2.02g Calculate mean and standard deviation

4 The table shows information about the time, \(t\) minutes correct to the nearest minute, taken by 50 people to complete a race.

1. In a histogram illustrating the data, the height of the block for the \(31 \leqslant t \leqslant 35\) class is 5.6 cm . Find the height of the block for the \(28 \leqslant t \leqslant 30\) class. (There is no need to draw the histogram.)
2. The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
3. Calculate estimates of the mean and standard deviation of the data.
4. It was found that the winner's time had been incorrectly recorded and that it was actually less than 27 minutes 30 seconds. State whether each of the following will increase, decrease or remain the same:
   1. the mean,
   2. the standard deviation,
   3. the median,
   4. the interquartile range.

2 The masses, \(x \mathrm {~kg}\), of 50 bags of flour were measured and the results were summarised as follows. $$n = 50 \quad \Sigma ( x - 1.5 ) = 1.4 \quad \Sigma ( x - 1.5 ) ^ { 2 } = 0.05$$ Calculate the mean and standard deviation of the masses of these bags of flour.

3 The table shows information about the numbers of people per household in 280900 households in the northwest of England in 2001.

1. Taking ' 5 or more' to mean ' 5 or 6 ', calculate estimates of the mean and standard deviation of the number of people per household.
2. State the values of the median and upper quartile of the number of people per household.

2 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 Dwayne is a car salesman. The numbers of cars, \(x\), sold by Dwayne each month during the year 2008 are summarised by $$n = 12 , \quad \Sigma x = 126 , \quad \Sigma x ^ { 2 } = 1582 .$$

1. Calculate the mean and standard deviation of the monthly numbers of cars sold.
2. Dwayne earns \(\pounds 500\) each month plus \(\pounds 100\) commission for each car sold. Show that the mean of Dwayne's monthly earnings is \(\pounds 1550\). Find the standard deviation of Dwayne's monthly earnings.
3. Marlene is a car saleswoman and is paid in the same way as Dwayne. During 2008 her monthly earnings have mean \(\pounds 1625\) and standard deviation \(\pounds 280\). Briefly compare the monthly numbers of cars sold by Marlene and Dwayne during 2008.

5 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.

6 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?

1 The weights, \(x\) grams, of 100 potatoes are summarised as follows. $$n = 100 \quad \sum x = 24940 \quad \sum x ^ { 2 } = 6240780$$

1. Calculate the mean and standard deviation of \(x\).
2. The weights, \(y\) grams, of the potatoes after they have been peeled are given by the formula \(y = 0.9 x - 15\). Deduce the mean and standard deviation of the weights of the potatoes after they have been peeled.

2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.

1. State the exact distribution of \(X\), the number of four-leaf clovers in the sample.
2. Explain why \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
4. Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
5. Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.

   Number of four-leaf clovers	0	1	2	\(> 2\)
   Number of samples	11	7	2	0

6. Calculate the mean and variance of the data in the table.
7. Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).

2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.

1. Show that the sample mean is 1.84 and calculate the sample variance.
2. Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit. For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
3. Find the probability of
   (A) no sultanas in a biscuit,
   (B) at least two sultanas in a biscuit.
4. Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
5. Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
6. Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.

3 The masses, in kilograms, of a random sample of 100 chickens on sale in a large supermarket were recorded as follows.

1. Assuming that the first and last classes are the same width as the other classes, calculate an estimate of the sample mean and show that the corresponding estimate of the sample standard deviation is 0.2227 kg . A Normal distribution using the mean and standard deviation found in part (i) is to be fitted to these data. The expected frequencies for the classes are as follows.

