Calculate statistics from raw data

4 The six faces of a fair die are numbered \(1,1,1,2,3,3\). The score for a throw of the die, denoted by the random variable \(W\), is the number on the top face after the die has landed.

1. Find the mean and standard deviation of \(W\).
2. The die is thrown twice and the random variable \(X\) is the sum of the two scores. Draw up a probability distribution table for \(X\).
3. The die is thrown \(n\) times. The random variable \(Y\) is the number of times that the score is 3 . Given that \(\mathrm { E } ( Y ) = 8\), find \(\operatorname { Var } ( Y )\).

1 Rani and Diksha go shopping for clothes.

1. Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs \(
   ) 5.50\( and the coat costs \)\\( 90\). The mean cost of Rani's 8 items is \(
   ) 29\(. Find the cost of a sweater.
2. Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha's 5 items is \)\\( 26\) and the standard deviation is \(
   ) 0\(. Explain how you can tell that Diksha spends \)\\( 104\) on shirts.

1 Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded below. $$\begin{array} { l l l l l l l l l l } 32 & 35 & 45 & 37 & 38 & 44 & 33 & 39 & 36 & 45 \end{array}$$ Find the mean and standard deviation of the lengths of these leaves.

1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$

1 The following are the times, in minutes, taken by 11 runners to complete a 10 km run.
\(\begin{array} { l l l l l l l l l l l } 48.3 & 55.2 & 59.9 & 67.7 & 60.5 & 75.6 & 62.5 & 57.4 & 53.4 & 49.2 & 64.1 \end{array}\)
Find the mean and standard deviation of these times.

6 Last year Eleanor played 11 rounds of golf. Her scores were as follows: \(79 , \quad 71 , \quad 80 , \quad 67 , \quad 67 , \quad 74 , \quad 66 , \quad 65 , \quad 71 , \quad 66 , \quad 64\).

1. Calculate the mean of these scores and show that the standard deviation is 5.31 , correct to 3 significant figures.
2. Find the median and interquartile range of the scores. This year, Eleanor also played 11 rounds of golf. The standard deviation of her scores was 4.23, correct to 3 significant figures, and the interquartile range was the same as last year.
3. Give a possible reason why the standard deviation of her scores was lower than last year although her interquartile range was unchanged. In golf, smaller scores mean a better standard of play than larger scores. Ken suggests that since the standard deviation was smaller this year, Eleanor's overall standard has improved.
4. Explain why Ken is wrong.
5. State what the smaller standard deviation does show about Eleanor's play.

7 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.

1. Calculate the mean GCSE score, \(X\), of a student who has the following GCSE grades: $$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$ 60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.

   Mean GCSE score	Number of students
   \(4.5 \leqslant X < 5.5\)	8
   \(5.5 \leqslant X < 6.0\)	14
   \(6.0 \leqslant X < 6.5\)	19
   \(6.5 \leqslant X < 7.0\)	13
   \(7.0 \leqslant X \leqslant 8.0\)	6

2. Draw a histogram to illustrate this information.
3. Calculate estimates of the sample mean and the sample standard deviation. The scoring system for AS grades is shown in the table below.

   AS Grade	A	B	C	D	E	U
   Score	60	50	40	30	20	0

   The Mathematics department at the college predicts each student's AS score, \(Y\), using the formula \(Y = 13 X - 46\), where \(X\) is the student's average GCSE score.
4. What AS grade would the department predict for a student with an average GCSE score of 7.4 ?
5. What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.
6. Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.

3 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.

1. Calculate the mean GCSE score, \(X\), of a student who has the following GCSE grades: $$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$ 60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.

   Mean GCSE score	Number of students
   \(4.5 \leqslant X < 5.5\)	8
   \(5.5 \leqslant X < 6.0\)	14
   \(6.0 \leqslant X < 6.5\)	19
   \(6.5 \leqslant X < 7.0\)	13
   \(7.0 \leqslant X \leqslant 8.0\)	6

2. Draw a histogram to illustrate this information.
3. Calculate estimates of the sample mean and the sample standard deviation. The scoring system for AS grades is shown in the table below.

