1- The height of adult women in the United States is normally distributed with mean 64.5 inches and standard deviation 2.4 inches. Find the probability that a randomly chosen woman is

(a) Less than 63 inches tall
(b) Less than 70 inches tall
(c) Between 63 and 70 inches tall
(d) Alice is 72 inches tall. What percentages of women are shorter than Alice?

you can play around with Z table stuff here:

http://davidmlane.com/hyperstat/z_table.html

0.0125

To solve these problems, we'll use the standard normal distribution. First, let's convert the given values for the mean and standard deviation to the standard normal distribution by standardizing the values.

(a) To find the probability that a randomly chosen woman is less than 63 inches tall, we need to find the area under the standard normal curve to the left of the standardized value of 63 inches.

Standardized value = (63 - mean) / standard deviation
= (63 - 64.5) / 2.4
= -0.625

Using a standard normal distribution table or a calculator, we find that the area to the left of -0.625 is 0.267.

Therefore, the probability that a randomly chosen woman is less than 63 inches tall is 0.267, or 26.7%.

(b) To find the probability that a randomly chosen woman is less than 70 inches tall, we need to find the area under the standard normal curve to the left of the standardized value of 70 inches.

Standardized value = (70 - mean) / standard deviation
= (70 - 64.5) / 2.4
= 2.29

Using a standard normal distribution table or a calculator, we find that the area to the left of 2.29 is 0.988.

Therefore, the probability that a randomly chosen woman is less than 70 inches tall is 0.988, or 98.8%.

(c) To find the probability that a randomly chosen woman is between 63 and 70 inches tall, we need to find the difference between the areas under the standard normal curve to the left of the standardized values of 63 and 70 inches.

Standardized value for 63 inches = (63 - mean) / standard deviation
= (63 - 64.5) / 2.4
= -0.625

Standardized value for 70 inches = (70 - mean) / standard deviation
= (70 - 64.5) / 2.4
= 2.29

Using a standard normal distribution table or a calculator, we find that the area to the left of -0.625 is 0.267. The area to the left of 2.29 is 0.988. Therefore, the area between these two values is 0.988 - 0.267 = 0.721.

Therefore, the probability that a randomly chosen woman is between 63 and 70 inches tall is 0.721, or 72.1%.

(d) To find the percentage of women shorter than Alice (72 inches tall), we first need to find the standardized value for 72 inches.

Standardized value = (72 - mean) / standard deviation
= (72 - 64.5) / 2.4
= 3.125

Using a standard normal distribution table or a calculator, we find that the area to the left of 3.125 is 0.999.

Therefore, the percentage of women shorter than Alice (72 inches tall) is 99.9%.

To find the probabilities in this normal distribution problem, we can use the concept of Z-scores. A Z-score measures the number of standard deviations an individual value is from the mean. We can then use the Z-score to find the corresponding probability using the standard normal distribution table or a calculator.

(a) To find the probability that a randomly chosen woman is less than 63 inches tall, we need to calculate the Z-score for 63 inches.

Z = (X - μ) / σ
Z = (63 - 64.5) / 2.4
Z ≈ -0.625

Looking up the Z-score of -0.625 in a standard normal distribution table or using a calculator, we find that the corresponding probability is about 0.2660. Therefore, the probability that a randomly chosen woman is less than 63 inches tall is approximately 0.2660.

(b) To find the probability that a randomly chosen woman is less than 70 inches tall, we need to calculate the Z-score for 70 inches.

Z = (X - μ) / σ
Z = (70 - 64.5) / 2.4
Z ≈ 2.292

Again, looking up the Z-score of 2.292 in a standard normal distribution table or using a calculator, we find that the corresponding probability is about 0.9880. Therefore, the probability that a randomly chosen woman is less than 70 inches tall is approximately 0.9880.

(c) To find the probability that a randomly chosen woman is between 63 and 70 inches tall, we need to calculate the Z-scores for both 63 inches and 70 inches.

For 63 inches:
Z = (X - μ) / σ
Z = (63 - 64.5) / 2.4
Z ≈ -0.625

For 70 inches:
Z = (X - μ) / σ
Z = (70 - 64.5) / 2.4
Z ≈ 2.292

The probability of being between two values is the difference between the probabilities of being less than those values. So, the probability that a randomly chosen woman is between 63 and 70 inches tall is approximately 0.9880 - 0.2660 ≈ 0.7220.

(d) To find the percentage of women shorter than Alice, who is 72 inches tall, we need to calculate the Z-score for 72 inches.

Z = (X - μ) / σ
Z = (72 - 64.5) / 2.4
Z ≈ 3.125

Using a standard normal distribution table or a calculator, we find that the probability associated with a Z-score of 3.125 is about 0.9993. Therefore, approximately 99.93% of women are shorter than Alice, who is 72 inches tall.