Edexcel S1 (Statistics 1) 2023 January

1. The histogram shows the times taken, \(t\) minutes, by each of 100 people to swim 500 metres.
   \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-02_986_1070_342_424}
   1. Use the histogram to complete the frequency table for the times taken by the 100 people to swim 500 metres.

   Time taken ( \(\boldsymbol { t }\) minutes)	\(5 - 10\)	\(10 - 14\)	\(14 - 18\)	\(18 - 25\)	\(25 - 40\)
   Frequency ( \(\boldsymbol { f }\) )	10	16	24

2. Estimate the number of people who took less than 16 minutes to swim 500 metres.
3. Find an estimate for the mean time taken to swim 500 metres. Given that \(\sum f t ^ { 2 } = 41033\)
4. find an estimate for the standard deviation of the times taken to swim 500 metres. Given that \(Q _ { 3 } = 23\)
5. use linear interpolation to estimate the interquartile range of the times taken to swim 500 metres.

1. Two bags, \(\boldsymbol { X }\) and \(\boldsymbol { Y }\), each contain green marbles (G) and blue marbles (B) only.

• Bag \(\boldsymbol { X }\) contains 5 green marbles and 4 blue marbles
• Bag \(\boldsymbol { Y }\) contains 6 green marbles and 5 blue marbles

A marble is selected at random from bag \(\boldsymbol { X }\) and placed in bag \(\boldsymbol { Y }\)
A second marble is selected at random from bag \(\boldsymbol { X }\) and placed in bag \(\boldsymbol { Y }\)
A third marble is then selected, this time from bag \(\boldsymbol { Y }\)

1. Use this information to complete the tree diagram shown on page 7
2. Find the probability that the 2 marbles selected from bag \(\boldsymbol { X }\) are of different colours.
3. Find the probability that all 3 marbles selected are the same colour. Given that all three marbles selected are the same colour,
4. find the probability that they are all green. 2nd Marble (from bag \(\boldsymbol { X }\) ) \section*{3rd Marble (from bag Y)} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{1st Marble (from bag \(\boldsymbol { X }\) )} \includegraphics[alt={},max width=\textwidth]{c316fa29-dedc-4890-bd82-31eb0bb819f9-07_1694_1312_484_310}
   \end{figure}

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

where \(a\) is a constant.

1. Find, in terms of \(a , \mathrm { E } ( X )\)
2. Find the range of the possible values of \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 0.56\)
3. find the possible values of \(a\)

1. 1. In the Venn diagram below, \(A\) and \(B\) represent events and \(p , q , r\) and \(s\) are probabilities.
      \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-12_400_789_347_639}

$$\mathrm { P } ( A ) = \frac { 7 } { 25 } \quad \mathrm { P } ( B ) = \frac { 1 } { 5 } \quad \mathrm { P } \left[ \left( A \cap B ^ { \prime } \right) \cup \left( A ^ { \prime } \cap B \right) \right] = \frac { 8 } { 25 }$$

1. Use algebra to show that \(2 p + 2 q + 2 r = \frac { 4 } { 5 }\)
2. Find the value of \(p\), the value of \(q\), the value of \(r\) and the value of \(s\)
   (ii) Two events, \(C\) and \(D\), are such that $$\mathrm { P } ( C ) = \frac { x } { x + 5 } \quad \mathrm { P } ( D ) = \frac { 5 } { x }$$ where \(x\) is a positive constant.
   By considering \(\mathrm { P } ( C ) + \mathrm { P } ( D )\) show that \(C\) and \(D\) cannot be mutually exclusive.

1. The lengths, \(L \mathrm {~mm}\), of housefly wings are normally distributed with \(L \sim \mathrm {~N} \left( 4.5,0.4 ^ { 2 } \right)\)
   1. Find the probability that a randomly selected housefly has a wing length of less than 3.86 mm .
   2. Find
      1. the upper quartile ( \(Q _ { 3 }\) ) of \(L\)
      2. the lower quartile ( \(Q _ { 1 }\) ) of \(L\)
   A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
2. Find these two outlier limits. A housefly is selected at random.
3. Using standardisation, show that the probability that this housefly is not an outlier is 0.993 to 3 decimal places. Given that this housefly is not an outlier,
4. showing your working, find the probability that the wing length of this housefly is greater than 5 mm .

1. A research student is investigating the maximum weight, \(y\) grams, of sugar that will dissolve in 100 grams of water at various temperatures, \(x ^ { \circ } \mathrm { C }\), where \(10 \leqslant x \leqslant 80\)

The research student calculated the regression line of \(y\) on \(x\) and found it to be $$y = 151.2 + 2.72 x$$

1. Give an interpretation of the gradient of the regression line.
2. Use the regression line to estimate the maximum weight of sugar that will dissolve in 100 grams of water when the temperature is \(90 ^ { \circ } \mathrm { C }\).
3. Comment on the reliability of your estimate, giving a reason for your answer. Using the regression line of \(y\) on \(x\) and the following summary statistics $$\sum y = 3119 \quad \sum y ^ { 2 } = 851093 \quad \sum x ^ { 2 } = 24500 \quad n = 12$$
4. show that the product moment correlation coefficient for these data is 0.988 to 3 decimal places. The research student's supervisor plotted the original data on a scatter diagram, shown on page 23 With reference to both the scatter diagram and the correlation coefficient,
5. discuss the suitability of a linear regression model to describe the relationship between \(x\) and \(y\).
   \includegraphics[max width=\textwidth, alt={}]{c316fa29-dedc-4890-bd82-31eb0bb819f9-23_990_1138_205_356}