Answer:
A: 16%
B: 95%
C: 97.5%
Step-by-step explanation:
According to the empirical rule:
68% of a normal distribution is between -1 and +1 standard deviations.
95% of a normal distribution is between -2 and +2 standard deviations.
99.7% of a normal distribution is between -3 and +3 standard deviations.
Given μ = 27 and σ = 2.
Part A
29 is one standard deviation above the mean. We can show this by calculating the z-score:
z = (x − μ) / σ
z = (29 − 27) / 2
z = 1
We know that 68% is between -1 and +1 standard deviations. Since normal distributions are symmetrical, we can also say that 34% is between 0 and +1 standard deviations.
P(0 < Z < 1) = 68%/2 = 34%
We can also say that 50% is less than 0 standard deviations.
P(Z < 0) = 50%
Therefore, P(Z < 1) = 34% + 50% = 84%.
Which means P(Z > 1) = 100% − 84% = 16%.
Part B
Like before, calculate the z-scores:
z₁ = (23 − 27) / 2
z₁ = -2
z₂ = (31 − 27) / 2
z₂ = 2
From the empirical rule, we know this is 95% of the normal distribution.
Part C
We found in part B that the z-score is 2.
P(0 < Z < 2) = 95%/2 = 47.5%
P(Z < 2) = 50% + 47.5% = 97.5%
