2.02g Calculate mean and standard deviation

6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.

1. Draw a histogram on graph paper to illustrate the data.
2. Calculate estimates of the mean and standard deviation of these heights.

4 The ages of a group of 12 people at an Art class have mean 48.7 years and standard deviation 7.65 years. The ages of a group of 7 people at another Art class have mean 38.1 years and standard deviation 4.2 years.

1. Find the mean age of all 19 people.
2. The individual ages in years of people in the first Art class are denoted by \(x\) and those in the second Art class by \(y\). By first finding \(\Sigma x ^ { 2 }\) and \(\Sigma y ^ { 2 }\), find the standard deviation of the ages of all 19 people.

1 Andy counts the number of emails, \(x\), he receives each day and notes that, over a period of \(n\) days, \(\Sigma ( x - 10 ) = 27\) and the mean number of emails is 11.5 . Find the value of \(n\).

2 Tien measured the arm lengths, \(x \mathrm {~cm}\), of 20 people in his class. He found that \(\Sigma x = 1218\) and the standard deviation of \(x\) was 4.2. Calculate \(\Sigma ( x - 45 )\) and \(\Sigma ( x - 45 ) ^ { 2 }\).

5 The Quivers Archery club has 12 Junior members and 20 Senior members. For the Junior members, the mean age is 15.5 years and the standard deviation of the ages is 1.2 years. The ages of the Senior members are summarised by \(\Sigma y = 910\) and \(\Sigma y ^ { 2 } = 42850\), where \(y\) is the age of a Senior member in years.

1. Find the mean age of all 32 members of the club.
2. Find the standard deviation of the ages of all 32 members of the club.

7 The heights, in cm, of the 11 members of the Anvils athletics team and the 11 members of the Brecons swimming team are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Anvils on the left-hand side of the diagram and Brecons on the right-hand side.
2. Find the median and the interquartile range for the heights of the Anvils.
   The heights of the 11 members of the Anvils are denoted by \(x \mathrm {~cm}\). It is given that \(\Sigma x = 1923\) and \(\Sigma x ^ { 2 } = 337221\). The Anvils are joined by 3 new members whose heights are \(166 \mathrm {~cm} , 172 \mathrm {~cm}\) and 182 cm .
3. Find the standard deviation of the heights of all 14 members of the Anvils.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 The mean and standard deviation of 20 values of \(x\) are 60 and 4 respectively.

1. Find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
   Another 10 values of \(x\) are such that their sum is 550 and the sum of their squares is 40500 .
2. Find the mean and standard deviation of all these 30 values of \(x\).

3 The speeds, in \(\mathrm { km } \mathrm { h } ^ { - 1 }\), of 90 cars as they passed a certain marker on a road were recorded, correct to the nearest \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results are summarised in the following table.

1. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{5307cf3d-3d3a-441a-83d7-4adad917e168-04_1594_1198_657_516}
2. Calculate an estimate for the mean speed of these 90 cars as they pass the marker.

5 Last Saturday, 200 drivers entering a car park were asked the time, in minutes, that it had taken them to travel from home to the car park. The results are summarised in the following cumulative frequency table.

1. On the grid, draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{06f6c8dd-170c-4e94-a960-0c649a7363a1-08_1198_1399_735_415}
2. Use your graph to estimate the median of the data.
3. For 80 of the drivers, the time taken was at least \(T\) minutes. Use your graph to estimate the value of \(T\).
4. Calculate an estimate of the mean time taken by all 200 drivers to travel to the car park.

4 Delip measured the speeds, \(x \mathrm {~km}\) per hour, of 70 cars on a road where the speed limit is 60 km per hour. His results are summarised by \(\Sigma ( x - 60 ) = 245\).

1. Calculate the mean speed of these 70 cars. His friend Sachim used values of \(( x - 50 )\) to calculate the mean.
2. Find \(\Sigma ( x - 50 )\).
3. The standard deviation of the speeds is 10.6 km per hour. Calculate \(\Sigma ( x - 50 ) ^ { 2 }\).

5 The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower. \includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-3_1031_1326_372_406}

1. Copy and complete the following frequency table for the data.

   Time \(( t\) minutes \()\)	\(2 < t \leqslant 4\)	\(4 < t \leqslant 6\)	\(6 < t \leqslant 7\)	\(7 < t \leqslant 8\)	\(8 < t \leqslant 10\)	\(10 < t \leqslant 16\)
   Frequency

2. Calculate an estimate of the mean time to take a shower.
3. Two of these students are chosen at random. Find the probability that exactly one takes between 7 and 10 minutes to take a shower.

2 The values, \(x\), in a particular set of data are summarised by $$\Sigma ( x - 25 ) = 133 , \quad \Sigma ( x - 25 ) ^ { 2 } = 3762 .$$ The mean, \(\bar { x }\), is 28.325 .

1. Find the standard deviation of \(x\).
2. Find \(\Sigma x ^ { 2 }\).

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.

2 The amounts of money, \(x\) dollars, that 24 people had in their pockets are summarised by \(\Sigma ( x - 36 ) = - 60\) and \(\Sigma ( x - 36 ) ^ { 2 } = 227.76\). Find \(\Sigma x\) and \(\Sigma x ^ { 2 }\).

7 The number of cars caught speeding on a certain length of motorway is 7.2 per day, on average. Speed cameras are introduced and the results shown in the following table are those from a random selection of 40 days after this.

