2.02g Calculate mean and standard deviation

The birth weights, in kg, of 1500 babies are summarised in the table below.

Weight (kg)	Midpoint, \(x\)kg	Frequency, \(f\)
\(0.0 - 1.0\)	\(0.50\)	\(1\)
\(1.0 - 2.0\)	\(1.50\)	\(6\)
\(2.0 - 2.5\)	\(2.25\)	\(60\)
\(2.5 - 3.0\)		\(280\)
\(3.0 - 3.5\)	\(3.25\)	\(820\)
\(3.5 - 4.0\)	\(3.75\)	\(320\)
\(4.0 - 5.0\)	\(4.50\)	\(10\)
\(5.0 - 6.0\)		\(3\)

[You may use \(\sum fx = 4841\) and \(\sum fx^2 = 15889.5\)]

1. Write down the missing midpoints in the table above. [2]
2. Calculate an estimate of the mean birth weight. [2]
3. Calculate an estimate of the standard deviation of the birth weight. [3]
4. Use interpolation to estimate the median birth weight. [2]
5. Describe the skewness of the distribution. Give a reason for your answer. [2]

A class of students had a sudoku competition. The time taken for each student to complete the sudoku was recorded to the nearest minute and the results are summarised in the table below. (You may use \(\sum fx^2 = 8603.75\))

1. Write down the mid-point for the 9 - 12 interval. [1]
2. Use linear interpolation to estimate the median time taken by the students. [2]
3. Estimate the mean and standard deviation of the times taken by the students. [5]

The teacher suggested that a normal distribution could be used to model the times taken by the students to complete the sudoku.

1. Give a reason to support the use of a normal distribution in this case. [1]

On another occasion the teacher calculated the quartiles for the times taken by the students to complete a different sudoku and found \(Q_1 = 8.5 \quad Q_2 = 13.0 \quad Q_3 = 21.0\)

1. Describe, giving a reason, the skewness of the times on this occasion. [2]

The following stem and leaf diagram shows the aptitude scores \(x\) obtained by all the applicants for a particular job.

1. Write down the modal aptitude score. [1]
2. Find the three quartiles for these data. [3]

Outliers can be defined to be outside the limits \(Q_1 - 1.0(Q_3 - Q_1)\) and \(Q_3 + 1.0(Q_3 - Q_1)\).

1. On a graph paper, draw a box plot to represent these data. [7]

For these data, \(\Sigma x = 3363\) and \(\Sigma x^2 = 238305\).

1. Calculate, to 2 decimal places, the mean and the standard deviation for these data. [3]
2. Use two different methods to show that these data are negatively skewed. [4]

A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3. The daisies are distributed randomly throughout the field. Find the probability that, in a randomly chosen square there will be

1. more than 2 daisies, [3]
2. either 5 or 6 daisies. [2]

The botanist decides to count the number of daisies, \(x\), in each of 80 randomly selected squares within the field. The results are summarised below $$\sum x = 295 \quad \sum x^2 = 1386$$

1. Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places. [3]
2. Explain how the answers from part (c) support the choice of a Poisson distribution as a model. [1]
3. Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square. [2]

A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:

1. Show that the mean number of errors per page in this sample of pages is 2. [2]
2. Find the variance of the number of errors per page in this sample. [2]
3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution. [1]

Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2.

1. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a 5\% level of significance. [6]

The continuous random variable \(X\) has probability density function f(\(x\)) given by $$\text{f}(x) = \begin{cases} \frac{1}{20}x^3, & 1 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases}$$

1. Sketch f(\(x\)) for all values of \(x\). [3]
2. Calculate E(\(X\)). [3]
3. Show that the standard deviation of \(X\) is 0.459 to 3 decimal places. [3]
4. Show that for \(1 \leq x \leq 3\), P(\(X \leq x\)) is given by \(\frac{1}{80}(x^4 - 1)\) and specify fully the cumulative distribution function of \(X\). [5]
5. Find the interquartile range for the random variable \(X\). [4]

Some statisticians use the following formula to estimate the interquartile range: $$\text{interquartile range} = \frac{4}{3} \times \text{standard deviation}.$$

1. Use this formula to estimate the interquartile range in this case, and comment. [2]

The variable \(X\) represents the marks out of 150 scored by a group of students in an examination. The following ten values of \(X\) were obtained: 60, 66, 76, 80, 94, 106, 110, 116, 124, 140.

