The height of women in Dewaria follows a normal distribution with mean 160 cm and standard deviation of 6 cm. In a normal distribution, only 0.0063% of the population is not within 4 standard deviations of the mean. If 5 women are more than 184 cm tall, approximately how many women live in Dewaria?

the answer doesn't add up.... 0.0063 of 160,000 is actually 1008 not 10 as suggested in the answer explanation, can you help ?

Since the heights of these women follow a normal distribution, which is symmetric about the mean, there are 5 + 5 = 10 women whose heights are not within 4 standard deviations of the mean. A standard deviation is 6, so 4 standard deviations is 24. There are 5 women below 136 (160 –the mean– minus 24), and 5 above 184 (24 + 160).

These 10 women account for 0.0063 % of the total number of women. So if the number of Dewarian women is n, we can write: n(6.3)/100,000 = 10, so n = (10/6.3) × 100,000 = 160,000 approximately.

The data of women's heights can be presented by the following diagram:

In the normal distribution, only 0.0063% of the population is not within 4 standard deviations of the mean. Those are shown as green and lie to the right of 184 cm or to the left of 136 cm.

Since any normal distribution is symmetric, then (0.0063/2)% lie to the right of 184 cm (5 pers.) and (0.0063/2)% lie to the left of 136 cm (must be 5 pers. as well). 5+ 5 = 10 .

Note, that 99.9937% of the data are within 4 standard deviations of the mean. Those are shown as red area and lie between 136 cm and 184 cm.

Another graphical interpretation of this question is following:

In the normal distribution, only 0.0063% of the population is not within 4 standard deviations of the mean. Those are shown as red points and lie above 184 cm or below 136 cm.

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