Edexcel S1 (Statistics 1) 2022 October

1. The stem lengths of a sample of 120 tulips are recorded in the grouped frequency table below.

A histogram is drawn to represent these data.
The area of the bar representing the \(40 \leqslant x < 42\) class is \(16.5 \mathrm {~cm} ^ { 2 }\)

1. Calculate the exact area of the bar representing the \(42 \leqslant x < 45\) class. The height of the tallest bar in the histogram is 10 cm .
2. Find the exact height of the second tallest bar.
   \(Q _ { 1 }\) for these data is 45 cm .
3. Use linear interpolation to find an estimate for
   1. \(Q _ { 2 }\)
   2. the interquartile range. One measure of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
4. By calculating this measure, describe the skewness of these data.

1. The production cost, \(\pounds c\) million, of a film and the total ticket sales, \(\pounds t\) million, earned by the film are recorded for a sample of 40 films.

Some summary statistics are given below. $$\sum c = 1634 \quad \sum t = 1361 \quad \sum t ^ { 2 } = 82873 \quad \sum c t = 83634 \quad \mathrm {~S} _ { c c } = 28732.1$$

1. Find the exact value of \(\mathrm { S } _ { t t }\) and the exact value of \(\mathrm { S } _ { c t }\)
2. Calculate the value of the product moment correlation coefficient for these data.
3. Give an interpretation of your answer to part (b)
4. Show that the equation of the linear regression line of \(t\) on \(c\) can be written as $$t = - 5.84 + 0.976 c$$ where the values of the intercept and gradient are given to 3 significant figures.
5. Find the expected total ticket sales for a film with a production cost of \(\pounds 90\) million. Using the regression line in part (d)
6. find the range of values of the production cost of a film for which the total ticket sales are less than \(80 \%\) of its production cost.

1. Morgan is investigating the body length, \(b\) centimetres, of squirrels.

A random sample of 8 squirrels is taken and the data for each squirrel is coded using $$x = \frac { b - 21 } { 2 }$$ The results for the coded data are summarised below $$\sum x = - 1.2 \quad \sum x ^ { 2 } = 5.1$$

1. Find the mean of \(b\)
2. Find the standard deviation of \(b\) A 9th squirrel is added to the sample. Given that for all 9 squirrels \(\sum x = 0\)
3. find
   1. the body length of the 9th squirrel,
   2. the standard deviation of \(x\) for all 9 squirrels.

1. The cumulative distribution function of the discrete random variable \(W\), which takes only the values 6,7 and 8 , is given by

$$F ( W ) = \frac { ( w + 3 ) ( w - 1 ) } { 77 } \text { for } w = 6,7,8$$ Find \(\mathrm { E } ( W )\)

1. The weights, \(W\) grams, of kiwi fruit grown on a farm are normally distributed with mean 80 grams and standard deviation 8 grams.

The table shows the classifications of the kiwi fruit by their weight, where \(k\) is a positive constant. One kiwi fruit is selected at random from those grown on the farm.

1. Find the probability that this kiwi fruit is Large. 35\% of the kiwi fruit are Jumbo.
2. Find the value of \(k\) to one decimal place. 75\% of Tiny kiwi fruit weigh more than \(y\) grams.
3. Find the value of \(y\) giving your answer to one decimal place.

1. The Venn diagram shows the events \(A , B , C\) and \(D\), where \(p , q , r\) and \(s\) are probabilities.
   \includegraphics[max width=\textwidth, alt={}, center]{1fda59cb-059e-4850-810f-cc3e69bc058e-20_504_826_296_621}
   1. Write down the value of
      1. \(\mathrm { P } ( A )\)
      2. \(\mathrm { P } ( A \mid B )\)
      3. \(\mathrm { P } ( A \mid C )\)
   Given that \(\mathrm { P } \left( B ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid D ) = \frac { 3 } { 5 }\)
2. find the exact value of \(q\) and the exact value of \(r\) Given also that \(\mathrm { P } \left( B \cup C ^ { \prime } \right) = \frac { 5 } { 8 }\)
3. find the exact value of \(s\)

1. Adana selects one number at random from the distribution of \(X\) which has the following probability distribution.

1. Given that the number selected by Adana is not 5 , write down the probability it is 0
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 75\)
3. Find \(\operatorname { Var } ( X )\)
4. Find \(\operatorname { Var } ( 4 - 3 X )\) Bruno and Charlie each independently select one number at random from the distribution of \(X\)
5. Find the probability that the number Bruno selects is greater than the number Charlie selects. Devika multiplies Bruno's number by Charlie's number to obtain a product, \(D\)
6. Determine the probability distribution of \(D\)