Test scores on a university admissions test are normally distributed, with a mean of 500 and a standard deviation of 100.
a. What is the probability that a randomly selected applicant scores between 425 and 575?
b. What is the probability that a randomly selected applicant scores 625 or more?
c. What is the probability that a randomly selected applicant scores less than 500?
d. Twenty per cent of test scores exceed what value?
plug in your numbers and play around with Z table stuff at
http://davidmlane.com/hyperstat/z_table.html
OR… just in case your computer crashes, you could use this equation:
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
For d, just reverse the process. Find .20, insert its Z score and find the raw score.
To solve these questions, we'll use the normal distribution formula and z-scores.
Part a:
To find the probability that a randomly selected applicant scores between 425 and 575, we'll first calculate the z-scores for both values.
z1 = (425 - 500) / 100
z2 = (575 - 500) / 100
Then we'll use a standard normal distribution table or calculator to find the probabilities associated with these z-scores:
P(425 < X < 575) = P(z1 < Z < z2)
Part b:
To find the probability that a randomly selected applicant scores 625 or more, we'll calculate the z-score for 625:
z3 = (625 - 500) / 100
Then we'll use the normal distribution table or calculator to find the probability associated with this z-score:
P(X ≥ 625) = P(Z ≥ z3)
Part c:
To find the probability that a randomly selected applicant scores less than 500, we can calculate the z-score for 500:
z4 = (500 - 500) / 100 = 0
This means that the applicant's score is at the mean of the distribution. The probability of selecting an applicant with a score less than 500 is the area under the curve to the left of z4:
P(X < 500) = P(Z < 0)
Part d:
To find the value beyond which 20% of test scores exceed, we'll need to find the z-score associated with the 80th percentile.
P(X > x) = 0.20
We'll then use the inverse normal distribution (also known as the z-score to value) to find the value corresponding to this percentile.
Now let's calculate the answers step-by-step.
a. What is the probability that a randomly selected applicant scores between 425 and 575?
First, let's calculate the z-scores:
z1 = (425 - 500) / 100 = -0.75
z2 = (575 - 500) / 100 = 0.75
Using the standard normal distribution table or calculator, we can find the probabilities associated with these z-scores:
P(425 < X < 575) = P(-0.75 < Z < 0.75)
b. What is the probability that a randomly selected applicant scores 625 or more?
Calculate the z-score for 625:
z3 = (625 - 500) / 100 = 1.25
Using the normal distribution table or calculator, we can find the probability associated with this z-score:
P(X ≥ 625) = P(Z ≥ 1.25)
c. What is the probability that a randomly selected applicant scores less than 500?
Since 500 is at the mean of the distribution, the z-score is 0:
z4 = (500 - 500) / 100 = 0
We can find the probability of selecting an applicant with a score less than 500 by calculating the area under the curve to the left of z4:
P(X < 500) = P(Z < 0)
d. Twenty percent of test scores exceed what value?
To find the value beyond which 20% of test scores exceed, we need to find the z-score associated with the 80th percentile:
P(X > x) = 0.20
Using the inverse normal distribution (z-score to value), we can find the value corresponding to this percentile.
To solve these probability questions, we will use the standard normal distribution and the z-score.
The z-score formula is: z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
a. To find the probability that a randomly selected applicant scores between 425 and 575, we need to find the area under the normal curve between these two scores.
First, we need to find the z-scores for both values:
For 425, z1 = (425 - 500) / 100 = -0.75
For 575, z2 = (575 - 500) / 100 = 0.75
Next, we use a standardized normal distribution table (or a z-table) to find the area between these two z-scores. We look up the area corresponding to z = -0.75 and subtract the area corresponding to z = 0.75.
Using the z-table, the area corresponding to z = -0.75 is 0.2266, and the area corresponding to z = 0.75 is 0.7734.
Therefore, the probability that a randomly selected applicant scores between 425 and 575 is 0.7734 - 0.2266 = 0.5468, or 54.68%.
b. To find the probability that a randomly selected applicant scores 625 or more, we first find the z-score for 625:
z = (625 - 500) / 100 = 1.25
Using the z-table, we find the area corresponding to z = 1.25, which is 0.8944.
Therefore, the probability that a randomly selected applicant scores 625 or more is 1 - 0.8944 = 0.1056, or 10.56%.
c. To find the probability that a randomly selected applicant scores less than 500, we find the z-score for 500:
z = (500 - 500) / 100 = 0
Using the z-table, we find the area corresponding to z = 0, which is 0.5.
Therefore, the probability that a randomly selected applicant scores less than 500 is 0.5, or 50%.
d. To find the value that exceeds 20% of test scores, we need to find the z-score that corresponds to an area of 0.2.
Using the z-table, we look up the z-score that has an area of 0.2. We find that it is approximately -0.84.
Now, we can use the z-score formula to solve for the test score:
-0.84 = (x - 500) / 100
-0.84 * 100 = x - 500
-84 = x - 500
x = -84 + 500
x = 416
Therefore, 20% of test scores exceed the value of 416.