2.02g Calculate mean and standard deviation

4 A company sells sugar in bags which are labelled as containing 450 grams.
Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.

1. State two adjustments the company could make. The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
2. Given that \(\sum x = 11409\) and \(\sum x ^ { 2 } = 5206937\), calculate the sample mean and sample standard deviation of these weights.

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.

2 The numbers of absentees per day from Mrs Smith's reception class over a period of 50 days are summarised below.

1. Illustrate these data by means of a vertical line chart.
2. Calculate the mean and root mean square deviation of these data.
3. There are 30 children in Mrs Smith's class altogether. Find the mean and root mean square deviation of the number of children who are present during the 50 days.

4 A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.2 of finding oil, independently of all other sites. The random variable \(X\) represents the number of sites investigated. The probability distribution of \(X\) is shown below.

1. Find the expectation and variance of \(X\).
2. It costs \(\pounds 45000\) to investigate each site. Find the expected total cost of the investigation.
3. Draw a suitable diagram to illustrate the distribution of \(X\).

6 The maximum temperatures \(x\) degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below. These data are summarised by \(n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56\).

1. Calculate the mean and standard deviation of the data.
2. Determine whether there are any outliers.
3. The formula \(y = 1.8 x + 32\) is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
4. In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.

   Hours \(h\)	\(70 \leqslant h < 100\)	\(100 \leqslant h < 110\)	\(110 \leqslant h < 120\)	\(120 \leqslant h < 150\)	\(150 \leqslant h < 170\)	\(170 \leqslant h < 190\)
   Number of years	6	8	10	11	10	3

5. Draw a cumulative frequency graph for these data.
6. Use your graph to estimate the 90th percentile.

1 At a certain stage of a football league season, the numbers of goals scored by a sample of 20 teams in the league were as follows. \(\begin{array} { l l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}\)

1. Calculate the sample mean and sample variance, \(s ^ { 2 }\), of these data.
2. The three teams with the most goals appear to be well ahead of the other teams. Determine whether or not any of these three pieces of data may be considered outliers.

3 Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution $$\mathrm { P } ( X = r ) = \frac { 1 } { 40 } r ( r + 1 ) \quad \text { for } r = 1,2,3,4 .$$

1. Verify that \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 2 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days.

1 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

1. Draw a vertical line chart to illustrate the data.
2. State the type of skewness shown by your diagram.
3. Calculate the mean and the mean squared deviation of the data.
4. How many correct answers would George need to average over the next 6 days if he is to achieve an average of 5 correct answers for all 31 days of January?

3 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$

1. Calculate the mean and standard deviation of the marks.
2. The highest mark scored by any of the 56 students in the examination was 93 . Show that this result may be considered to be an outlier.
3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.

1 In a survey, a sample of 44 fields is selected. Their areas ( \(x\) hectares) are summarised in the grouped frequency table.

1. Calculate an estimate of the sample mean and the sample standard deviation.
2. Determine whether there could be any outliers at the upper end of the distribution.

4 The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.

1. Draw a histogram to illustrate these data.
2. Calculate estimates of the mean and standard deviation of \(w\).
3. Use your answers to part (ii) to investigate whether there are any outliers. The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x ^ { 2 } = 265416$$
4. Calculate the mean and standard deviation of \(x\).
5. Compare the central tendency and variation of the weights of varieties A and B .

2 The engine sizes \(x \mathrm {~cm} ^ { 3 }\) of a sample of 80 cars are summarised in the table below.

1. Draw a histogram to illustrate the distribution.
2. A student claims that the midrange is \(2750 \mathrm {~cm} ^ { 3 }\). Discuss briefly whether he is likely to be correct.
3. Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
4. Hence investigate whether there are any outliers in the sample.
5. A vehicle duty of \(\pounds 1000\) is proposed for all new cars with engine size greater than \(2000 \mathrm {~cm} ^ { 3 }\). Assuming that this sample of cars is representative of all new cars in Britain and that there are 2.5 million new cars registered in Britain each year, calculate an estimate of the total amount of money that this vehicle duty would raise in one year.
6. Why in practice might your estimate in part (v) turn out to be too high?

4 The incomes of a sample of 918 households on an island are given in the table below.

1. Draw a histogram to illustrate the data.
2. Calculate an estimate of the mean income.
3. Calculate an estimate of the standard deviation of the incomes.
4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation.
5. The incomes were converted into another currency using the formula \(y = 1.15 x\). Calculate estimates of the mean and variance of the incomes in the new currency.

