Question

The mean height of women in the United States (ages 20-29) is 64.2 inches with a standard deviation of 2.9 inches. A random sample of 60 women in this age group is selected. Assume that the distribution of these heights is normally distributed.

Are you more likely to randomly select 1 woman with a height more than 70 inches or are you more likely to select a random sample of 20 women with a mean height more than 70 inches? Show work necessary to answer this question: sketch both distributions and calculate each probability.

A) Sketch the distribution of women’s heights in the United States (age 20-29). Label the mean, label at least two standard deviations in each direction and shade the area in question. Calculate the probability that a randomly selected woman will have a height more than 70 inches. Show your work.

B) Sketch the sampling distribution of sample mean heights for random samples of 20 women in the United States (age 20-29). Label the mean, label at least two standard deviations in each direction and shade the area in question. Calculate the probability that a random sample of 20 women will have a mean height more than 70 inches. Show your work.

C) Which is more likely? Explain. Provide your answer as a sentence.

Answer 1

A) P(X > 70)

    = P((X - )/ > (70 - )/)

    = P(Z > (70 - 64.2)/2.9)

    = P(Z > 2)

   = 1 - P(Z < 2)

   = 1 - 0.9772

   = 0.0228

b) P( > 70)

= P(( - )/() > (70 - )/())

= P(Z > (70 - 64.2)/(2.9/))

= P(Z > 8.94)

= 1 - P(Z < 8.94)

= 1 - 1 = 0

c) Since the probability value in part-a is more than the probability value in part - b, so the probability in part - a is more likely.