The heights of young women are approximately normally distributed with a mean of 64.5 inches and a standard deviation of 2.5 inches. Determine the following:
what percent of young women would be acceptable as members if being a
member required a height of 70 inches or more?
For the population of young women determine the heights of the shortest 8%.
Diagrams required.
You need a z- score
70 - 64.5 divided by 2.5
Once you get the z-score, use a calculator or normal table to find the probability to the right of that z-score.
To determine the percentage of young women who would be acceptable as members if the height requirement is 70 inches or more, we need to calculate the area under the normal distribution curve to the right of 70 inches. This area represents the percentage of young women whose height is equal to or greater than 70 inches.
To calculate this, we will use a standard normal distribution table or a calculator with a normal distribution function. The standard normal distribution has a mean of 0 and a standard deviation of 1. Since our given distribution has a mean of 64.5 inches and a standard deviation of 2.5 inches, we need to convert the value 70 inches to a z-score using the formula:
z = (x - μ) / σ
where x is the given value (70 inches), μ is the mean (64.5 inches), and σ is the standard deviation (2.5 inches).
Plugging in the values, we get:
z = (70 - 64.5) / 2.5
z = 5.5 / 2.5
z = 2.2
Now, we can use the standard normal distribution table or a calculator to find the area to the right of 2.2. This area represents the percentage of young women whose height is equal to or greater than 70 inches.
Using the table or calculator, we find that the area to the right of 2.2 is approximately 0.0139 or 1.39%. Therefore, approximately 1.39% of young women would be acceptable as members if there is a height requirement of 70 inches or more.
Now, let's determine the heights of the shortest 8% of young women in the population.
To find the height value corresponding to the shortest 8% of young women, we need to find the z-score corresponding to the 8th percentile. The 8th percentile represents the cutoff below which 8% of the population will fall.
To do this, we can use the standard normal distribution table or a calculator to find the z-score that corresponds to the 8th percentile. The z-score represents how many standard deviations a particular value is from the mean.
The z-score for the 8th percentile is found by subtracting 0.08 (8%) from 1 since we want the area to the left of the z-score. Using the table or calculator, we find that the z-score is approximately -1.4051 (rounded to four decimal places).
Now, we can use the formula to convert this z-score back to the original distribution:
x = (z * σ) + μ
where x is the value we want to find (height in this case), z is the z-score (-1.4051), σ is the standard deviation (2.5 inches), and μ is the mean (64.5 inches).
Plugging in the values, we get:
x = (-1.4051 * 2.5) + 64.5
x = -3.51275 + 64.5
x = 61.98725
Therefore, the height corresponding to the shortest 8% of young women in the population is approximately 61.99 inches.
To summarize:
- Approximately 1.39% of young women would be acceptable as members if the height requirement is 70 inches or more.
- The height corresponding to the shortest 8% of young women in the population is approximately 61.99 inches.
To determine the percentage of young women that would be acceptable as members if being a member required a height of 70 inches or more, we can use the standard normal distribution.
Step 1: Find the Z-score corresponding to a height of 70 inches.
Z = (X - μ) / σ
Where:
Z is the Z-score,
X is the given height (70 inches),
μ is the mean height (64.5 inches),
σ is the standard deviation (2.5 inches).
Substituting the given values:
Z = (70 - 64.5) / 2.5
Z = 2.2
Step 2: Calculate the area under the standard normal curve to the right of the Z-score.
Using a Z-table or calculator, we can find that the area to the right of Z = 2.2 is approximately 0.0139.
Step 3: Convert the area to a percentage.
To get the percentage, we multiply the area by 100:
0.0139 * 100 = 1.39%
Therefore, approximately 1.39% of young women would be acceptable as members if being a member required a height of 70 inches or more.
For the population of young women, to determine the heights of the shortest 8%, we need to find the Z-score corresponding to the 8th percentile.
Step 1: Find the Z-score corresponding to the 8th percentile.
Using a Z-table or calculator, we can find that the Z-score for the 8th percentile is approximately -1.41.
Step 2: Calculate the corresponding height using the Z-score formula.
Z = (X - μ) / σ
Rearranging the formula, we have:
X = Z * σ + μ
Substituting the values:
X = -1.41 * 2.5 + 64.5
X ≈ 60.48 inches
Therefore, the height of the shortest 8% of young women is approximately 60.48 inches.
Note: Diagrams are not required for these calculations, but you can refer to a standard normal distribution diagram if you find it helpful in visualizing the concepts.