Although the heights of people of the same sex and similar ages follow Normal di

Question

Although the heights of people of the same sex and similar ages follow Normal distributions closely, weights, on the other hand, are not Normally distributed. Suppose that the weights of women aged 20 to 29 have mean 141.7 pounds and median 133.2 pounds. (a) (4 points) If two women in this age group are randomly selected and are independent, what is the probability that both of them weigh less than the median of 133.2 pounds? (b) (7 points) Suppose that 36 women in this age group are to be random selected. Assuming that the standard deviation of the population distribution of weights is 25 pounds, find the probability that the mean weight of these 36 women is 135 or lower. (c) (2 points) If you select 2 women instead of 36, can you still use the Central Limit Theorem to find the probability that the mean weight of these 2 women is 135 or lower? Explain why or why not.

Explanation / Answer

(a) Median means the value of 50th percentile.

So Pr(Person < median value ) = 0.50

so both women in this age group have weight less than median = 0.50 * 0.50 = 0.25

(b) Here n = 36 ; standard deviation = 25 pounds

Here we can say that by Central limit theorem that mean value of sample population will have a normal distribution with expected value of sample distribution is equal to population mean which is 141.7 pounds.

so mean weight is lower than 135 or lower

Here standard error of the mean se = s/n = 25/ 36 = 25/6 = 4.167

Pr ( xbar<= 135; 141.7 ; 4.167)

so Z - value = (135 - 141.7)/ 4.167 = - 1.61

so Pr ( xbar<= 135; 141.7 ; 4.167) = ( -1.61) = 0.0537

(c) If you select 2 women instead of 36, can you still use the Central Limit Theorem to find the probability that the mean weight of these 2 women is 135 or lower? Explain why or why not.

answer : No, i will not use the Central limit theorem if we have only 2 women as sample. It is because central limt theorem is mostly applicable to large sample size preferable n >= 30.

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