The Empirical Rule states that for bell-shaped distributions, about 68% of the v

Question

The Empirical Rule states that for bell-shaped distributions, about 68% of the values fall within 1 standard deviation of the mean. The heights of women at a large university are approximately bell-shaped, with a mean of 68 inches and standard deviation of 2 inches. Use this information to answer the questions.

(1) What is the probability that two randomly selected women from this university are both 70 inches or shorter? (Give the answer to four decimal places.)

(2) What is the probability that of two randomly selected women, one is 70 inches or shorter and the other is 70 inches or taller? (Give the answer to four decimal places.)

(3) What is the probability that three randomly selected women are all 68 inches or taller? (Give the answer to two decimal places.)

Explanation / Answer

The mean is 68 and with std deviations, it goes like this, draw a number line and label the mean.

Add the standard deviation to the mean so 68in---69in---70in---71in. The percentages are 34% within 1std dev (34% goes in between 68 and 70), 13.5% within 2, (68 and 70) and 2.5% within 3 (70 and 72). Make it big enough to show the percentages above the line.

It is the same percentages below the mean but the values are going down by 2.

Since the deviation is 2, within 62 and 66 are 1 deviation away from the mean, which is 64% chance of being within that distribution. 34% above mean and 34% below.

In general: The right side from the mean should add to 50% and so should left side. Each value should increase and decrease from the mean by the std deviation.

A) 70in or taller is > 1 std deviation so 13.5 + 3.5=16% chance
B) 70in or shorter is < 1 std deviation above the mean so 34+34+13.5+2.5=84% chance
C) .84 *.16=.1344

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