A normal distribution has a mean of 80 and a standard deviation of 14. Compute the probability of a value between 75.0 and 90.0.
a) .2022
b) .3595
c) .4017
d) None of the above
x1=75
x2=90
z1=(x1-mean)/standard dev. = -0.35
z2=(x2-mean)/standard dev. = 0.71
then look for the area table from your book, get for the equivalent value from z1 and z2, then get the difference of the two.. 0.7642-0.3632 = 0.401
thus, i assumed its C.
To compute the probability of a value between 75.0 and 90.0 in a normal distribution with a mean of 80 and a standard deviation of 14, we can use the Z-score formula and then find the area under the curve using a Z-table or a statistical calculator.
First, we need to convert the given values into Z-scores. The Z-score is calculated using the formula:
Z = (X - μ) / σ
where:
X is the given value,
μ is the mean, and
σ is the standard deviation.
For 75.0:
Z1 = (75.0 - 80) / 14 = -0.3571
For 90.0:
Z2 = (90.0 - 80) / 14 = 0.7143
Next, we need to find the area between these Z-scores. This represents the probability of a value falling between 75.0 and 90.0.
Using a Z-table or a statistical calculator, we can look up the probabilities associated with the Z-scores Z1 and Z2. Subtracting the cumulative probability of Z1 from the cumulative probability of Z2 will give us the desired probability.
Looking up the Z-scores in the Z-table, we find that the cumulative probability for Z1 (-0.3571) is approximately 0.3621 and the cumulative probability for Z2 (0.7143) is approximately 0.7631.
Therefore, the probability of a value falling between 75.0 and 90.0 is:
P(75.0 < X < 90.0) = P(Z1 < Z < Z2)
= P(Z < Z2) - P(Z < Z1)
= 0.7631 - 0.3621
= 0.4010
Now let's compare this result to the given answer choices:
a) .2022
b) .3595
c) .4017
d) None of the above
The closest answer to our calculated probability of 0.4010 is c) .4017.
Therefore, the correct answer is c) .4017.