The overhead reach distances of adult females are normally distributed with a me

Question

The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8.3cm Find the probability that an individual distance is greater than 218.40 cm. Find the probability that the mean for 20 randomly selected distances is greater than 202.80 cm Why can the normal distribution be used in part (b). even though the sample size does not 30? The probability is (Round to four decimal places as needed.) The probability is (Round to four decimal places as needed.) Choose the correct answer below. The normal distribution can be used because the probability is less than 0.5 The normal distribution can be used because the mean is large. The normal distribution can be used because the finite population correction factor is small. The normal distribution can be used because the original population has a normal distribution.

Explanation / Answer

Solution :-

Given, normally distributed data with,

Mean = 205 cm

Standard deviation = 8.3 cm

(a) Probability that an individual distance is greater than 218.40 cm.

Since = 205 and = 8.3 we have:

P ( X > 218.40 ) = P ( X >218.40 205 ) = P ( (X)/ > (218.40 205)/8.3)

Since Z=(x)/ and (218.40205)/8.3=1.61 we have:

P ( X > 218.40 ) = P ( Z > 1.61 )

Use the standard normal table to conclude that:

P (Z > 1.61) = 0.0537

(b) Probability that the Mean for 20 randomly selected distances is greater than 202.80 cm.

Standard deviation of means of samples of size 20 = 8.3/sqrt(20) = 8.3/4.472 = 1.856

z(202.80) = (202.8 - 205)/1.856 = -1.185

P(x-bar > 202.8) = P(z > -1.185) = 0.8821

(c) (d)

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