Calculate statistics from discrete frequency table

1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.

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.

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

1. Find the mean and standard deviation of the number of eggs laid per gull.
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?

6 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.

7 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 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.

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.

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

1. Find the mean and standard deviation of the number of eggs laid per gull.
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?

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.

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

4 A survey of the number of cars per household in a certain village generated the data in Fig. 4. \begin{table}[h] \end{table}

1. Calculate the mean number of cars per household.
2. Calculate the standard deviation of the number of cars per household.

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.

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)

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 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 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]

3. 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.

1. Calculate the mean and standard deviation of the values chosen. It is suggested that the value chosen could be modelled by a discrete uniform distribution.
2. Write down the mean that this model would predict. Given also that the standard deviation according to this model would be 2.58,
3. explain why this model is not suitable and suggest why this is the case.

1. Helen is studying the daily mean wind speed for Camborne using the large data set from 1987. The data for one month are summarised in Table 1 below.

\begin{table}[h] \end{table}

1. Calculate the mean for these data.
2. Calculate the standard deviation for these data and state the units. The means and standard deviations of the daily mean wind speed for the other months from the large data set for Camborne in 1987 are given in Table 2 below. The data are not in month order. \begin{table}[h]

   Month	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
   Mean	7.58	8.26	8.57	8.57	11.57
   Standard Deviation	2.93	3.89	3.46	3.87	4.64

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
3. Using your knowledge of the large data set, suggest, giving a reason, which month had a mean of 11.57 The data for these months are summarised in the box plots on the opposite page. They are not in month order or the same order as in Table 2.
4. 1. State the meaning of the * symbol on some of the box plots.
   2. Suggest, giving your reasons, which of the months in Table 2 is most likely to be summarised in the box plot marked \(Y\). \includegraphics[max width=\textwidth, alt={}, center]{2edcf965-9c93-4a9b-9395-2d3c023801af-11_1177_1216_324_427}

3 The discrete random variable \(A\) has the following probability distribution function $$\mathrm { P } ( A = a ) = \begin{cases} 0.45 & a = 0
0.25 & a = 1
0.3 & a = 2
0 & \text { otherwise } \end{cases}$$ 3

1. Find the median of \(A\)
   3
2. Find the standard deviation of \(A\), giving your answer to three significant figures.
   3
3. \(\quad\) Find \(\operatorname { Var } ( 9 A - 2 )\)

4 The discrete random variable \(Y\) has probability distribution The standard deviation of \(Y\) is \(s\) 4

1. Show that \(s = 10.53\) correct to two decimal places.
   [0pt] [4 marks]
   4
2. The median of \(Y\) is \(m\) Find \(\mathrm { P } ( Y > m - 1.5 s )\)