October

1. The heights above sea level ( \(h\) hundred metres) and the temperatures ( \(t ^ { \circ } \mathrm { C }\) ) at 12 randomly selected places in France, at 7 am on July 31st, were recorded.
   The data are summarised as follows
   1. Find the value of \(S _ { t t }\)
   2. Calculate the product moment correlation coefficient for these data.
   3. Interpret the relationship between \(t\) and \(h\).
   4. Find an equation of the regression line of \(t\) on \(h\).
   At 7 am on July 31st Yinka is on holiday in South Africa. He uses the regression equation to estimate the temperature when the height above sea level is 500 m .
2. Find the estimated temperature Yinka calculates.
3. Comment on the validity of your answer in part (e). $$\sum h = 112 \quad \sum t = 136 \quad \sum t ^ { 2 } = 1828 \quad S _ { h t } = - 236 \quad S _ { h h } = 297$$
4. Find the value of \(S\) (2)

1. The weights, to the nearest kilogram, of a sample of 33 female spotted hyenas living in the Serengeti are summarised in the stem and leaf diagram below.

\begin{table}[h] \end{table} Totals

1. Find the median and quartiles for the weights of the female spotted hyenas. An outlier is defined as any value greater than \(c\) or any value less than \(d\) where $$\begin{aligned} & c = Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right)
   & d = Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \end{aligned}$$
2. Showing your working clearly, identify any outliers for these data.
   (3) The weights, to the nearest kilogram, of a sample of male spotted hyenas living in the Serengeti are summarised below.
   \includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-06_755_1568_1537_185}
3. In the space provided in the grid above, draw a box and whisker plot to represent the weights of female spotted hyenas living in the Serengeti. Indicate clearly any outliers. (A copy of this grid is on page 9 if you need to redraw your box and whisker plot.)
4. Compare the weights of male and female spotted hyenas living in the Serengeti. Key: 3|2 means 32
   \includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-09_2658_101_107_9}

3. The parking times, \(t\) hours, for cars in a car park are summarised below. $$\text { (You may use } \sum \mathrm { fm } = 182 \text { and } \sum \mathrm { fm } ^ { 2 } = 883 \text { ) }$$ A histogram is drawn to represent these data.
The bar representing the time \(1 \leqslant t < 2\) has a width of 1.5 cm and a height of 6 cm .

1. Calculate the width and the height of the bar representing the time \(4 \leqslant t < 6\)
2. Use linear interpolation to estimate the median parking time for the cars in the car park.
3. Estimate the mean and the standard deviation of the parking time for the cars in the car park.
4. Describe, giving a reason, the skewness of the data. One of these cars is selected at random.
5. Estimate the probability that this car is parked for more than 75 minutes.

4. Pieces of wood cladding are produced by a timber merchant. There are three types of fault, \(A , B\) and \(C\), that can appear in each piece of wood cladding. The Venn diagram shows the probabilities of a piece of wood cladding having the various types of fault.
\includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-14_602_1120_497_413} A piece of wood cladding is chosen at random.

1. Find the probability that the piece of wood cladding has more than one type of fault. Fault types \(A\) and \(C\) occur independently.
2. Find the probability that the piece of wood cladding has no faults. Given that the piece of wood cladding has fault \(A\),
3. find the probability that it also has fault \(B\) but not fault \(C\). Two pieces of the wood cladding are selected at random.
4. Find the probability that both have exactly 2 types of fault.

1. The discrete random variable \(X\) is defined by the cumulative distribution function

where \(k\) is a constant.

1. Find the probability distribution of \(X\).
2. Find \(\mathrm { P } ( 1.5 < X \leqslant 3.5 )\) The random variable \(Y = 12 - 7 X\)
3. Calculate Var(Y)
4. Calculate \(\mathrm { P } ( 4 X \leqslant | Y | )\)

1. A machine makes bolts such that the length, \(L \mathrm {~cm}\), of a bolt has distribution \(L \sim \mathrm {~N} \left( 4.1,0.125 ^ { 2 } \right)\)

A bolt is selected at random.

1. Find the probability that the length of this bolt is more than 4.3 cm .
2. Show that \(\mathrm { P } ( 3.9 < L < 4.3 )\) is 0.890 correct to 3 decimal places. The machine makes 500 bolts.
   The cost to make each bolt is 5 pence.
   Only bolts with length between 3.9 cm and 4.3 cm can be used. These are sold for 9 pence each. All the bolts that cannot be used are recycled with a scrap value of 1 pence each.
3. Calculate an estimate for the profit made on these 500 bolts. Following adjustments to the machine, the length of a bolt, \(B \mathrm {~cm}\), made by the machine is such that \(B \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( B > 4.198 ) = 0.025\) and \(\mathrm { P } ( B < 4.065 ) = 0.242\)
4. find the value of \(\mu\) and the value of \(\sigma\)
5. State, giving a reason, whether the adjustments to the machine will result in a decrease or an increase in the profit made on 500 bolts.