   Mass \(( m \mathrm {~kg} )\)	\(m < 1.6\)	\(1.6 \leqslant m < 1.8\)	\(1.8 \leqslant m < 2.0\)	\(2.0 \leqslant m < 2.2\)	\(2.2 \leqslant m < 2.4\)	\(2.4 \leqslant m < 2.6\)	\(2.6 \leqslant m\)
   Expected frequency	2.17	10.92	\(f\)	33.85	19.22	5.13	0.68

2. Use the Normal distribution to find \(f\).
3. Carry out a goodness of fit test of this Normal model using a significance level of 5\%.
4. Discuss the outcome of the test with reference to the contributions to the test statistic and to the possibility of other significance levels.

1 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.

1. Find the mean and standard deviation of the prize money per ticket.
2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.

6 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]{7b92607f-1bf9-45f0-997b-fe76c88b5fcd-4_1054_1649_539_248}

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.

1 The stem and leaf diagram illustrates the weights in grams of 20 house sparrows. Key: \(\quad 27 \quad \mid \quad 7 \quad\) represents 27.7 grams

1. Find the median and interquartile range of the data.
2. Determine whether there are any outliers.

6 An online store has a total of 930 different types of women's running shoe on sale. The prices in pounds of the types of women's running shoe are summarised in the table below.

1. Calculate estimates of the mean and standard deviation of the shoe prices.
2. Calculate an estimate of the percentage of types of shoe that cost at least \(\pounds 100\).
3. Draw a histogram to illustrate the data. The corresponding histogram below shows the prices in pounds of the 990 types of men's running shoe on sale at the same online store. \includegraphics[max width=\textwidth, alt={}, center]{aff0c5b2-011b-49a0-bf05-6d905f890eba-4_643_1192_340_440}
4. State the type of skewness shown by the histogram for men's running shoes.
5. Martin is investigating the percentage of types of shoe on sale at the store that cost more than \(\pounds 100\). He believes that this percentage is greater for men's shoes than for women's shoes. Estimate the percentage for men's shoes and comment on whether you can be certain which percentage is higher.
6. You are given that the mean and standard deviation of the prices of men's running shoes are \(\pounds 68.83\) and \(\pounds 42.93\) respectively. Compare the central tendency and variation of the prices of men's and women's running shoes at the store.

7 The cumulative frequency graph below illustrates the distances that 176 children live from their primary school. \begin{figure}[h] \end{figure}

1. Use the graph to estimate, to the nearest 10 metres,
   (A) the median distance from school,
   (B) the lower quartile, upper quartile and interquartile range.
2. Draw a box and whisker plot to illustrate the data. The graph on page 4 used the following grouped data.

   Distance (metres)	200	400	600	800	1000	1200
   Cumulative frequency	20	64	118	150	169	176

3. Copy and complete the grouped frequency table below describing the same data.

   Distance ( \(d\) metres)	Frequency
   \(0 < d \leqslant 200\)	20
   \(200 < d \leqslant 400\)

4. Hence estimate the mean distance these children live from school. It is subsequently found that none of the 176 children lives within 100 metres of the school.
5. Calculate the revised estimate of the mean distance.
6. Describe what change needs to be made to the cumulative frequency graph.

8 The stem-and-leaf diagram shows the heights, in centimetres, of 17 plants, measured correct to the nearest centimetre. Key: 5 | 6 means 56

1. Find the median and inter-quartile range of these heights.
2. Calculate the mean and standard deviation of these heights.
3. State one advantage of using the median rather than the mean as a measure of average for these heights.

1. Joshua is investigating the daily total rainfall in Hurn for May to October 2015

Using the information from the large data set, Joshua wishes to calculate the mean of the daily total rainfall in Hurn for May to October 2015

1. Using your knowledge of the large data set, explain why Joshua needs to clean the data before calculating the mean. Using the information from the large data set, he produces the grouped frequency table below.

