show me Police academy acceptance exams to qualify for a police academy, applicants are given a test of physical fitness. the scores are norally distributed with a mean of 64 and standard deviation of 9. if only the top 20% of the applicants are selected find the cutoff score?

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.20) and its Z score. Insert into above equation to calculate raw score.

56.44

To find the cutoff score for the top 20% of applicants, we need to determine the z-score associated with the 80th percentile. The z-score represents how many standard deviations an observation is from the mean in a normal distribution.

Step 1: Determine the z-score using the percentile value:
To find the z-score, we need to use the inverse normal distribution or the Z-table. The inverse normal distribution for the 80th percentile is 0.8416.

Step 2: Convert the z-score to the actual score:
Once we have the z-score, we can use the formula:
z = (x - mean) / standard deviation

Rearranging the formula, we can solve for the actual score, x:
x = (z * standard deviation) + mean

Step 3: Substitute the values into the formula:
Using the given mean of 64 and standard deviation of 9:
x = (0.8416 * 9) + 64

Step 4: Calculate the cutoff score:
x = 7.5744 + 64
x ≈ 71.57

Therefore, the cutoff score for the top 20% of applicants is approximately 71.57.

To find the cutoff score for the top 20% of applicants, you can use the concept of z-scores and the standard normal distribution. Here's how to calculate it step by step:

Step 1: Determine the z-score for the desired percentile.
The percentile corresponds to the area under the standard normal curve. In this case, we want to find the z-score that corresponds to the top 20% of applicants. Since the distribution is symmetric, the bottom 80% corresponds to the z-score of 0.20. We can find this value using a standard normal distribution table or a calculator. The z-score corresponding to the top 20% is approximately 0.84.

Step 2: Apply the z-score formula.
The formula for a z-score is: z = (x - μ) / σ
where x is the raw score, μ is the mean, and σ is the standard deviation.

Step 3: Rearrange the formula to solve for x.
Reorganize the formula to find the cutoff score, x: x = z * σ + μ

Now let's calculate the cutoff score:

x = 0.84 * 9 + 64
x ≈ 7.56 + 64
x ≈ 71.56

Therefore, the cutoff score for the top 20% of applicants is approximately 71.56. Applicants scoring above this value will be selected for the police academy.