1. Calculate unbiased estimates of the population mean and variance of the number of cars per day caught speeding after the speed cameras were introduced.
2. Taking the null hypothesis \(\mathrm { H } _ { 0 }\) to be \(\mu = 7.2\), test at the \(5 \%\) level whether there is evidence that the introduction of speed cameras has resulted in a reduction in the number of cars caught speeding.
3. State what is meant by a Type I error in words relating to the context of the test in part (ii). Without further calculation, illustrate on a suitable diagram the region representing the probability of this Type I error.

6 The daily takings, \(\\) x\(, for a shop were noted on 30 randomly chosen days. The takings are summarised by \)\Sigma x = 31500 , \Sigma x ^ { 2 } = 33141816$.

1. Calculate unbiased estimates of the population mean and variance of the shop's daily takings.
2. Calculate a \(98 \%\) confidence interval for the mean daily takings. The mean daily takings for a random sample of \(n\) days is found.
3. Estimate the value of \(n\) for which it is approximately \(95 \%\) certain that the sample mean does not differ from the population mean by more than \(\\) 6$.

1 A magazine conducted a survey about the sleeping time of adults. A random sample of 12 adults was chosen from the adults travelling to work on a train.

1. Give a reason why this is an unsatisfactory sample for the purposes of the survey.
2. State a population for which this sample would be satisfactory. A satisfactory sample of 12 adults gave numbers of hours of sleep as shown below. \(4.6 \quad 6.8\) 5.2
   6.2
   5.7 \(\begin{array} { l l } 7.1 & 6.3 \end{array}\) 5.6
   7.0 \(5.8 \quad 6.5\) 7.2
3. Calculate unbiased estimates of the mean and variance of the sleeping times of adults.

1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4

8 In the 2001 census, the household size (the number of people living in each household) was recorded. The percentages of households of different sizes were then calculated. The table shows the percentages for two wards, Withington and Old Moat, in Manchester.

1. Calculate the median and interquartile range of the household size for Withington.
2. Making an appropriate assumption for the last class, which should be stated, calculate the mean and standard deviation of the household size for Withington. Give your answers to an appropriate degree of accuracy. The corresponding results for Old Moat are as follows.

   Median	Interquartile range	Mean	Standard deviation
   2	2	2.4	1.5

3. State one advantage of using the median rather than the mean as a measure of the average household size.
4. By comparing the values for Withington with those for Old Moat, explain briefly why the interquartile range may be less suitable than the standard deviation as a measure of the variation in household size.
5. For one of the above wards, the value of Spearman's rank correlation coefficient between household size and percentage is - 1 . Without any calculation, state which ward this is. Explain your answer.

8 The stem-and-leaf diagram shows the age in completed years of the members of a sports club. \section*{Male} \begin{table}[h] \end{table} Key: 1 | 4 | 0 represents a male aged 41 and a female aged 40.

1. Find the median and interquartile range for the males.
2. The median and interquartile range for the females are 27 and 15 respectively. Make two comparisons between the ages of the males and the ages of the females.
3. The mean age of the males is 30.7 and the mean age of the females is 27.5 , each correct to 1 decimal place. Give one advantage of using the median rather than the mean to compare the ages of the males with the ages of the females. A record was kept of the number of hours, \(X\), spent by each member at the club in a year. The results were summarised by $$n = 49 , \quad \Sigma ( x - 200 ) = 245 , \quad \Sigma ( x - 200 ) ^ { 2 } = 9849 .$$
4. Calculate the mean and standard deviation of \(X\).

7 In a UK government survey in 2000, smokers were asked to estimate the time between their waking and their having the first cigarette of the day. For heavy smokers, the results were as follows. Times are given correct to the nearest minute.

1. Assuming that 'At least 60 minutes' means 'At least 60 minutes but less than 240 minutes', calculate estimates for the mean and standard deviation of the time between waking and first cigarette for these smokers.
2. Find an estimate for the interquartile range of the time between waking and first cigarette for these smokers. Give your answer correct to the nearest minute.
3. The meaning of 'At least 60 minutes' is now changed to 'At least 60 minutes but less than 480 minutes'. Without further calculation, state whether this would cause an increase, a decrease or no change in the estimated value of
   1. the mean,
   2. the standard deviation,
   3. the interquartile range.

3 The masses, \(m\) grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$

1. Find the mean and variance of the masses of these 52 apples.
2. Use your answers from part (i) to find the exact value of \(\Sigma m ^ { 2 }\). The masses of the apples are illustrated in the box-and-whisker plot below. \includegraphics[max width=\textwidth, alt={}, center]{b5ce3230-7528-439c-9e85-ef159a49cba3-3_250_1310_662_383}
3. How many apples have masses in the interval \(130 \leqslant m < 140\) ?
4. An 'outlier' is a data item that lies more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile. Explain whether any of the masses of these apples are outliers.

1 Janet and John wanted to compare their daily journey times to work, so they each kept a record of their journey times for a few weeks.

1. Janet's daily journey times, \(x\) minutes, for a period of 25 days, were summarised by \(\Sigma x = 2120\) and \(\Sigma x ^ { 2 } = 180044\). Calculate the mean and standard deviation of Janet's journey times.
2. John's journey times had a mean of 79.7 minutes and a standard deviation of 6.22 minutes. Describe briefly, in everyday terms, how Janet and John's journey times compare.

2 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. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$

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

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.