Edexcel S1 (Statistics 1) 2022 January

1. A factory produces shoes.

A quality control inspector at the factory checks a sample of 120 shoes for each of three types of defect. The Venn diagram represents the inspector's results. A represents the event that a shoe has defective stitching \(B\) represents the event that a shoe has defective colouring \(C\) represents the event that a shoe has defective soles
\includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-02_684_935_607_566} One of the shoes in the sample is selected at random.

1. Find the probability that it does not have defective soles.
2. Find \(\mathrm { P } \left( A \cap B \cap C ^ { \prime } \right)\)
3. Find \(\mathrm { P } \left( A \cup B \cup C ^ { \prime } \right)\)
4. Find the probability that the shoe has at most one type of defect.
5. Given the selected shoe has at most one type of defect, find the probability it has defective stitching. The random variable \(X\) is the number of the events \(A , B , C\) that occur for a randomly selected shoe.
6. Find \(\mathrm { E } ( X )\) \section*{This is a copy of the Venn diagram for this question.} \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-05_684_940_388_566}

2. Tom's car holds 50 litres of petrol when the fuel tank is full. For each of 10 journeys, each starting with 50 litres of petrol in the fuel tank, Tom records the distance travelled, \(d\) kilometres, and the amount of petrol used, \(p\) litres. The summary statistics for the 10 journeys are given below. $$\sum d = 1029 \quad \sum p = 50.8 \quad \sum d p = 5240.8 \quad \mathrm {~S} _ { d d } = 344.9 \quad \mathrm {~S} _ { p p } = 0.576$$

1. Calculate the product moment correlation coefficient between \(d\) and \(p\) The amount of petrol remaining in the fuel tank for each journey, \(w\) litres, is recorded.
2. 1. Write down an equation for \(w\) in terms of \(p\)
   2. Hence, write down the value of the product moment correlation coefficient between \(w\) and \(p\)
3. Write down the value of the product moment correlation coefficient between \(d\) and \(w\)

1. The stem and leaf diagram shows the number of deliveries made by Pat each day for 24 days

\begin{table}[h] \end{table} where \(a\), \(b\) and \(c\) are positive integers with \(a < b < c\)
An outlier is defined as any value greater than \(1.5 \times\) interquartile range above the upper quartile. Given that there is only one outlier for these data,

1. show that \(c = 9\) The number of deliveries made by Pat each day is represented by \(d\)
   The data in the stem and leaf diagram are coded using $$x = d - 125$$ and the following summary statistics are obtained $$\sum x = - 96 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1306$$
2. Find the mean number of deliveries.
3. Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable \(D\) represents the number of deliveries made by Pat on this day. The random variable \(X = D - 125\)
4. Find \(\mathrm { P } ( D > 118 \mid X < 0 )\)

1. The random variable \(W\) has a discrete uniform distribution where

$$\mathrm { P } ( W = w ) = \frac { 1 } { 5 } \quad \text { for } w = 1,2,3,4,5$$

1. Find \(\mathrm { P } ( 2 \leqslant W < 3.5 )\) The discrete random variable \(X = 5 - 2 W\)
2. Find \(\mathrm { E } ( X )\)
3. Find \(\mathrm { P } ( X < W )\) The discrete random variable \(\mathrm { Y } = \frac { 1 } { W }\)
4. Find
   1. the probability distribution of \(Y\)
   2. \(\operatorname { Var } ( Y )\), showing your working.
5. Find \(\operatorname { Var } ( 2 - 3 Y )\)

1. Jia writes a computer program that randomly generates values from a normal distribution. He sets the mean as 40 and the standard deviation as 2.4
   1. Find the probability that a particular value generated by the computer program is less than 37
   Jia changes the mean to \(m\) but leaves the standard deviation as 2.4
   The computer program then randomly generates 2 independent values from this normal distribution. The probability that both of these values are greater than 32 is 0.16
2. Find the value of \(m\), giving your answer to 2 decimal places. Jia now changes the mean to 4 and the standard deviation to 8
   The computer program then randomly generates 5 independent values from this normal distribution.
3. Find the probability that at least one of these values is negative.

1. Students on a psychology course were given a pre-test at the start of the course and a final exam at the end of the course. The teacher recorded the number of marks achieved on the pre-test, \(p\), and the number of marks achieved on the final exam, \(f\), for 34 students and displayed them on the scatter diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-22_1121_1136_447_438}

The equation of the least squares regression line for these data is found to be $$f = 10.8 + 0.748 p$$ For these students, the mean number of marks on the pre-test is 62.4

1. Use the regression model to find the mean number of marks on the final exam.
2. Give an interpretation of the gradient of the regression line. Considering the equation of the regression line, Priya says that she would expect someone who scored 0 marks on the pre-test to score 10.8 marks on the final exam.
3. Comment on the reliability of Priya's statement.
4. Write down the number of marks achieved on the final exam for the student who exceeded the expectation of the regression model by the largest number of marks.
5. Find the range of values of \(p\) for which this regression model, \(f = 10.8 + 0.748 p\), predicts a greater number of marks on the final exam than on the pre-test. Later the teacher discovers an error in the recorded data. The student who achieved a score of 98 on the pre-test, scored 92 not 29 on the final exam. The summary statistics used for the model \(f = 10.8 + 0.748 p\) are corrected to include this information and a new least squares regression line is found. Given the original summary statistics were, $$n = 34 \quad \sum p = 2120 \quad \sum p f = 133486 \quad \mathrm {~S} _ { p p } = 15573.76 \quad \mathrm {~S} _ { p f } = 11648.35$$
6. calculate the gradient of the new regression line. Show your working clearly.

7. A bag contains \(n\) marbles of which 7 are green. From the bag, 3 marbles are selected at random.
The random variable \(X\) represents the number of green marbles selected.
The cumulative distribution function of \(X\) is given by

1. Show that \(n ( n - 1 ) ( n - 2 ) = 7980\)
2. Verify that \(n = 21\) satisfies the equation in part (a). Given that \(n = 21\)
3. find the exact value of \(a\) and the exact value of \(b\)
   \includegraphics[max width=\textwidth, alt={}]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-28_2655_1947_114_116}