3 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears. \includegraphics[max width=\textwidth, alt={}, center]{56f1bd5c-4b45-4e36-a324-e7e0edbb5bdd-2_825_1634_467_295}

1. Calculate the number of pears which are between 90 and 100 mm long.
2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
3. Calculate an estimate of the standard deviation.
4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
5. Name the type of skewness of the distribution.
6. Illustrate the data using a cumulative frequency diagram.

1 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-1_1052_1647_549_289}

1. Use the diagram to estimate the median and interquartile range of the data.
2. Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
3. Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
4. Copy and complete the frequency table below for these data.

   Temperature \(( t\) degrees Celsius \()\)	\(3.0 \leqslant t \leqslant 3.4\)	\(3.4 < t \leqslant 3.8\)	\(3.8 < t \leqslant 4.2\)	\(4.2 < t \leqslant 4.6\)	\(4.6 < t \leqslant 5.0\)
   Frequency			243	157

5. Use your table to calculate an estimate of the mean.
6. The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula \(F = 1.8 C + 32\). Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.

4 The numbers of absentees per day from Mrs Smith's reception class over a period of 50 days are summarised below.

1. Illustrate these data by means of a vertical line chart.
2. Calculate the mean and root mean square deviation of these data.
3. There are 30 children in Mrs Smith's class altogether. Find the mean and root mean square deviation of the number of children who are present during the 50 days.

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.

4 At a certain stage of a football league season, the numbers of goals scored by a sample of 20 teams in the league were as follows. \(\begin{array} { l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}\)

1. Calculate the sample mean and sample variance, \(s ^ { 2 }\), of these data.
2. The three teams with the most goals appear to be well ahead of the other teams. Determine whether or not any of these three pieces of data may be considered outliers.

3 The stem and leaf diagram illustrates the heights in metres of 25 young oak trees. Key: 4 |2 represents 4.2

1. State the type of skewness of the distribution.
2. Use your calculator to find the mean and standard deviation of these data.
3. Determine whether there are any outliers.

1 The maximum temperatures \(x\) degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below. These data are summarised by \(n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56\).

1. Calculate the mean and standard deviation of the data.
2. Determine whether there are any outliers.
3. The formula \(y = 1.8 x + 32\) is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
4. In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.

   Hours \(h\)	\(70 \leqslant h < 100\)	\(100 \leqslant h < 110\)	\(110 \leqslant h < 120\)	\(120 \leqslant h < 150\)	\(150 \leqslant h < 170\)	\(170 \leqslant h < 190\)
   Number of years	6	8	10	11	10	3

5. Draw a cumulative frequency graph for these data.
6. Use your graph to estimate the 90th percentile.

1 The amounts of electricity, \(x \mathrm { kWh }\) (kilowatt hours), used by 40 households in a three-month period are summarised as follows. $$n = 40 \quad \sum x = 59972 \quad \sum x ^ { 2 } = 96767028$$

1. Calculate the mean and standard deviation of \(x\).
2. The formula \(y = 0.163 x + 14.5\) gives the cost in pounds of the electricity used by each household. Use your answers to part (i) to deduce the mean and standard deviation of the costs of the electricity used by these 40 households.

1 The hourly wages, \(\pounds x\), of a random sample of 60 employees working for a company are summarised as follows. $$n = 60 \quad \sum x = 759.00 \quad \sum x ^ { 2 } = 11736.59$$

1. Calculate the mean and standard deviation of \(x\).
2. The workers are offered a wage increase of \(2 \%\). Use your answers to part (i) to deduce the new mean and standard deviation of the hourly wages after this increase.
3. As an alternative the workers are offered a wage increase of 25 p per hour. Write down the new mean and standard deviation of the hourly wages after this 25p increase.

3 The numbers of eggs laid by a sample of 70 female herring gulls are shown in the table.

1. Find the mean and standard deviation of the number of eggs laid per gull.
2. The sample did not include female herring gulls that laid no eggs. How would the mean and standard deviation change if these gulls were included?

6 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.

1. Calculate the mean and standard deviation of the numbers of loaves bought per person.
2. Each loaf costs \(\pounds 1.04\). Calculate the mean and standard deviation of the amount spent on loaves per person.