The probability that an observation, following a normal distribution, will lie within μ
± 1.3σ is
a. 9.5 %
b. 19.0%
c. 40.3%
d. 50.0%
e. 80.6%
µ ± 1.0 SD = approximately 68%
To find the probability that an observation following a normal distribution will lie within μ ± 1.3σ, you can use the standard normal distribution and the Z-score formula.
1. The Z-score formula is given by:
Z = (X - μ) / σ
Where:
- X is the value of the observation
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
2. In this case, since we want to find the probability within μ ± 1.3σ, we need to calculate the Z-score for both X = μ + 1.3σ and X = μ - 1.3σ.
3. Once we have the Z-score values, we can look up the corresponding probabilities in the standard normal distribution table.
4. The probability between the two Z-scores will give us the desired probability.
Let's calculate the Z-scores and find the probability:
Z1 = (μ + 1.3σ - μ) / σ = 1.3
Z2 = (μ - 1.3σ - μ) / σ = -1.3
Looking up the Z-scores in the standard normal distribution table, we can find the probabilities corresponding to these Z-scores:
P(Z ≤ Z1) = P(Z ≤ 1.3) ≈ 0.9032
P(Z ≤ Z2) = P(Z ≤ -1.3) ≈ 0.0968
To find the probability between Z1 and Z2, we need to subtract the smaller probability from the larger one:
P(Z2 < Z < Z1) = P(Z ≤ Z1) - P(Z ≤ Z2)
= 0.9032 - 0.0968
≈ 0.8064
Therefore, the probability that an observation will lie within μ ± 1.3σ is approximately 80.6%.
So the correct answer is:
e. 80.6%