2.02f Measures of average and spread

2. An estate agent recorded the price per square metre, \(p \pounds / \mathrm { m } ^ { 2 }\), for 7 two-bedroom houses. He then coded the data using the coding \(q = \frac { p - a } { b }\), where \(a\) and \(b\) are positive constants. His results are shown in the table below.

1. Find the value of \(a\) and the value of \(b\) The estate agent also recorded the distance, \(d \mathrm {~km}\), of each house from the nearest train station. The results are summarised below. $$\mathrm { S } _ { d d } = 1.02 \quad \mathrm {~S} _ { q q } = 8.22 \quad \mathrm {~S} _ { d q } = - 2.17$$
2. Calculate the product moment correlation coefficient between \(d\) and \(q\)
3. Write down the value of the product moment correlation coefficient between \(d\) and \(p\) The estate agent records the price and size of 2 additional two-bedroom houses, \(H\) and \(J\).

   House	Price \(( \pounds )\)	Size \(\left( \mathrm { m } ^ { 2 } \right)\)
   \(H\)	156400	85
   \(J\)	172900	95

4. Suggest which house is most likely to be closer to a train station. Justify your answer.

3. Before going on holiday to Seapron, Tania records the weekly rainfall ( \(x \mathrm {~mm}\) ) at Seapron for 8 weeks during the summer. Her results are summarised as $$\sum x = 86.8 \quad \sum x ^ { 2 } = 985.88$$

1. Find the standard deviation, \(\sigma _ { x }\), for these data.
   (3) Tania also records the number of hours of sunshine ( \(y\) hours) per week at Seapron for these 8 weeks and obtains the following $$\bar { y } = 58 \quad \sigma _ { y } = 9.461 \text { (correct to } 4 \text { significant figures) } \quad \sum x y = 4900.5$$
2. Show that \(\mathrm { S } _ { y y } = 716\) (correct to 3 significant figures)
3. Find \(\mathrm { S } _ { x y }\)
4. Calculate the product moment correlation coefficient, \(r\), for these data. During Tania's week-long holiday at Seapron there are 14 mm of rain and 70 hours of sunshine.
5. State, giving a reason, what the effect of adding this information to the above data would be on the value of the product moment correlation coefficient.

2. A botany student counted the number of daisies in each of 42 randomly chosen areas of 1 m by 1 m in a large field. The results are summarised in the following stem and leaf diagram.

1. Write down the modal value of these data.
2. Find the median and the quartiles of these data.
3. On graph paper and showing your scale clearly, draw a box plot to represent these data.
4. Comment on the skewness of this distribution. The student moved to another field and collected similar data from that field.
5. Comment on how the student might summarise both sets of raw data before drawing box plots.
   (1 mark)

6. A travel agent sells holidays from his shop. The price, in \(\pounds\), of 15 holidays sold on a particular day are shown below. For these data, find

1. the mean and the standard deviation,
2. the median and the inter-quartile range. An outlier is an observation that falls either more than \(1.5 \times\) (inter-quartile range) above the upper quartile or more than \(1.5 \times\) (inter-quartile range) below the lower quartile.
3. Determine if any of the prices are outliers. The travel agent also sells holidays from a website on the Internet. On the same day, he recorded the price, \(\pounds x\), of each of 20 holidays sold on the website. The cheapest holiday sold was \(\pounds 98\), the most expensive was \(\pounds 2400\) and the quartiles of these data were \(\pounds 305 , \pounds 1379\) and \(\pounds 1805\). There were no outliers.
4. On graph paper, and using the same scale, draw box plots for the holidays sold in the shop and the holidays sold on the website.
5. Compare and contrast sales from the shop and sales from the website. \section*{END}

1. As part of their job, taxi drivers record the number of miles they travel each day. A random sample of the mileages recorded by taxi drivers Keith and Asif are summarised in the back-toback stem and leaf diagram below.

Key: 0184 means 180 for Keith and 184 for Asif
The quartiles for these two distributions are summarised in the table below.

1. Find the values of \(a , b\) and \(c\). Outliers are values that lie outside the limits $$Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \text { and } Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) .$$
2. On graph paper, and showing your scale clearly, draw a box plot to represent Keith's data.
3. Comment on the skewness of the two distributions.
   

6. Students in Mr Brawn's exercise class have to do press-ups and sit-ups. The number of press-ups \(x\) and the number of sit-ups \(y\) done by a random sample of 8 students are summarised below. $$\begin{array} { l l } \Sigma x = 272 , & \Sigma x ^ { 2 } = 10164 , \quad \Sigma x y = 11222 , \\ \Sigma y = 320 , & \Sigma y ^ { 2 } = 13464 . \end{array}$$

1. Evaluate \(S _ { x x } , S _ { y y }\) and \(S _ { x y }\).
2. Calculate, to 3 decimal places, the product moment correlation coefficient between \(x\) and \(y\).
3. Give an interpretation of your coefficient.
4. Calculate the mean and the standard deviation of the number of press-ups done by these students. Mr Brawn assumes that the number of press-ups that can be done by any student can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Assuming that \(\mu\) and \(\sigma\) take the same values as those calculated in part (d),
5. find the value of \(a\) such that \(\mathrm { P } ( \mu - a < X < \mu + a ) = 0.95\).
6. Comment on Mr Brawn's assumption of normality.

