Question

The height of women in the United States is normally distributed with a mean of 165 cm and standard deviation of 7 cm. Show all work for full credit! What is the probability that a randomly selected woman in the United States is taller than 167 cm? What is the probability that a randomly selected sample of 50 women in the United States has an average height greater than 167 cm? bove

Answer 1

$\mu$ = 165 cm

$\sigma$ = 7 cm

P(X < A) = P(Z < (A - $\mu$)/$\sigma$)

P(a randomly selected woman is taller than 167 cm), P(X > 167) = 1 - P(X < 167)

= 1 - P(Z < (167 - 165)/7)

= 1 - P(Z < 0.286)

= 1 - 0.6126

= 0.3874

For a sampling distribution of mean with sample size n = 50,

$\mu_\bar{x}$ = $\mu$ = 165 cm

$\sigma_\bar{x}$ = $\sigma/\sqrt{n}$

= $7/\sqrt{50}$

= 0.99 cm

P($\bar{x}$ < A) = P(Z < (A - $\mu_\bar{x}$)/$\sigma_\bar{x}$)

P(a sample of 50 women has an average height greater than 167 cm), P($\bar{x}$ > 167) = 1 - P(X < 167)

= 1 - P(Z < (167 - 165)/0.99)

= 1 - P(Z < 2.02)

= 1 - 0.9783

= 0.0217