Scores on a test are Normally distributed with mean 78 and standard deviation 8. Find the probability that a randomly chosen student scored between 70 and 80 on the test.
70 = µ - 1σ
80 = µ + 0.125σ
so, what does your Z table tell you?
You can play around with Z-table stuff at
https://davidmlane.com/hyperstat/z_table.html
To find the probability that a randomly chosen student scored between 70 and 80 on the test, we need to calculate the area under the normal curve between these two scores.
Step 1: Standardize the scores
To do this, we can use the z-score formula:
z = (x - μ) / σ
where
x = score
μ = mean
σ = standard deviation
For the lower score of 70:
z1 = (70 - 78) / 8
For the upper score of 80:
z2 = (80 - 78) / 8
Step 2: Look up the z-scores
We can use a standard normal distribution table or a calculator to determine the proportions of the area under the curve associated with the z-scores.
Looking up z1 and z2 in the standard normal distribution table, we find:
z1 = -1.00, z2 = 0.25
Step 3: Calculate the probability
To find the probability between z1 and z2, we need to calculate the difference between the two areas:
P(70 ≤ x ≤ 80) = P(z1 ≤ z ≤ z2)
= P(z ≤ z2) - P(z ≤ z1)
From the standard normal distribution table, we find the corresponding probabilities:
P(z ≤ z2) = 0.5987
P(z ≤ z1) = 0.1587
Therefore,
P(70 ≤ x ≤ 80) = P(z ≤ z2) - P(z ≤ z1)
= 0.5987 - 0.1587
= 0.4400
So, the probability that a randomly chosen student scored between 70 and 80 on the test is 0.4400 or 44.00%.
To find the probability that a randomly chosen student scored between 70 and 80 on the test, we need to calculate the area under the normal curve between these two scores.
First, let's standardize the scores by using the z-score formula:
z = (x - μ) / σ
where:
z = z-score
x = score
μ = mean of the distribution
σ = standard deviation of the distribution
For the score 70:
z1 = (70 - 78) / 8 = -1
For the score 80:
z2 = (80 - 78) / 8 = 0.25
Now, we need to find the area under the normal curve between these two z-scores.
Using a Z-table or a statistical calculator, we can find the corresponding probabilities for these z-scores.
The area to the left of z = -1 is 0.1587, and the area to the left of z = 0.25 is 0.5987.
To find the probability between these two scores, we subtract the smaller area from the larger area:
P(70 ≤ X ≤ 80) = P(z1 ≤ Z ≤ z2)
= P(Z ≤ z2) - P(Z ≤ z1)
= 0.5987 - 0.1587
= 0.44
Therefore, the probability that a randomly chosen student scored between 70 and 80 on the test is 0.44 or 44%.