A population of values has a normal distribution with μ=76.6 and σ=43.9
You intend to draw a random sample of size n=142.
Find P75, which is the score separating the bottom 75% scores from the top 25% scores.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.75) and its Z score. Insert data into first equation and solve for score.
That would be for finding the distribution of means. Use the equation:
Z = (score-mean)/SD
To find P75, which is the score separating the bottom 75% scores from the top 25% scores, we can use the z-score formula and the standard normal distribution.
Step 1: Find the z-score corresponding to the desired percentile.
Since we want to find the score separating the bottom 75% scores, we can find the z-score corresponding to the percentile 75% using the standard normal distribution table or calculator.
The z-score corresponding to the 75th percentile is approximately 0.674.
Step 2: Use the formula to find the raw score (x) corresponding to the z-score.
We can use the formula:
z = (x - μ) / σ
Rearranging the formula, we have:
x = z * σ + μ
Substituting in the given values:
x = 0.674 * 43.9 + 76.6
Step 3: Calculate the value of x.
x ≈ 106.8
Therefore, the score separating the bottom 75% scores from the top 25% scores (P75) is approximately 106.8.
To find the score separating the bottom 75% scores from the top 25% scores, we need to first find the corresponding z-score and then calculate the x-value (score) from the z-score.
Step 1: Find the z-score corresponding to the desired percentile.
The z-score can be calculated using the formula:
z = (x - μ) / σ
where x is the score, μ is the population mean, and σ is the population standard deviation.
Step 2: Use the z-score to find the x-value (score).
The x-value can be calculated using the formula:
x = z * σ + μ
Now let's calculate the score separating the bottom 75% from the top 25%.
Step 1: Find the z-score corresponding to the 75th percentile.
To find the z-score, we can use a standard normal distribution table or a calculator.
Step 1 (continued): Look up the z-score corresponding to the 75th percentile.
Using a standard normal distribution table, the z-score corresponding to the 75th percentile is approximately 0.675.
Step 2: Calculate the x-value (score).
Using the formula mentioned earlier:
x = z * σ + μ
x = 0.675 * 43.9 + 76.6
x ≈ 105.7865
Therefore, the score separating the bottom 75% scores from the top 25% scores is approximately 105.79.