Coding to simplify calculation

1 Twelve values of \(x\) are shown below. Find the mean and standard deviation of \(( x - 1760 )\). Hence find the mean and standard deviation of \(x\). [4]

4. A researcher recorded the time, \(t\) minutes, spent using a mobile phone during a particular afternoon, for each child in a club. The researcher coded the data using \(v = \frac { t - 5 } { 10 }\) and the results are summarised in the table below. $$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$

1. Write down the value of \(a\) and the value of \(b\).
2. Calculate an estimate of the mean of \(v\).
3. Calculate an estimate of the standard deviation of \(v\).
4. Use linear interpolation to estimate the median of \(v\).
5. Hence describe the skewness of the distribution. Give a reason for your answer.
6. Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club.

1. A researcher recorded the time, \(t\) minutes, spent using a mobile phone during a particular afternoon, for each child in a club.

The researcher coded the data using \(v = \frac { t - 5 } { 10 }\) and the results are summarised in the table below. $$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$

1. Write down the value of \(a\) and the value of \(b\).
2. Calculate an estimate of the mean of \(v\).
3. Calculate an estimate of the standard deviation of \(v\).
4. Use linear interpolation to estimate the median of \(v\).
5. Hence describe the skewness of the distribution. Give a reason for your answer.
6. Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club. \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
   VJYV SIHI NI JIIIM ION OC
   VJYV SIHI NI JLIYM ION OC

3. The variable \(X\) represents the marks out of 150 scored by a group of students in an examination. The following ten values of \(X\) were obtained: $$60,66,76,80,94,106,110,116,124,140 .$$

1. Write down the median, \(M\), of the ten marks.
2. Using the coding \(y = \frac { x - M } { 2 }\), and showing all your working clearly, find the mean and the standard deviation of the marks.
3. Find \(\mathrm { E } ( 3 X - 5 )\).

1. Using the coding \(y = \frac { x - 90 } { 5 }\), and showing each step in your working clearly, calculate the mean and the standard deviation of the 20 observations of a variable \(X\) given by the following table:

3. A magazine collected data on the total cost of the reception at each of a random sample of 80 weddings. The data is grouped and coded using \(y = \frac { C - 3250 } { 250 }\), where \(C\) is the mid-point in pounds of each class, giving \(\sum f y = 37\) and \(\sum f y ^ { 2 } = 2317\).

1. Using these values, calculate estimates of the mean and standard deviation of the cost of the receptions in the sample.
2. Explain why your answers to part (a) are only estimates. The median of the data was \(\pounds 3050\).
3. Comment on the skewness of the data and suggest a reason for it.

5. An antiques shop recorded the value of items stolen to the nearest pound during each week for a year giving the data in the table below. Letting \(x\) represent the mid-point of each group and using the coding \(y = \frac { x - 699.5 } { 200 }\),

1. find \(\sum\) fy.
2. estimate to the nearest pound the mean and standard deviation of the value of the goods stolen each week using your value for \(\sum f y\) and \(\sum f y ^ { 2 } = 424\).
   (6 marks)
   The median for these data is \(\pounds 82\).
3. Explain why the manager of the shop might be reluctant to use either the mean or the median in summarising these data.
   (3 marks)

2. 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 grouped frequency table. The mid-point of each class in the table was represented by \(x\) and the corresponding frequency for that class by \(f\). The data were then coded using: $$y = \frac { ( x - 755.0 ) } { 2.5 }$$ and summarised as follows: $$\sum f y = - 467 , \sum f y ^ { 2 } = 9179$$ Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
(4 marks)
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