1. Write down the median, \(M\), of the ten marks. [1 mark]
2. Using the coding \(y = \frac{x - M}{2}\), and showing all your working clearly, find the mean and the standard deviation of the marks. [6 marks]
3. Find E\((3X - 5)\). [3 marks]

The heights, \(h\) m, of eight children were measured, giving the following values of \(h\): 1.20, 1.12, 1.43, 0.98, 1.31, 1.26, 1.02, 1.41.

1. Find the mean height of the children. [2 marks]
2. Calculate the variance of the heights. [3 marks]

The children were also weighed. It was found that their masses, \(w\) kg, were such that $$\sum w = 324, \quad \sum w^2 = 13532, \quad \sum wh = 403.$$

1. Calculate the product-moment correlation coefficient between \(w\) and \(h\). [4 marks]
2. Comment briefly on the value you have obtained. [1 mark]

Using the coding \(y = \frac{x-90}{5}\), and showing each step in your working clearly, calculate the mean and the standard deviation of the 20 observations of a variable \(X\) given by the following table:

\(x\)	75	80	85	90	95	100	105	110
Frequency	1	2	3	6	4	2	1	1

[8 marks]

The following table gives the weights, in grams, of 60 items delivered to a company in a day.

Weight (g)	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 80
No. of items	2	11	18	12	9	6	2

1. Use interpolation to calculate estimated values of
   1. the median weight,
   2. the interquartile range,
   3. the thirty-third percentile.
   [7 marks]

Outliers are defined to be outside the range from \(2.5Q_1 - 1.5Q_2\) to \(2.5Q_2 - 1.5Q_1\).

1. Given that the lightest item weighed 3 g and the two heaviest weighed 65 g and 79 g, draw on graph paper an accurate box-and-whisker plot of the data. Indicate any outliers clearly. [5 marks]
2. Describe the skewness of the distribution. [1 mark]

The mean weight was 32.0 g and the standard deviation of the weights was 14.9 g.

1. State, with a reason, whether you would choose to summarise the data by using the mean and standard deviation or the median and interquartile range. [2 marks]

On another day, items were delivered whose weights ranged from 14 g to 58 g; the median was 32 g, the lower quartile was 24 g and the interquartile range was 26 g.

1. Draw a further box plot for these data on the same diagram. Briefly compare the two sets of data using your plots. [6 marks]

40 people were asked to guess the length of a certain road. Each person gave their guess, \(l\) km, correct to the nearest kilometre. The results are summarised below.

\(l\)	10-12	13-15	16-20	21-30
Frequency	1	13	20	6

1. 1. Use appropriate formulae to calculate estimates of the mean and standard deviation of \(l\). [6]
   2. Explain why your answers are only estimates. [1]
2. A histogram is to be drawn to illustrate the data. Calculate the frequency density of the block for the 16-20 class. [2]
3. Explain which class contains the median value of \(l\). [2]
4. Later, the person whose guess was between 10 km and 12 km changed his guess to between 13 km and 15 km. Without calculation state whether the following will increase, decrease or remain the same:
   1. the mean of \(l\), [1]
   2. the standard deviation of \(l\). [1]

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

1. Calculate the mean of these scores and show that the standard deviation is 5.31, correct to 3 significant figures. [4]
2. Find the median and interquartile range of the scores. [4]

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.

1. Give a possible reason why the standard deviation of her scores was lower than last year although her interquartile range was unchanged. [1]

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.

1. Explain why Ken is wrong. [1]
2. State what the smaller standard deviation does show about Eleanor's play. [1]

At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m\) kg, and for each person the value of \((m - 5)\) was recorded. The mean and standard deviation of \((m - 5)\) were found to be 0.74 and 0.13 respectively.

1. Write down the mean and standard deviation of \(m\). [2]

The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.

1. Calculate the mean of all 25 estimates. [2]
2. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg, comment on this claim. [2]

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{figure_7}

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

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. [5]
2. Calculate an estimate of the mean income. [3]
3. Calculate an estimate of the standard deviation of the incomes. [4]
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. [4]
5. The incomes were converted into another currency using the formula \(y = 1.15x\). Calculate estimates of the mean and variance of the incomes in the new currency. [3]

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

Number of eggs	1	2	3	4
Frequency	10	40	15	5

1. Find the mean and standard deviation of the number of eggs laid per gull. [5]
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? [2]

The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

Heating quality (\(x\))	9.1 \(\leqslant x <\) 9.3	9.3 \(< x \leqslant\) 9.5	9.5 \(< x \leqslant\) 9.7	9.7 \(< x \leqslant\) 9.9	9.9 \(< x \leqslant\) 10.1
Frequency	5	7	15	16	7