   Daily total rainfall ( \(r \mathrm {~mm}\) )	Frequency	Midpoint ( \(\boldsymbol { x } \mathbf { m m }\) )
   \(0 \leqslant r < 0.5\)	121	0.25
   \(0.5 \leqslant r < 1.0\)	10	0.75
   \(1.0 \leqslant r < 5.0\)	24	3.0
   \(5.0 \leqslant r < 10.0\)	12	7.5
   \(10.0 \leqslant r < 30.0\)	17	20.0

   $$\text { You may use } \sum \mathrm { f } x = 539.75 \text { and } \sum \mathrm { f } x ^ { 2 } = 7704.1875$$
2. Use linear interpolation to calculate an estimate for the upper quartile of the daily total rainfall.
3. Calculate an estimate for the standard deviation of the daily total rainfall in Hurn for May to October 2015
4. 1. State the assumption involved with using class midpoints to calculate an estimate of a mean from a grouped frequency table.
   2. Using your knowledge of the large data set, explain why this assumption does not hold in this case.
   3. State, giving a reason, whether you would expect the actual mean daily total rainfall in Hurn for May to October 2015 to be larger than, smaller than or the same as an estimate based on the grouped frequency table.

1. A lake contains three different types of carp.

There are an estimated 450 mirror carp, 300 leather carp and 850 common carp.
Tim wishes to investigate the health of the fish in the lake.
He decides to take a sample of 160 fish.

1. Give a reason why stratified random sampling cannot be used.
2. Explain how a sample of size 160 could be taken to ensure that the estimated populations of each type of carp are fairly represented. You should state the name of the sampling method used. As part of the health check, Tim weighed the fish.
   His results are given in the table below.

   Weight (wkg)	Frequency (f)	Midpoint (m kg)
   \(2 \leqslant w < 3.5\)	8	2.75
   \(3.5 \leqslant w < 4\)	32	3.75
   \(4 \leqslant w < 4.5\)	64	4.25
   \(4.5 \leqslant w < 5\)	40	4.75
   \(5 \leqslant w < 6\)	16	5.5

   $$\left( \text { You may use } \sum \mathrm { f } m = 692 \quad \text { and } \quad \sum \mathrm { f } m ^ { 2 } = 3053 \right)$$
3. Calculate an estimate for the standard deviation of the weight of the carp. Tim realised that he had transposed the figures for 2 of the weights of the fish.
   He had recorded in the table 2.3 instead of 3.2 and 4.6 instead of 6.4
4. Without calculating a new estimate for the standard deviation, state what effect
   1. using the correct figure of 3.2 instead of 2.3
   2. using the correct figure of 6.4 instead of 4.6
      would have on your estimated standard deviation.
      Give a reason for each of your answers.
      

The histogram and its frequency polygon below give information about the weights, in grams, of 50 plums. \includegraphics[max width=\textwidth, alt={}, center]{854568d2-b32d-44de-8a9c-26372e509c20-02_908_1307_328_386}

1. Show that an estimate for the mean weight of the 50 plums is 63.72 grams.
2. Calculate an estimate for the standard deviation of the 50 plums. Later it was discovered that the scales used to weigh the plums were broken.
   Each plum actually weighs 5 grams less than originally thought.
3. State the effect this will have on the estimate of the standard deviation in part (b). Give a reason for your answer.

2. \begin{figure}[h] \end{figure} The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 An outlier is defined as a value
more than \(1.5 \times\) IQR below \(Q _ { 1 }\) or
more than \(1.5 \times\) IQR above \(Q _ { 3 }\) The three lowest air temperatures in the data set are \(7.6 ^ { \circ } \mathrm { C } , 8.1 ^ { \circ } \mathrm { C }\) and \(9.1 ^ { \circ } \mathrm { C }\) The highest air temperature in the data set is \(32.5 ^ { \circ } \mathrm { C }\)

