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.

3. The birth weights, in kg, of 1500 babies are summarised in the table below. $$\text { [You may use } \sum \mathrm { f } x = 4841 \text { and } \sum \mathrm { f } x ^ { 2 } = 15889.5 \text { ] }$$

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

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.
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   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. A random sample of distances travelled to work for 120 commuters from a train station in Devon is recorded. The distances travelled, to the nearest mile, are summarised below.

For this distribution:
a estimate the median. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\). The mid-point of the lowest class was taken to be 4.75 giving: $$\Sigma f x = 3552.5 \text { and } \Sigma f x ^ { 2 } = 138043.125$$ b Estimate the mean and the standard deviation of this distribution.
c Explain why the median is less than the mean for these data.
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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.