Answer:
a. The probability of a value between 75.0 and 90.0 is 0.40173
b. The probability of a value of 75.0 or less is 0.35942
c. The probability of a value between 55.0 and 70.0 is 0.19712
Explanation:
To solve for this we make use of the z score formula.
z = (x-μ)/σ,
where
x = raw score
μ = the population mean
σ = the population standard deviation.
a. Compute the probability of a value between 75.0 and 90.0.
When x = 75
μ =80.0 and σ = 14.0.
z = (x - μ)/σ
z = 75 - 80/ 14
z = -0.35714
z = -0.36 to 2 decimal places
Using the z score table to find the probability
P(x = 75) = P(z = -0.36)
= 0.35942
For x = 90
z = 90 - 80/14
z = 0.71429
z = 0.71 to 2 decimal place
Using the z score table to find the probability
P(x = 90) = P(z = 0.71)
= 0.76115
The probability of a value between 75.0 and 90.0 is:
75 < x < 90
= P( x = 90) - P(x = 75)
= 0.76115 - 0.35942
= 0.40173
Therefore, probability of a value between 75.0 and 90.0 is 0.40173
b. Compute the probability of a value of 75.0 or less.
For x = 75
From the question, we know that
mean of 80.0 and a standard deviation of 14.0.
z = (x - μ)/σ
z = 75 - 80/ 14
z = -0.35714
z = -0.36 approximately to 2 decimal places.
P-value from Z-Table:
P(x ≤ 75) = 0.35942
c. Compute the probability of a value between 55.0 and 70.0.
For x = 55
From the question, we know that
mean of 80.0 and a standard deviation of 14.0.
z = (x - μ)/σ
z = 55 - 80/ 14
z = -1.78571
z = -1.79 approximately to 2 decimal places
Using the z score table to find the probability
P(x = 55) = P(z = -1.79)
= 0.036727
For x = 70
z = 70 - 80/14
z = -0.71429
z = - 0.71 approximately to 2 decimal place.
Using the z score table to find the probability
P(x = 70) = P(z = -0.71)
= 0.23885
The probability of a value between 55.0 and 70.0 is:
55 < x < 70
= P( x = 70) - P(x = 55)
= P( z = -0.71) - P(z = -1.79)
= 0.23885 - 0.03673
= 0.19712