1. Complete the box plot in Figure 1 showing clearly any outliers.
2. Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come. Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, \(x ^ { \circ } \mathrm { C }\), for Beijing in 2015 $$n = 184 \quad \sum x = 4153.6 \quad \mathrm {~S} _ { x x } = 4952.906$$
3. Show that, to 3 significant figures, the standard deviation is \(5.19 ^ { \circ } \mathrm { C }\) Simon decides to model the air temperatures with the random variable $$T \sim \mathrm {~N} \left( 22.6,5.19 ^ { 2 } \right)$$
4. Using Simon's model, calculate the 10th to 90th interpercentile range. Simon wants to model another variable from the large data set for Beijing using a normal distribution.
5. State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer. \includegraphics[max width=\textwidth, alt={}, center]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-09_473_1813_2161_127}
   (Total for Question 2 is 11 marks)

Dian uses the large data set to investigate the Daily Total Rainfall, \(r \mathrm {~mm}\), for Camborne.

1. Write down how a value of \(0 < r \leqslant 0.05\) is recorded in the large data set. Dian uses the data for the 31 days of August 2015 for Camborne and calculates the following statistics $$n = 31 \quad \sum r = 174.9 \quad \sum r ^ { 2 } = 3523.283$$
2. Use these statistics to calculate
   1. the mean of the Daily Total Rainfall in Camborne for August 2015,
   2. the standard deviation of the Daily Total Rainfall in Camborne for August 2015. Dian believes that the mean Daily Total Rainfall in August is less in the South of the UK than in the North of the UK.
      The mean Daily Total Rainfall in Leuchars for August 2015 is 1.72 mm to 2 decimal places.
3. State, giving a reason, whether this provides evidence to support Dian's belief. Dian uses the large data set to estimate the proportion of days with no rain in Camborne for 1987 to be 0.27 to 2 decimal places.
4. Explain why the distribution \(\mathrm { B } ( 14,0.27 )\) might not be a reasonable model for the number of days without rain for a 14-day summer event.

1. Ben is studying the Daily Total Rainfall, \(x \mathrm {~mm}\), in Leeming for 1987

He used all the data from the large data set and summarised the information in the following table.

1. Explain how the data will need to be cleaned before Ben can start to calculate statistics such as the mean and standard deviation. Using all 184 of these values, Ben estimates \(\sum x = 390\) and \(\sum x ^ { 2 } = 4336\)
2. Calculate estimates for
   1. the mean Daily Total Rainfall,
   2. the standard deviation of the Daily Total Rainfall. Ben suggests using the statistic calculated in part (b)(i) to estimate the annual mean Daily Total Rainfall in Leeming for 1987
3. Using your knowledge of the large data set,
   1. give a reason why these data would not be suitable,
   2. state, giving a reason, how you would expect the estimate in part (b)(i) to differ from the actual annual mean Daily Total Rainfall in Leeming for 1987

1. Ming is studying the large data set for Perth in 2015

He intended to use all the data available to find summary statistics for the Daily Mean Air Temperature, \(x { } ^ { \circ } \mathrm { C }\).
Unfortunately, Ming selected an incorrect variable on the spreadsheet.
This incorrect variable gave a mean of 5.3 and a standard deviation of 12.4

1. Using your knowledge of the large data set, suggest which variable Ming selected. The correct values for the Daily Mean Air Temperature are summarised as $$n = 184 \quad \sum x = 2801.2 \quad \sum x ^ { 2 } = 44695.4$$
2. Calculate the mean and standard deviation for these data. One of the months from the large data set for Perth in 2015 has
   • mean \(\bar { X } = 19.4\)
   • standard deviation \(\sigma _ { x } = 2.83\) for Daily Mean Air Temperature.
   • Suggest, giving a reason, a month these data may have come from.

1. Each member of a group of 27 people was timed when completing a puzzle.

The time taken, \(x\) minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot. \includegraphics[max width=\textwidth, alt={}, center]{2b63aa7f-bc50-4422-8dc0-e661b521c221-08_353_1436_458_319}

1. Find the range of the times.
2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
3. calculate the mean time taken to complete the puzzle,
4. calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
5. State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\).
   When their times are included with the data of the other 27 people
   • the median time increases
   • the mean time does not change
   • Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions.
   • Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).