Edexcel S1 (Statistics 1) 2016 October

1. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)

Given that \(\mathrm { P } ( X > \mu + a ) = 0.35\) where \(a\) is a constant, find

1. \(\mathrm { P } ( X > \mu - a )\)
2. \(\mathrm { P } ( \mu - a < X < \mu + a )\)
3. \(\mathrm { P } ( X < \mu + a \mid X > \mu - a )\)

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

where \(a\) and \(b\) are constants.

1. Write down an equation for \(a\) and \(b\).
2. Calculate \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 3.5\)
3. 1. find a second equation in \(a\) and \(b\),
   2. hence find the value of \(a\) and the value of \(b\).
4. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 5 - 3 X\)
5. Find \(\mathrm { P } ( Y > 0 )\).
6. Find
   1. \(\mathrm { E } ( Y )\),
   2. \(\operatorname { Var } ( Y )\).

1. Hugo recorded the purchases of 80 customers in the ladies fashion department of a large store. His results were as follows

20 customers bought a coat
12 customers bought a coat and a scarf
23 customers bought a pair of gloves
13 customers bought a pair of gloves and a scarf no customer bought a coat and a pair of gloves 14 customers did not buy a coat nor a scarf nor a pair of gloves.

1. Draw a Venn diagram to represent all of this information.
2. One of the 80 customers is selected at random.
   1. Find the probability that the customer bought a scarf.
   2. Given that the customer bought a coat, find the probability that the customer also bought a scarf.
   3. State, giving a reason, whether or not the event 'the customer bought a coat' and the event 'the customer bought a scarf' are statistically independent. Hugo had asked the member of staff selling coats and the member of staff selling gloves to encourage customers also to buy a scarf.
3. By considering suitable conditional probabilities, determine whether the member of staff selling coats or the member of staff selling gloves has the better performance at selling scarves to their customers. Give a reason for your answer.

1. A doctor is studying the scans of 30 -week old foetuses. She takes a random sample of 8 scans and measures the length, \(f \mathrm {~mm}\), of the leg bone called the femur. She obtains the following results.

$$\begin{array} { l l l l l l l l } 52 & 53 & 56 & 57 & 57 & 59 & 60 & 62 \end{array}$$

1. Show that \(\mathrm { S } _ { f f } = 80\) The doctor also measures the head circumference, \(h \mathrm {~mm}\), of each foetus and her results are summarised as $$\sum h = 2209 \quad \sum h ^ { 2 } = 610463 \quad \mathrm {~S} _ { f h } = 182$$
2. Find \(\mathrm { S } _ { h h }\)
3. Calculate the product moment correlation coefficient between the length of the femur and the head circumference for these data. The doctor believes that there is a linear relationship between the length of the femur and the head circumference of 30-week old foetuses.
4. State, giving a reason, whether or not your calculation in part (c) supports the doctor's belief.
5. Find an equation of the regression line of \(h\) on \(f\). The doctor plans in future to measure the femur length, \(f\), and then use the regression line to estimate the corresponding head circumference, \(h\). A statistician points out that there will always be the chance of an error between the true head circumference and the estimated value of the head circumference. Given that the error, \(E \mathrm {~mm}\), has the normal distribution \(\mathrm { N } \left( 0,4 ^ { 2 } \right)\)
6. find the probability that the estimate is within 3 mm of the true value.

1. The label on a jar of Amy’s jam states that the jar contains about 400 grams of jam. For each jar that contains less than 388 grams of jam, Amy will be fined \(\pounds 100\). If a jar contains more than 410 grams of jam then Amy makes a loss of \(\pounds 0.30\) on that jar.

The weight of jam, \(A\) grams, in a jar of Amy's jam has a normal distribution with mean \(\mu\) grams and standard deviation \(\sigma\) grams. Amy chooses \(\mu\) and \(\sigma\) so that \(\mathrm { P } ( A < 388 ) = 0.001\) and \(\mathrm { P } ( A > 410 ) = 0.0197\)

1. Find the value of \(\mu\) and the value of \(\sigma\). Amy can sell jars of jam containing between 388 grams and 410 grams for a profit of \(\pounds 0.25\)
2. Calculate the expected amount, in £s, that Amy receives for each jar of jam.

1. The stem and leaf diagram gives the blood pressure, \(x \mathrm { mmHg }\), for a random sample of 19 female patients.

Key: 10 | 1 means blood pressure of 101 mmHg

1. Find the median and the quartiles for these data.
2. Find the interquartile range ( \(Q _ { 3 } - Q _ { 1 }\) ) An outlier is a value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or less than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\)
3. Showing your working clearly, identify any outliers for these data.
4. On the grid on page 21 draw a box and whisker plot to represent these data. Show any outliers clearly. The above data can be summarised by $$\sum x = 2299 \text { and } \sum x ^ { 2 } = 279709$$
5. Calculate the mean and the standard deviation for these data. For a random sample taken from a normal distribution, a rule for determining outliers is: an outlier is more than \(2.7 \times\) standard deviation above or below the mean.
6. Find the limits to determine outliers using this rule.
7. State, giving a reason based on some of the above calculations, whether or not a normal distribution is a suitable model for these data. \includegraphics[max width=\textwidth, alt={}, center]{8ff7539e-fa44-4388-af8c-80656f081528-21_2281_73_308_15}
   Turn over for a spare diagram if you need to redraw your plot.
   \includegraphics[max width=\textwidth, alt={}]{8ff7539e-fa44-4388-af8c-80656f081528-24_2639_1830_121_121}