Question

Assume that women's heights are normally distributed with a mean given by u = 64.3 in, and a standard deviation given by 0 = 2.7 in. (a) f 1 woman is randomly selected, find the probability that her height is less than 65 in. (b) if 43 women are randomly selected, find the probability that they have a mean heightless than 65 in. (a) The probability is approximately I. (Round to four decimal places as needed.) 4

Answer 1

0.9533

Explanation:

(a) Probability that a randomly selected woman's height is less than 65 inches:

Using the z-score formula:

�

=

�

−

�

�

Z=

σ

X−μ

​

Where:

�

X = 65 inches

�

μ = 64.3 inches

�

σ = 2.7 inches

�

=

65

−

64.3

2.7

≈

0.2593

Z=

2.7

65−64.3

​

≈0.2593

Now, find the probability associated with this z-score, which is approximately 0.6010 (rounded to four decimal places).

(b) Probability that the mean height of 43 randomly selected women is less than 65 inches:

Using the Central Limit Theorem:

�

μ (mean of the sample means) remains 64.3 inches.

�

sample mean

σ

sample mean

​

(standard deviation of the sample means) is calculated as

2.7

43

≈

0.4115

43

​

2.7

​

≈0.4115.

Now, find the z-score for a sample mean of 65 inches:

�

=

65

−

64.3

0.4115

≈

1.6924

Z=

0.4115

65−64.3

​

≈1.6924

The probability associated with this z-score is approximately 0.9533 (rounded to four decimal places).