Calculate statistics from grouped frequency table

4 In a survey, the percentage of meat in a certain type of take-away meal was found. The results, to the nearest integer, for 193 take-away meals are summarised in the table.

1. Calculate estimates of the mean and standard deviation of the percentage of meat in these take-away meals.
2. Draw, on graph paper, a histogram to illustrate the information in the table.

1 In a survey, a sample of 44 fields is selected. Their areas ( \(x\) hectares) are summarised in the grouped frequency table.

1. Calculate an estimate of the sample mean and the sample standard deviation.
2. Determine whether there could be any outliers at the upper end of the distribution.

6 In a survey, a sample of 44 fields is selected. Their areas ( \(x\) hectares) are summarised in the grouped frequency table.

1. Calculate an estimate of the sample mean and the sample standard deviation.
2. Determine whether there could be any outliers at the upper end of the distribution.

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.

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.
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3. A soccer fan collected data on the number of minutes of league football, \(m\), played by each team in the four main divisions before first scoring a goal at the start of a new season. Her results are shown in the table below.

1. Calculate estimates of the mean and standard deviation of these data.
2. Explain why the mean and standard deviation might not be the best summary statistics to use with these data.
3. Suggest alternative summary statistics that would better represent these data.

1. Nina weighed a random sample of 50 carrots from her shop and recorded the weight, in grams to the nearest gram, for each carrot. The results are summarised below.

$$\text { (You may use } \sum \mathrm { f } x ^ { 2 } = 192102.5 \text { ) }$$

1. Use linear interpolation to estimate the median weight of these carrots.
2. Find an estimate for the mean weight of these carrots.
3. Find an estimate for the standard deviation of the weights of these carrots. A carrot is selected at random from Nina's shop.
4. Estimate the probability that the weight of this carrot is more than 70 grams.

120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table.

Time (seconds)	1 -- 25	26 -- 35	36 -- 45	46 -- 55	56 -- 90
Number of people	4	24	38	34	20

Calculate estimates of the mean and standard deviation of the reading times. [5]

The birth weights, in kg, of 1500 babies are summarised in the table below.

Weight (kg)	Midpoint, \(x\)kg	Frequency, \(f\)
\(0.0 - 1.0\)	\(0.50\)	\(1\)
\(1.0 - 2.0\)	\(1.50\)	\(6\)
\(2.0 - 2.5\)	\(2.25\)	\(60\)
\(2.5 - 3.0\)		\(280\)
\(3.0 - 3.5\)	\(3.25\)	\(820\)
\(3.5 - 4.0\)	\(3.75\)	\(320\)
\(4.0 - 5.0\)	\(4.50\)	\(10\)
\(5.0 - 6.0\)		\(3\)

[You may use \(\sum fx = 4841\) and \(\sum fx^2 = 15889.5\)]

1. Write down the missing midpoints in the table above. [2]
2. Calculate an estimate of the mean birth weight. [2]
3. Calculate an estimate of the standard deviation of the birth weight. [3]
4. Use interpolation to estimate the median birth weight. [2]
5. Describe the skewness of the distribution. Give a reason for your answer. [2]

A class of students had a sudoku competition. The time taken for each student to complete the sudoku was recorded to the nearest minute and the results are summarised in the table below. (You may use \(\sum fx^2 = 8603.75\))

1. Write down the mid-point for the 9 - 12 interval. [1]
2. Use linear interpolation to estimate the median time taken by the students. [2]
3. Estimate the mean and standard deviation of the times taken by the students. [5]

The teacher suggested that a normal distribution could be used to model the times taken by the students to complete the sudoku.

1. Give a reason to support the use of a normal distribution in this case. [1]

On another occasion the teacher calculated the quartiles for the times taken by the students to complete a different sudoku and found \(Q_1 = 8.5 \quad Q_2 = 13.0 \quad Q_3 = 21.0\)

1. Describe, giving a reason, the skewness of the times on this occasion. [2]

In March 2020, the coronavirus pandemic caused major disruption to the lives of individuals across the world. A newspaper published the following graph from the gov.uk website, along with an article which included the following excerpt. "The daily number of vaccines administered continues to fall. In order to get control of the virus, we need the number of people receiving a second dose of the vaccine to keep rocketing. The fear is it will start to drop off soon, which will leave many people still unprotected." \includegraphics{figure_5}

1. By referring to the graph, explain how the quote could be misleading. [1]

The daily numbers of second dose vaccines, in thousands, over the period April 1st 2021 to May 31st 2021 are shown in the table below.

1. 1. Calculate estimates of the mean and standard deviation for the daily number of second dose vaccines given over this period. You may use \(\sum x^2 f = 8272500\). [4]
   2. Comment on the skewness of these data. [1]
2. Give a possible reason for the pattern observed in this graph. [1]
3. State, with a reason, whether or not you think the data for April 15th to April 18th are incorrect. [1]

Employees at a company were asked how long their average commute to work was. The table below gives information about their answers.

Time taken (\(t\) minutes)	Number of people
\(0 < t \leq 10\)	\(x\)
\(10 < t \leq 20\)	30
\(20 < t \leq 30\)	35
\(30 < t \leq 50\)	28
\(50 < t \leq 90\)	12

The company estimates that the mean time for employees commuting to work is 28 minutes. Work out the value of \(x\), showing your working clearly. [4]

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

Mass (\(x\) kg)	\(1.6 \leqslant x < 1.8\)	\(1.8 \leqslant x < 2.0\)	\(2.0 \leqslant x < 2.2\)	\(2.2 \leqslant x < 2.4\)	\(2.4 \leqslant x < 2.6\)
Number of chickens	16	27	28	18	11

Calculate estimates of the mean and standard deviation of the masses of these chickens. [5]

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

Mass (\(x\) kg)	\(1.6 \leq x < 1.8\)	\(1.8 \leq x < 2.0\)	\(2.0 \leq x < 2.2\)	\(2.2 \leq x < 2.4\)	\(2.4 \leq x < 2.6\)
Number of chickens	16	27	28	18	11

Calculate estimates of the mean and standard deviation of the masses of these chickens. [5]