The minus sign indicates that it is the left-hand tail of the distribution, below the mean, which is being considered. Since the distribution is symmetric, it is the same as if it were the right-hand tail, so the minus sign may be ignored when consulting the table. Looking up z = 0.83 in Table A2 gives an area of 0.2033 in the tail, so area C is therefore 0.5 - 0.2033 = 0.2967. Adding areas C and D gives 0.1628 + 0.2967 = 0.4595. So nearly half of all men are between 166 and 178 cm in height.

An alternative interpretation of the results obtained above is that if a man is drawn at random from the adult population, the probability that he is over 180 cm tall is 26.43%. This is in line with the frequentist school of thought. Since 26.43% of the population is over 180 cm in height, that is the probability of a man over 180 cm being drawn at random.

(a) The random variable x is distributed Normally, with x ~ N(40, 36). Find the probability that x > 50.

Exercise 3.4

The mean +/- 0.67 standard deviations cuts off 25% in each tail of the Normal distribution. Hence the middle 50% of the distribution lies within +/- 0.67 standard deviations of the mean. Use this fact to calculate the inter-quartile range for the distribution x ~ N(200, 256).

Exercise 3.5

As suggested in the text, the logarithm of income is approximately Normally distributed. Suppose the log (to the base 10) of income has the distribution x ~ N(4.18, 2.56). Calculate the inter-quartile range for x and then take anti-logs to find the inter-quartile range of income.

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