Answer:
A) P(21.4317 < ¯x < 22.7561) = 0.2975
B) Q1 for the ¯x distribution = 21.9844
Explanation:
The Central Limit theorem allows us to say that
Sample mean = Population mean = 21.1 points
Mean of sampling distribution = σₓ = (σ/√n)
σ = population standard deviation = 8.9 points
n = sample size = 46
σₓ = (8.9/√46) = 1.3122334098 = 1.3122
A) P(21.4317 < ¯x < 22.7561) =
This is a normal distribution problem
To find this probability, we will use the normal probability tables
We first normalize/standardize 21.4317 and 22.7561.
The standardized score of any value is that value minus the mean divided by the standard deviation.
For 21.4317
z = (x - μ)/σ = (21.4317 - 21.1)/1.3122 = 0.25
For 22.7561
z = (x - μ)/σ = (22.7561 - 21.1)/1.3122 = 1.26
The required probability
P(21.4317 < ¯x < 22.7561) = P(0.25 < z < 1.26)
Checking the tables
P(21.4317 < ¯x < 22.7561) = P(0.25 < z < 1.26)
= P(z < 1.26) - P(z < 0.25)
= 0.89617 - 0.59871
= 0.29746 = 0.2975 to 4 d.p.
B) Q1 for the distribution is the first quartile. The first quartile is greater than 25% of the distribution.
P(x > Q1) = 0.25
Let the z-score that corresponds to Q1 be z'
P(x > Q1) = P(z > z') = 0.25
But P(z > z') = 1 - P(z ≤ z') = 0.25
P(z ≤ z') = 1 - 0.25 = 0.75
From the normal distribution tables,
z' = 0.674
z' = (Q1 - μ)/σ
0.674 = (Q1 - 21.1)/1.3122
Q1 = 0.674×1.3122 + 21.1 = 21.9844228 = 21.9844 to 4 d.p.
Hope this Helps!!!
