2.02g Calculate mean and standard deviation

3. The number of accidents on a particular stretch of motorway was recorded each day for 200 consecutive days. The results are summarised in the following table.

1. Show that the mean number of accidents per day for these data is 1.6 A motorway supervisor believes that the number of accidents per day on this stretch of motorway can be modelled by a Poisson distribution. She uses the mean found in part (a) to calculate the expected frequencies for this model. Her results are given in the following table.

   Number of accidents	0	1	2	3	4	5 or more
   Frequency	40.38	64.61	\(r\)	27.57	11.03	\(s\)

2. Find the value of \(r\) and the value of \(s\), giving your answers to 2 decimal places.
3. Stating your hypotheses clearly, use a \(10 \%\) level of significance to test the motorway supervisor's belief. Show your working clearly.
   

4. Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, \(x\) minutes, are summarised in Table 1. \begin{table}[h] \end{table}

1. Show that an estimate of \(\bar { x } = 5.49\) and an estimate of \(s _ { x } ^ { 2 } = 6.88\) The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
   Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]

   Waiting Time	\(x < 3\)	\(3 - 5\)	\(5 - 6\)	\(6 - 8\)	\(x > 8\)
   Expected Frequency	8.56	12.73	7.56	\(a\)	\(b\)

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Find the value of \(a\) and the value of \(b\).
3. Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.

3 When an alarm is raised at a market town's fire station, the fire engine cannot leave until at least five fire-fighters arrive at the station. The call-out time, \(X\) minutes, is the time between an alarm being raised and the fire engine leaving the station. The value of \(X\) was recorded on a random sample of 50 occasions. The results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 286.5 \quad \sum ( x - \bar { x } ) ^ { 2 } = 45.16$$

1. Find values for the mean and standard deviation of this sample of 50 call-out times.
2. Hence construct a \(99 \%\) confidence interval for the mean call-out time.
3. The fire and rescue service claims that the station's mean call-out time is less than 5 minutes, whereas a parish councillor suggests that it is more than \(6 \frac { 1 } { 2 }\) minutes. Comment on each of these claims.

4 The time, \(x\) seconds, spent by each of a random sample of 100 customers at an automatic teller machine (ATM) is recorded. The times are summarised in the table.

1. Calculate estimates for the mean and standard deviation of the time spent at the ATM by a customer.
2. The mean time spent at the ATM by a random sample of \(\mathbf { 3 6 }\) customers is denoted by \(\bar { Y }\).
   1. State why the distribution of \(\bar { Y }\) is approximately normal.
   2. Write down estimated values for the mean and standard error of \(\bar { Y }\).
   3. Hence estimate the probability that \(\bar { Y }\) is less than \(1 \frac { 1 } { 2 }\) minutes.

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.)

5 The times taken by new recruits to complete an assault course may be modelled by a normal distribution with a standard deviation of 8 minutes. A group of 30 new recruits takes a total time of 1620 minutes to complete the course.

1. Calculate the mean time taken by these 30 new recruits.
2. Assuming that the 30 recruits may be considered to be a random sample, construct a \(98 \%\) confidence interval for the mean time taken by new recruits to complete the course.
3. Construct an interval within which approximately \(98 \%\) of the times taken by individual new recruits to complete the course will lie.
4. State where, if at all, in this question you made use of the Central Limit Theorem.

3 The volume, \(X\) litres, of orange juice in a 1-litre carton may be modelled by a normal distribution with unknown mean \(\mu\). The volumes, \(x\) litres, recorded to the nearest 0.01 litre, in a random sample of 100 cartons are shown in the table.