4. Explain what you understand by

1. a sampling unit,
2. a sampling frame,
3. a sampling distribution.

6 For each of the Premiership football seasons 2004/05 and 2005/06, a count is made of the number of goals scored in each of the 380 matches. The results are shown in the table.

1. For the number of goals scored in a match during the 2004/05 season:
   1. determine the median and the interquartile range;
   2. calculate the mean and the standard deviation.
2. Two statistics students, Jole and Katie, independently analyse the data on the number of goals scored in a match during the 2005/06 season.
   • Jole determines correctly that the median is 2 and that the interquartile range is also 2.
   • Katie calculates correctly, to two decimal places, that the mean is 2.48 and that the standard deviation is 1.59 .
     1. Use your answers from part (a), together with Jole's and Katie's results, to compare briefly the two seasons with regard to the average and the spread of the number of goals scored in a match.
     2. Jole claims that Katie's results must be wrong as \(95 \%\) of values always lie within 2 standard deviations of the mean and \(( 2.48 - 2 \times 1.59 ) < 0\) which is nonsense.
   Explain why Jole's claim is incorrect. (You are not expected to confirm Katie's results.)

1 Ms N Parker always reads the columns of announcements in her local weekly newspaper. During each week of 2008, she notes the number of births announced. Her results are summarised in the table.

1. Calculate the mean, median and modes of these data.
2. State, with a reason, which of the three measures of average in part (a) you consider to be the least appropriate for summarising the number of births.

1 Giles, a keen gardener, rents a council allotment. During early April 2011, he planted 27 seed potatoes. When he harvested his potato crop during the following August, he counted the number of new potatoes that he obtained from each seed potato. He recorded his results as follows.

1. Calculate values for the median and the interquartile range of these data.
2. Advise Giles on how to record his corresponding data for 2012 so that it would then be possible to calculate the mean number of new potatoes per seed potato.

4 A library allows each member to have up to 15 books on loan at any one time. The table shows the numbers of books currently on loan to a random sample of 95 members of the library.

1. For these data:
   1. state values for the mode and range;
   2. determine values for the median and interquartile range;
   3. calculate estimates of the mean and standard deviation.
2. Making reference to your answers to part (a), give a reason for preferring:
   1. the median and interquartile range to the mean and standard deviation for summarising the given data;
   2. the mean and standard deviation to the mode and range for summarising the given data.
      (1 mark)

4 The runs scored by a cricketer in 11 innings during the 2006 season were as follows. $$\begin{array} { l l l l l l l l l l l } 47 & 63 & 0 & 28 & 40 & 51 & a & 77 & 0 & 13 & 35 \end{array}$$ The exact value of \(a\) was unknown but it was greater than 100 .

1. Calculate the median and the interquartile range of these 11 values.
2. Give a reason why, for these 11 values:
   1. the mode is not an appropriate measure of average;
   2. the range is not an appropriate measure of spread.

2 Hermione, who is studying reptiles, measures the length, \(x \mathrm {~cm}\), and the weight, \(y\) grams, of a sample of 11 adult snakes of the same type. Her results are shown in the table.

5 A survey of all the households on an estate is undertaken to provide information on the number of children per household. The results, for the 99 households with children, are shown in the table.

1. For these 99 households, calculate values for:
   1. the median and the interquartile range;
   2. the mean and the standard deviation.
2. In fact, 163 households were surveyed, of which 64 contained no children.
   1. For all 163 households, calculate the value for the mean number of children per household.
   2. State whether the value for the standard deviation, when calculated for all 163 households, will be smaller than, the same as, or greater than that calculated in part (a)(ii).
   3. It is claimed that, for all 163 households on the estate, the number of children per household may be modelled approximately by a normal distribution. Comment, with justification, on this claim. Your comment should refer to a fact other than the discrete nature of the data.
      
      \includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-11_2484_1709_223_153}

1 The number of matches in each of a sample of 85 boxes is summarised in the table.

1. For these data:
   1. state the modal value;
   2. determine values for the median and the interquartile range.
2. Given that, on investigation, the 2 extreme values in the above table are 227 and 271 :
   1. calculate the range;
   2. calculate estimates of the mean and the standard deviation.
3. For the numbers of matches in the 85 boxes, suggest, with a reason, the most appropriate measure of spread.

