F71SM STATISTICAL METHODS
Tutorial on Section 1 DATA SUMMARY
1. The sums insured (£), in order, under 7 policies are as follows.
1,216 1,543 1,616 1,684 1,823 2,015 15,042
Calculate the sample mean and standard deviation of the sums insured (a) including the observation 15,042 [x ̄ = £3562.7, s = £5067.9]
(b) excluding the observation 15,042 [x ̄ = £1649.5, s = £270.3] and comment briefly on the results.
2. A random sample of 100 observations (sorted) is as follows:
16.6 18.3 19.9 19.9 20.3 20.6 20.9 20.9 21.1 21.2 21.5 21.6 21.8 22.1 22.1 22.1 22.3 22.4 22.6 22.6 22.7 22.7 22.8 22.9 22.9 23.0 23.5 23.8 23.9 23.9 23.9 23.9 23.9 24.0 24.1 24.2 24.3 24.4 24.4 24.6 24.6 24.7 24.8 25.0 25.0 25.0 25.1 25.1 25.1 25.1 25.2 25.4 25.5 25.5 25.6 25.6 25.6 25.7 25.8 25.9 26.1 26.2 26.2 26.4 26.7 26.8 26.8 26.9 27.0 27.0 27.0 27.1 27.3 27.3 27.3 27.4 27.4 27.5 27.5 27.5 27.7 27.7 27.7 27.8 27.9 27.9 28.1 28.1 28.4 28.8 28.8 29.1 29.2 29.3 30.0 30.4 31.0 31.4 31.7 33.9
For these data x = 2, 524.2, x2 = 64, 608.82.
(a) Calculate the sample mean and median. [x ̄ = 25.24, median = 25.15]
(b) Calculate the sample IQR and standard deviation. [IQR = 4.45, s = 3.00]
3. Shortly before close of business on a particular day, a local insurance agency has sold 8 new policies. The sample mean and standard deviation of the sums assured for these policies have been calculated (in units of £1,000) as 41.825 and 27.236, respectively. Just before the office closes another policy is sold, with sum assured £60,400.
Calculate the sample mean and standard deviation of the sums assured for the full set of 9 policies sold that day. [x ̄ = 43.3889(£43, 890), s = 26.219(£26, 220)]
4. A plumber has a callout charge of £70 and in addition he charges £30 per hour for all his jobs. In a particular week the mean and standard deviation of the lengths of his jobs are 3.5hrs and 0.5hrs respectively.
Calculate the mean and standard deviation of the invoice amounts for this week. [y ̄ = £175, sy = £15]
5. A set of data has sample mean 62 and standard deviation 6. Find a change of origin and scale (that is, a linear transformation) that will result in the new data set having sample mean 50 and standard deviation 12. [Let y = ax+b. a = −74,b = 2, or a = 174,b = −2]
6. The gross incomes in tax year 2005–2006 of a random sample of 10 school teachers were recorded (to the nearest £100 and in units of £1,000) and gave the following summary statistics:
x = 321.6, x2 = 10,628.31
Income tax is charged at a single rate of 30% on that part of a person’s income above
Calculate the sample mean and standard deviation of the 10 net incomes (that is, the incomes after deduction of tax). You may assume that every teacher in the sample has a gross income over £10,000. [y ̄ = 25.512(£25, 510), sy = 3.944(£3, 940)]
7. Consider the following two frequency distributions. Ax1234 Bx1234
f 80 60 40 20 f 110 50 30 10
Without making full detailed calculations (but you may want to draw quick pictures) state which distribution has (a) the higher mean, (b) the higher standard deviation, (c) the higher skew.
8. Food packages, after filling, are weighed on an automatic device that rejects all under- weight packages. All rejected packages are carefully weighed and their weights recorded. The table below gives a frequency distribution for a sample of 200 such weights.
weight (kg) of rejected 2.075 – 2.085 2.085 – 2.095 2.095 – 2.105 2.105 – 2.115 2.115 – 2.125 2.125 – 2.135 2.135 – 2.145 2.145 – 2.155
frequency 1
(a) Calculate the sample mean and standard deviation of the weights for the sample of rejected packages. [x ̄ = 2.14 kg, s = 0.014 kg]
(b) The mean weight of all packages is known to be 2.342kg. It is also known that rejected packages represent 8% of the total distribution of packages’ weights.
Obtain an estimate of the mean weight of packages that are not rejected. [y ̄ = 2.360 kg]