   AS Grade	A	B	C	D	E	U
   Score	60	50	40	30	20	0

   The Mathematics department at the college predicts each student's AS score, \(Y\), using the formula \(Y = 13 X - 46\), where \(X\) is the student's average GCSE score.
4. What AS grade would the department predict for a student with an average GCSE score of 7.4 ?
5. What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.
6. Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.

2. Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below. $$\text { [You may use } \sum x ^ { 2 } = 724 \text { 961] }$$

1. Find the mean and standard deviation of the level of cotinine in a patient's blood.
2. Find the median, upper and lower quartiles of these data. A doctor suspects that some of his patients have been smoking more than a packet of cigarettes per day. He decides to use \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) to determine if any of the cotinine results are far enough away from the upper quartile to be outliers.
3. Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly. Research suggests that cotinine levels in the blood form a skewed distribution.
   One measure of skewness is found using \(\frac { \left( Q _ { 1 } - 2 Q _ { 2 } + Q _ { 3 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }\).
4. Evaluate this measure and describe the skewness of these data.

4. The attendance at college of a group of 18 students was recorded for a 4-week period. The number of students actually attending each of 16 classes are shown below.

1. 1. Calculate the mean and the standard deviation of the number of students attending these classes.
   2. Express the mean as a percentage of the 18 students in the group. In the same 4-week period, the attendance of a different group of 20, students is shown below.

      20	16	18	19
      15	14	14	15
      18	15	16	17
      16	18	15	14

2. Construct a back-to-back stem and leaf diagram to represent the attendance in both groups.
3. Find the mode, median and inter-quartile range for each group of students. The mean percentage attendance and standard deviation for the second group of students are 81.25 and 1.82 respectively.
4. Compare and contrast the attendance of these 2 groups of students.

4. The marks, \(x\) out of 100 , scored by 30 candidates in an examination were as follows: Given that \(\sum x = 1600\) and \(\sum x ^ { 2 } = 102400\),

1. find the median, the mean and the standard deviation of these marks. The marks were scaled to give modified scores, \(y\), using the formula \(y = \frac { 4 x } { 5 } + 20\).
2. Find the median, the mean and the standard deviation of the modified scores. \section*{STATISTICS 1 (A) TEST PAPER 1 Page 2}

4. The heights, \(h \mathrm {~m}\), of eight children were measured, giving the following values of \(h\) : $$1 \cdot 20,1 \cdot 12,1 \cdot 43,0 \cdot 98,1 \cdot 31,1 \cdot 26,1 \cdot 02,1 \cdot 41$$

1. Find the mean height of the children.
2. Calculate the variance of the heights. The children were also weighed. It was found that their masses, \(w \mathrm {~kg}\), were such that $$\sum w = 324 , \quad \sum w ^ { 2 } = 13532 , \quad \sum w h = 403 .$$
3. Calculate the product-moment correlation coefficient between \(w\) and \(h\).
4. Comment briefly on the value you have obtained.

1 The times, in seconds, taken by 20 people to solve a simple numerical puzzle were

1. Calculate the mean and the standard deviation of these times.
2. In fact, 23 people solved the puzzle. However, 3 of them failed to solve it within the allotted time of 60 seconds. Calculate the median and the interquartile range of the times taken by all 23 people.
   (4 marks)
3. For the times taken by all 23 people, explain why:
   1. the mode is not an appropriate numerical measure;
   2. the range is not an appropriate numerical measure.

11 The table below shows the temperature on Mount Everest on the first day of each month. Calculate the standard deviation of these temperatures.
Circle your answer.
-6.75
5.82
8.24
67.85 \begin{center} \begin{tabular}{|l|l|l|l|} \hline \multicolumn{4}{|c|}{\multirow[t]{2}{*}{\begin{tabular}{l}