2 Katy works as a clerical assistant for a small company. Each morning, she collects the company's post from a secure box in the nearby Royal Mail sorting office. Katy's supervisor asks her to keep a daily record of the number of letters that she collects. Her records for a period of 175 days are summarised in the table.

1. For these data:
   1. state the modal value;
   2. determine values for the median and the interquartile range.
2. The most letters that Katy collected on any of the 175 days was 54. Calculate estimates of the mean and the standard deviation of the daily number of letters collected by Katy.
3. During the same period, a total of 280 letters was also delivered to the company by private courier firms. Calculate an estimate of the mean daily number of all letters received by the company during the 175 days.

1 The average maximum monthly temperatures, \(u\) degrees Fahrenheit, and the average minimum monthly temperatures, \(v\) degrees Fahrenheit, in New York City are as follows.

1. 1. Calculate, to one decimal place, the mean and the standard deviation of the 12 values of the average maximum monthly temperature.
   2. For comparative purposes with a UK city, it was necessary to convert the temperatures from degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ) to degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ). The formula used to convert \(f ^ { \circ } \mathrm { F }\) to \(c ^ { \circ } \mathrm { C }\) is: $$c = \frac { 5 } { 9 } ( f - 32 )$$ Use this formula and your answers in part (a)(i) to calculate, in \({ } ^ { \circ } \mathbf { C }\), the mean and the standard deviation of the 12 values of the average maximum monthly temperature.
      (3 marks)
2. The value of the product moment correlation coefficient, \(r _ { u v }\), between the above 12 values of \(u\) and \(v\) is 0.997 , correct to three decimal places. State, giving a reason, the corresponding value of \(r _ { x y }\), where \(x\) and \(y\) are the exact equivalent temperatures in \({ } ^ { \circ } \mathrm { C }\) of \(u\) and \(v\) respectively.
   (2 marks)

1 The weights, in kilograms, of a random sample of 15 items of cabin luggage on an aeroplane were as follows. \section*{\(\begin{array} { l l l l l l l l l l l l l l l } 4.6 & 3.8 & 3.9 & 4.5 & 4.9 & 3.6 & 3.7 & 5.2 & 4.0 & 5.1 & 4.1 & 3.3 & 4.7 & 5.0 & 4.8 \end{array}\)} For these data:

1. find values for the median and the interquartile range;
2. find the value for the range;
3. state why the mode is not an appropriate measure of average.

7 For the year 2014, the table below summarises the weights, \(x\) kilograms, of a random sample of 160 women residing in a particular city who are aged between 18 years and 25 years.

1. Calculate estimates of the mean and the standard deviation of these 160 weights.
2. 1. Construct a 98\% confidence interval for the mean weight of women residing in the city who are aged between 18 years and 25 years.
   2. Hence comment on a claim that the mean weight of women residing in the city who are aged between 18 years and 25 years has increased from that of 61.7 kg in 1965.
      [0pt] [2 marks]
      
      \includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-28_2488_1728_219_141}

1 Henrietta lives on a small farm where she keeps some hens.
For a period of 35 weeks during the hens' first laying season, she records, each week, the total number of eggs laid by the hens. Her records are shown in the table.

1. For these data:
   1. state values for the mode and the range;
   2. find values for the median and the interquartile range;
   3. calculate values for the mean and the standard deviation.
2. Each week, for the 35 weeks, Henrietta sells 60 eggs to a local shop, keeping the remainder for her own use. State values for the mean and the standard deviation of the number of eggs that she keeps.
   [0pt] [2 marks]

2 A small chapel was open to visitors for 55 days during the summer of 2015. The table summarises the daily numbers of visitors.

1. For these data:
   1. state the modal value;
   2. find values for the median and the interquartile range.
2. Name one measure of average and one measure of spread that cannot be calculated exactly from the data in the table.
   [0pt] [2 marks]
3. Reference to the raw data revealed that the 3 unknown exact values in the table were 13,37 and 58. Making use of this additional information, together with the data in the table, calculate the value of each of the two measures that you named in part (b).
   [0pt] [3 marks]

7 Customers buying euros ( €) at a travel agency must pay for them in pounds ( \(\pounds\) ). The amounts paid, \(\pounds x\), by a sample of 40 customers were, in ascending order, as follows.

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}

3. The frequency distribution for the lengths of 108 fish in an aquarium is given by the following table. The lengths of the fish ranged from 5 cm to 90 cm .

1. Calculate estimates of the three quartiles of the distribution.
2. On graph paper, draw a box and whisker plot of the data.
3. Hence describe the skewness of the distribution.
4. If the data were represented by a histogram, what would be the ratio of the heights of the shortest and highest bars?

1. An adult evening class has 14 students. The ages of these students have a mean of 31.2 years and a standard deviation of 7.4 years.

A new student who is exactly 42 years old joins the class. Calculate the mean and standard deviation of the 15 students now in the group.