1. Draw a cumulative frequency diagram to illustrate these data. [5]
2. Use the diagram to estimate the median and interquartile range of the data. [3]
3. Show that there are no outliers in the sample. [3]
4. Three of these 50 sacks are selected at random. Find the probability that
   1. in all three, the heating quality \(x\) is more than 9.5, [3]
   2. in at least two, the heating quality \(x\) is more than 9.5. [4]

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

Weight	\(30 \leqslant w < 50\)	\(50 \leqslant w < 60\)	\(60 \leqslant w < 70\)	\(70 \leqslant w < 80\)	\(80 \leqslant w < 90\)
Frequency	11	10	18	14	7

1. Draw a histogram to illustrate these data. [5]
2. Calculate estimates of the mean and standard deviation of \(w\). [4]
3. Use your answers to part (ii) to investigate whether there are any outliers. [3]

The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x^2 = 265416$$

1. Calculate the mean and standard deviation of \(x\). [3]
2. Compare the central tendency and variation of the weights of varieties A and B. [2]

A group of 60 children were each asked to choose an integer value between 1 and 9 inclusive. Their choices are summarised in the table below.

Value chosen	1	2	3	4	5	6	7	8	9
Number of children	3	4	5	10	12	13	7	4	2

1. Calculate the mean and standard deviation of the values chosen. [6]

It is suggested that the value chosen could be modelled by a discrete uniform distribution.

1. Write down the mean that this model would predict. [2]

Given also that the standard deviation according to this model would be 2.58,

1. explain why this model is not suitable and suggest why this is the case. [2]

Jane and Tahira play together in a basketball team. The list below shows the number of points that Jane scored in each of 30 games.

1. Construct a stem and leaf diagram for these data. [3 marks]
2. Find the median and quartiles for these data. [4 marks]
3. Represent these data with a boxplot. [3 marks]

Tahira played in the same 30 games and her lowest and highest points total in a game were 19 and 41 respectively. The quartiles for Tahira were 27, 31 and 35 respectively.

1. Using the same scale draw a boxplot for Tahira's points totals. [2 marks]
2. Compare and contrast the number of points scored per game by Jane and Tahira. [3 marks]

A College offers evening classes in GCSE Mathematics and English. In order to assess which age groups were reluctant to use the classes, the College collected data on the age in completed years of those currently attending each course. The results are shown in this back-to-back stem and leaf diagram. \includegraphics{figure_4} Key: \(1 | 3 | 2\) means age 31 doing Mathematics and age 32 doing English

1. Find the median and quartiles of the age in completed years of those attending the Mathematics classes. [4 marks]
2. On graph paper, draw a box plot representing the data for the Mathematics class. [3 marks]

The median and quartiles of the age in completed years of those attending the English classes are 25, 41 and 57 years respectively.

1. Draw a box plot representing the data for the English class using the same scale as for the data from the Mathematics class. [3 marks]
2. Using your box plots, compare and contrast the ages of those taking each class. [4 marks]

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. [2]

The weights, \(x\) grams, of a random sample of 25 bags are now recorded.

1. Given that \(\sum x = 11409\) and \(\sum x^2 = 5206937\), calculate the sample mean and sample standard deviation of these weights. [3]

A sprinter runs many 100-metre trials, and the time, \(x\) seconds, for each is recorded. A sample of eight of these times is taken, as follows. 10.53 \quad 10.61 \quad 10.04 \quad 10.49 \quad 10.63 \quad 10.55 \quad 10.47 \quad 10.63

1. Calculate the sample mean, \(\bar{x}\), and sample standard deviation, \(s\), of these times. [3]
2. Show that the time of 10.04 seconds may be regarded as an outlier. [2]
3. Discuss briefly whether or not the time of 10.04 seconds should be discarded. [2]

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

Weight	\(30 \leqslant w < 50\)	\(50 \leqslant w < 60\)	\(60 \leqslant w < 70\)	\(70 \leqslant w < 80\)	\(80 \leqslant w < 90\)
Frequency	11	10	18	14	7

1. Draw a histogram to illustrate these data. [5]
2. Calculate estimates of the mean and standard deviation of \(w\). [4]
3. Use your answers to part (ii) to investigate whether there are any outliers. [3]

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 \(\sum x^2 = 265416\)

1. Calculate the mean and standard deviation of \(x\). [3]
2. Compare the central tendency and variation of the weights of varieties A and B. [2]

In a packet of 40 biscuits, the number of currants in each biscuit is as follows

1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit. [4 marks]
2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. [2 marks]

Another machine produces biscuits with a mean of 1.9 currants per biscuit.

1. Determine which machine is more likely to produce a biscuit with at least two currants. [5 marks]