Edexcel S1 (Statistics 1) 2017 January

1. Ralph records the weights, in grams, of 100 tomatoes. This information is displayed in the histogram below.
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Given that 5 of the tomatoes have a weight between 2 and 3 grams,

1. find the number of tomatoes with a weight between 0 and 2 grams. One of the tomatoes is selected at random.
2. Find the probability that it weighs more than 3 grams.
3. Estimate the proportion of the tomatoes with a weight greater than 6.25 grams.
4. Using your answer to part (c), explain whether or not the median is greater than 6.25 grams. Given that the mean weight of these tomatoes is 6.25 grams and using your answer to part (d),
5. describe the skewness of the distribution of the weights of these tomatoes. Give a reason for your answer. Two of these 100 tomatoes are selected at random.
6. Estimate the probability that both tomatoes weigh within 0.75 grams of the mean.

1. An integer is selected at random from the integers 1 to 50 inclusive.
   \(A\) is the event that the integer selected is prime.
   \(B\) is the event that the integer selected ends in a 3
   \(C\) is the event that the integer selected is greater than 20
   The Venn diagram shows the number of integers in each region for the events \(A , B\) and \(C\)
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   1. Describe in words the event \(( A \cap B )\)
   2. Write down the probability that the integer selected is prime.
   3. Find \(\mathrm { P } \left( [ A \cup B \cup C ] ^ { \prime } \right)\)
   Given that the integer selected is greater than 20
2. find the probability that it is prime. Using your answers to (b) and (d),
3. state, with a reason, whether or not the events \(A\) and \(C\) are statistically independent. Given that the integer selected is greater than 20 and prime,
4. find the probability that it ends in a 3

1. A scientist measured the salinity of water, \(x \mathrm {~g} / \mathrm { kg }\), and recorded the temperature at which the water froze, \(y ^ { \circ } \mathrm { C }\), for 12 different water samples. The summary statistics are listed below.

$$\begin{gathered} \sum x = 504 \quad \sum y = - 27 \quad \sum x ^ { 2 } = 22842 \quad \sum y ^ { 2 } = 62.98
\sum x y = - 1190.7 \quad \mathrm {~S} _ { x x } = 1674 \quad \mathrm {~S} _ { y y } = 2.23 \end{gathered}$$

1. Find the mean and variance of the recorded temperatures.
   (3) Priya believes that the higher the salinity of water, the higher the temperature at which the water freezes.
2. 1. Calculate the product moment correlation coefficient between \(x\) and \(y\)
   2. State, with a reason, whether or not this value supports Priya's belief.
3. Find the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\) Give the value of \(a\) and the value of \(b\) to 3 significant figures.
4. Estimate the temperature at which water freezes when the salinity is \(32 \mathrm {~g} / \mathrm { kg }\) The coding \(w = 1.8 y + 32\) is used to convert the recorded temperatures from \({ } ^ { \circ } \mathrm { C }\) to \({ } ^ { \circ } \mathrm { F }\)
5. Find an equation of the least squares regression line of \(w\) on \(x\) in the form \(w = c + d x\)
6. Find
   1. the variance of the recorded temperatures when converted to \({ } ^ { \circ } \mathrm { F }\)
   2. the product moment correlation coefficient between \(w\) and \(x\)
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1. In a game, the number of points scored by a player in the first round is given by the random variable \(X\) with probability distribution

Find

1. \(\mathrm { E } ( X )\)
2. \(\operatorname { Var } ( X )\)
3. \(\operatorname { Var } ( 3 - 2 X )\) The number of points scored by a player in the second round is given by the random variable \(Y\) and is independent of the number of points scored in the first round. The random variable \(Y\) has probability function $$\mathrm { P } ( Y = y ) = \frac { 1 } { 4 } \quad \text { for } y = 5,6,7,8$$
4. Write down the value of \(\mathrm { E } ( Y )\)
5. Find \(\mathrm { P } ( X = Y )\)
6. Find the probability that the number of points scored by a player in the first round is greater than the number of points scored by the player in the second round.

1. In a survey, people were asked if they use a computer every day.

Of those people under 50 years old, \(80 \%\) said they use computer every day. Of those people aged 50 or more, \(55 \%\) said they use computer every day. The proportion of people in the survey under 50 years old is \(p\)

1. Draw a tree diagram to represent this information. In the survey, 70\% of all people said they use computer every day.
2. Find the value of \(p\) One person is selected at random. Given that this person uses a computer every day,
3. find the probability that this person is under 50 years old.
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1. In a factory, a machine is used to fill bags of rice. The weights of bags of rice are modelled using a normal distribution with mean 250 g .

Only \(1 \%\) of the bags of rice weigh more than 256 g .

1. Write down the percentage of bags of rice with weights between 244 g and 256 g .
2. Find the standard deviation of the weights of the bags of rice. An inspection consists of selecting a bag of rice at random and checking if its weight is within 4 g of the mean. If the weight is more than 4 g away from the mean, then a second bag of rice is selected at random and checked. If the weight of each of the 2 bags of rice is more than 4 g away from the mean, then the machine is shut down.
3. Find the probability that the machine is shut down following an inspection.

1. The discrete random variable \(X\) can take only the values \(1,2,3\) and 4 . For these values, the probability function is given by

$$\mathrm { P } ( X = x ) = \frac { a x + b } { 60 } \quad \text { for } x = 1,2,3,4$$ where \(a\) and \(b\) are constants.

1. Show that \(5 a + 2 b = 30\) Given that \(\mathrm { F } ( 3 ) = \frac { 13 } { 20 }\)
2. find the value of \(a\) and the value of \(b\) Given also that \(Y = X ^ { 2 }\)
3. find the cumulative distribution function of \(Y\)