Edexcel S1 (Statistics 1) 2018 January

1. Two classes of students, class \(A\) and class \(B\), sat a test.

Class \(A\) has 10 students. Class \(B\) has 15 students. Each student achieved a score, \(x\), on the test and their scores are summarised in the table below. The mean score for Class \(A\) is 77 and the mean score for Class \(B\) is 61

1. Find the value of \(t\)
2. Calculate the variance of the test scores for each class. The highest score on the test was 95 and the lowest score was 45 These were each scored by students from the same class.
3. State, with a reason, which class you believe they were from. The two classes are combined into one group of 25 students.
4. 1. Find the mean test score for all 25 students.
   2. Find the variance of the test scores for all 25 students. The teacher of class \(A\) later realises that he added up the test scores for his class incorrectly. Each student's test score in class \(A\) should be increased by 3
5. Without further calculations, state, with a reason, the effect this will have on
   1. the variance of the test scores for class \(A\)
   2. the mean test score for all 25 students
   3. the variance of the test scores for all 25 students.

2. (a) Shade the region representing the event \(A \cup B ^ { \prime }\) on the Venn diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{01259350-0119-4500-a81b-bfa1b4234559-06_355_563_306_694} The two events \(C\) and \(D\) are mutually exclusive.
Given that \(\mathrm { P } ( C ) = \frac { 1 } { 5 }\) and \(\mathrm { P } ( D ) = \frac { 3 } { 10 }\) find
(b) (i) \(\quad \mathrm { P } ( C \cup D )\)
(ii) \(\mathrm { P } ( C \mid D )\) The two events \(F\) and \(G\) are independent.
Given that \(\mathrm { P } ( F ) = \frac { 1 } { 6 }\) and \(\mathrm { P } ( F \cup G ) = \frac { 3 } { 8 }\) find
(c) (i) \(\mathrm { P } ( G )\)
(ii) \(\mathrm { P } \left( F \mid G ^ { \prime } \right)\)

3. Martin is investigating the relationship between a person's daily caffeine consumption, \(c\) milligrams, and the amount of sleep they get, \(h\) hours, per night. He collected this information from 20 people and the results are summarised below. $$\begin{array} { c c } \sum c = 3660 \quad \sum h = 126 \quad \sum c ^ { 2 } = 973228
\sum c h = 20023.4 \quad S _ { c c } = 303448 \quad S _ { c h } = - 3034.6 \end{array}$$ Martin calculates the product moment correlation coefficient for these data and obtains - 0.833

1. Give a reason why this value supports a linear relationship between \(c\) and \(h\) The amount of sleep per night is the response variable.
2. Explain what you understand by the term 'response variable'. Martin says that for each additional 100 mg of caffeine consumed, the expected number of hours of sleep decreases by 1
3. Determine, by calculation, whether or not the data support this statement.
4. Use the data to calculate an estimate for the expected number of hours of sleep per night when no caffeine is consumed.
   

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

1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\) For this probability distribution, \(\operatorname { Var } ( X ) = \mathrm { E } \left( X ^ { 2 } \right)\)
2. 1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find the value of \(a\) and the value of \(b\)
3. Find \(\operatorname { Var } ( 1 - 3 X )\) Given that \(Y = 1 - X\), find
4. 1. \(\mathrm { P } ( Y < 0 )\)
   2. the largest possible value of \(k\) such that \(\mathrm { P } ( Y < k ) = 0.2\)

5. Franca is the manager of an accountancy firm. She is investigating the relationship between the salary, \(\pounds x\), and the length of commute, \(y\) minutes, for employees at the firm. She collected this information from 9 randomly selected employees. The salary of each employee was then coded using \(w = \frac { x - 20000 } { 1000 }\) The table shows the values of \(w\) and \(y\) for the 9 employees. (You may use \(\sum w = 81 \quad \sum y = 405 \quad \sum w y = 2490 \quad S _ { w w } = 660 \quad S _ { y y } = 2500\) )

1. Calculate the salary of the employee with \(w = - 2\)
2. Show that, to 3 significant figures, the value of the product moment correlation coefficient between \(w\) and \(y\) is - 0.899
3. State, giving a reason, the value of the product moment correlation coefficient between \(x\) and \(y\) The least squares regression line of \(y\) on \(w\) is \(y = 60.75 - 1.75 w\)
4. Find the equation of the least squares regression line of \(y\) on \(x\) giving your answer in the form \(y = a + b x\)
5. Estimate the length of commute for an employee with a salary of \(\pounds 21000\) Franca uses the regression line to estimate the length of commute for employees with salaries between \(\pounds 25000\) and \(\pounds 40000\)
6. State, giving a reason, whether or not these estimates are reliable.

1. Anju has a bag that contains 5 socks of which 2 are blue.

Anju randomly selects socks from the bag, one sock at a time. She does not replace any socks but continues to select socks at random until she has both blue socks. The discrete random variable \(S\) represents the total number of socks that Anju has selected.

1. Write down the value of \(\mathrm { P } ( S = 1 )\)
2. Find \(\mathrm { P } ( S > 2 )\)
3. Find \(\mathrm { P } ( S = 3 )\)
4. Given that the second sock selected is blue, find the probability that Anju selects exactly 3 socks.
5. Find \(\mathrm { P } ( S = 5 )\)

7. The weights, \(G\), of a particular breed of gorilla are normally distributed with mean 180 kg and standard deviation 15 kg .

1. Find the proportion of these gorillas whose weights exceed 174 kg .
2. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\) The weights, \(B\), of a particular breed of buffalo are normally distributed with mean 216 kg and standard deviation 30 kg . Given that \(\mathrm { P } ( G > w ) = \mathrm { P } ( B < w ) = p\)
3. 1. find the value of \(w\)
   2. find the value of \(p\) and standard deviation 15 kg .
4. Find the proportion of these gorillas whose weights exceed 174 kg .
5. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\)

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   Q7

   
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