Calculate variance/SD from coded sums

1 A summary of 50 values of \(x\) gives $$\Sigma ( x - q ) = 700 , \quad \Sigma ( x - q ) ^ { 2 } = 14235$$ where \(q\) is a constant.

1. Find the standard deviation of these values of \(x\).
2. Given that \(\Sigma x = 2865\), find the value of \(q\).

1 The length of time, \(t\) minutes, taken to do the crossword in a certain newspaper was observed on 12 occasions. The results are summarised below. $$\Sigma ( t - 35 ) = - 15 \quad \Sigma ( t - 35 ) ^ { 2 } = 82.23$$ Calculate the mean and standard deviation of these times taken to do the crossword.

1 A summary of 30 values of \(x\) gave the following information: $$\Sigma ( x - c ) = 234 , \quad \Sigma ( x - c ) ^ { 2 } = 1957.5 ,$$ where \(c\) is a constant.

1. Find the standard deviation of these values of \(x\).
2. Given that the mean of these values is 86 , find the value of \(c\).

2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \quad \text { and } \quad \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).

4

1. Amy measured her pulse rate while resting, \(x\) beats per minute, at the same time each day on 30 days. The results are summarised below. $$\Sigma ( x - 80 ) = - 147 \quad \Sigma ( x - 80 ) ^ { 2 } = 952$$ Find the mean and standard deviation of Amy's pulse rate.
2. Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute.

2 The values, \(x\), in a particular set of data are summarised by $$\Sigma ( x - 25 ) = 133 , \quad \Sigma ( x - 25 ) ^ { 2 } = 3762 .$$ The mean, \(\bar { x }\), is 28.325 .

1. Find the standard deviation of \(x\).
2. Find \(\Sigma x ^ { 2 }\).

2 The masses, \(x \mathrm {~kg}\), of 50 bags of flour were measured and the results were summarised as follows. $$n = 50 \quad \Sigma ( x - 1.5 ) = 1.4 \quad \Sigma ( x - 1.5 ) ^ { 2 } = 0.05$$ Calculate the mean and standard deviation of the masses of these bags of flour.

1. Morgan is investigating the body length, \(b\) centimetres, of squirrels.

A random sample of 8 squirrels is taken and the data for each squirrel is coded using $$x = \frac { b - 21 } { 2 }$$ The results for the coded data are summarised below $$\sum x = - 1.2 \quad \sum x ^ { 2 } = 5.1$$

1. Find the mean of \(b\)
2. Find the standard deviation of \(b\) A 9th squirrel is added to the sample. Given that for all 9 squirrels \(\sum x = 0\)
3. find
   1. the body length of the 9th squirrel,
   2. the standard deviation of \(x\) for all 9 squirrels.

3. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a group frequency table. The mid-point of each group in the table was represented by \(x\) and the corresponding frequency for that group by \(f\). The data were then coded using \(y = \frac { ( x - 755.0 ) } { 2.5 }\) and summarised as follows: $$\Sigma f y = - 467 , \Sigma f y ^ { 2 } = 9179 .$$

1. Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
   (9 marks)
   An estimate of the interquartile range for these data was 27.7 hours.
2. Explain, giving a reason, whether you would recommend the manufacturer to use the interquartile range or the standard deviation to represent the spread of lifetimes of the bulbs from this production line.
   (2 marks)

5. The masses, \(m\) grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$ Calculate the variance and thus the exact value of \(\sum m ^ { 2 }\)
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