1. For the group ' \(0.98 - 1.00\) ':
   1. show that it has a mid-point of 0.99 litres;
   2. state the minimum and the maximum values of \(x\) that could be included in this group.
2. Calculate, to three decimal places, estimates of the mean and the standard deviation of these 100 volumes.
3. 1. Construct an approximate \(99 \%\) confidence interval for \(\mu\).
   2. State why use of the Central Limit Theorem was not required when calculating this confidence interval.
   3. Give a reason why the confidence interval is approximate rather than exact.
4. Give a reason in support of the claim that:
   1. \(\mu > 1\);
   2. \(\mathrm { P } ( 0.94 < X < 1.16 )\) is approximately 1 .
      \includegraphics[max width=\textwidth, alt={}]{156f9453-ebc6-4406-b5bc-08d1918ebc62-10_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{156f9453-ebc6-4406-b5bc-08d1918ebc62-11_2486_1714_221_153}

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)

7 Vernon, a service engineer, is expected to carry out a boiler service in one hour.
One hour is subtracted from each of his actual times, and the resulting differences, \(x\) minutes, for a random sample of 100 boiler services are summarised in the table.

1. 1. Calculate estimates of the mean and the standard deviation of these differences.
      (4 marks)
   2. Hence deduce, in minutes, estimates of the mean and the standard deviation of Vernon's actual service times for this sample.
2. 1. Construct an approximate \(98 \%\) confidence interval for the mean time taken by Vernon to carry out a boiler service.
   2. Give a reason why this confidence interval is approximate rather than exact.
3. Vernon claims that, more often than not, a boiler service takes more than an hour and that, on average, a boiler service takes much longer than an hour. Comment, with a justification, on each of these claims.

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}

2 Before leaving for a tour of the UK during the summer of 2008, Eduardo was told that the UK price of a 1.5-litre bottle of spring water was about 50p. Whilst on his tour, Eduardo noted the prices, \(x\) pence, which he paid for 1.5-litre bottles of spring water from 12 retail outlets. He then subtracted 50 p from each price and his resulting differences, in pence, were $$\begin{array} { l l l l l l l l l l l l } - 18 & - 11 & 1 & 15 & 7 & - 1 & 17 & - 16 & 18 & - 3 & 0 & 9 \end{array}$$

1. 1. Calculate the mean and the standard deviation of these differences.
   2. Hence calculate the mean and the standard deviation of the prices, \(x\) pence, paid by Eduardo.
2. Based on an exchange rate of \(€ 1.22\) to \(\pounds 1\), calculate, in euros, the mean and the standard deviation of the prices paid by Eduardo.
   \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-05_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)

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}

1. Twelve observations are made of a random variable \(X\). This set of observations has mean 13 and variance \(10 \cdot 2\).

Another twelve observations of \(X\) are such that \(\sum x = 164\) and \(\sum x ^ { 2 } = 2372\).
Find the mean and the variance for all twenty-four observations.

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.

5. For a project, a student asked 40 people to draw two straight lines with what they thought was an angle of \(75 ^ { \circ }\) between them, using just a ruler and a pencil. She then measured the size of the angles that had been drawn and her data are summarised in this stem and leaf diagram.

1. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
2. determine if there are any outliers in these data,
3. draw a box plot representing these data on graph paper,
4. describe the skewness of the distribution and suggest a reason for it.

6. The number of people visiting a new art gallery each day is recorded over a three-month period and the results are summarised in the table below.

1. Draw a histogram on graph paper to illustrate these data. In order to calculate summary statistics for the data it is coded using \(y = \frac { x - 509.5 } { 10 }\), where \(x\) is the mid-point of each class.
2. Find \(\sum\) fy. You may assume that \(\sum f y ^ { 2 } = 2041\).
3. Using these values for \(\sum f y\) and \(\sum f y ^ { 2 }\), calculate estimates of the mean and standard deviation of the number of visitors per day.
   (6 marks)

4. The ages of 300 houses in a village are recorded giving the following table of results. Use linear interpolation to estimate for these data

1. the median,
2. the limits between which the middle \(80 \%\) of the ages lie. An estimate of the mean of these data is calculated to be 86.6 years.
3. Explain why the mean and median are so different and hence say which you consider best represents the data.