The heights of women in a certain population have a Normal distribution with mean 64 inches and standard deviation 3.5 inches. We select three women at random from this population. Assume that their heights are independent. Find the probability that the tallest of these three women will be taller than 67 inches.
My work:
(67-64)/3.5 = 0.857, using the z-score table I then get 0.8023, I then subtracted this from 1 to get 0.1977. But, it's not the right answer.
Appreciate any help I can get!
You are on the right track by finding the z-score of 67 inches. However, we want to find the probability that the tallest of the three women is taller than 67 inches.
The probability that a single woman is taller than 67 inches is 0.1977, which you have correctly found. We now need to find the probability that all three women are shorter than 67 inches. The probability that a single woman is shorter than 67 inches is 1 - 0.1977 = 0.8023. Since their heights are independent, the probability that all three women are shorter than 67 inches is 0.8023^3 = 0.5164.
Finally, the probability that at least one of the three women is taller than 67 inches (which is the same as the tallest of the three women being taller than 67 inches) is 1 - 0.5164 = 0.4836. This is your final answer.
To find the probability that the tallest of the three women will be taller than 67 inches, we need to use the cumulative distribution function (CDF) of the normal distribution.
Let's define X as the random variable representing the height of a randomly selected woman from the population. X follows a normal distribution with mean μ = 64 inches and standard deviation σ = 3.5 inches.
To find the probability that the tallest woman out of three will be taller than 67 inches, we can find the probability that no woman is taller than 67 inches and subtract it from 1.
P(tallest woman > 67 inches) = 1 - P(all women ≤ 67 inches)
First, let's find the probability that a single woman's height is less than or equal to 67 inches:
P(X ≤ 67) = Φ((67 - μ) / σ) = Φ((67 - 64) / 3.5)
Using the standard normal distribution table, we can find that Φ(0.857) is approximately 0.8032.
However, since we are interested in the probability that *all three* women have a height less than or equal to 67 inches, we need to raise this probability to the power of 3:
P(all women ≤ 67) = (0.8032)³
Finally, we can calculate the probability that at least one woman will be taller than 67 inches:
P(tallest woman > 67 inches) = 1 - P(all women ≤ 67) = 1 - (0.8032)³ ≈ 1 - 0.5143 ≈ 0.4857
So, the probability that the tallest of these three women will be taller than 67 inches is approximately 0.4857 or 48.57%.
It seems like you have made a mistake in calculating the probability. Let me explain how to correctly find the probability that the tallest woman is taller than 67 inches in this scenario.
To solve this problem, we need to find the probability that the height of the tallest woman exceeds 67 inches. We can break down this problem into two steps:
Step 1: Find the probability that a single woman's height exceeds 67 inches.
To do this, we need to standardize the height of 67 inches using the formula:
z = (value - mean) / standard deviation
In this case, the value is 67 inches, the mean is 64 inches, and the standard deviation is 3.5 inches. Plugging these values into the formula, we get:
z = (67 - 64) / 3.5 = 0.857
Step 2: Find the probability that all three women have heights less than or equal to 67 inches.
Since the heights of the women are independent, the probability of all three women having heights less than or equal to 67 inches is simply the probability of a single woman having a height less than or equal to 67 inches, raised to the power of 3 (since there are three women).
Using the standard normal distribution table, we can find the probability corresponding to a z-score of 0.857. The table may give you a value of 0.8051 for this z-score.
Then, the probability of all three women having heights less than or equal to 67 inches is:
P(height <= 67) = 0.8051^3 ≈ 0.5197
Finally, to find the probability that the tallest woman is taller than 67 inches, we subtract this probability from 1, since we want the complement of the event:
P(tallest woman > 67) = 1 - 0.5197 ≈ 0.4803
Therefore, the correct probability that the tallest of the three women will be taller than 67 inches is approximately 0.4803.