Answer:
Part 1
(a) 0.28434
(b) 0.43441
(c) 29.9 mm
Part 2
(a) 0.97722
Step-by-step explanation:
There are two questions here. We'll break them into two.
Part 1.
This is a normal distribution problem healthy children having the size of their left atrial diameters normally distributed with
Mean = μ = 26.4 mm
Standard deviation = σ = 4.2 mm
a) proportion of healthy children have left atrial diameters less than 24 mm
P(x < 24)
We first normalize/standardize 24 mm
The standardized score for any value is the value minus the mean then divided by the standard deviation.
z = (x - μ)/σ = (24 - 26.4)/4.2 = -0.57
The required probability
P(x < 24) = P(z < -0.57)
We'll use data from the normal probability table for these probabilities
P(x < 24) = P(z < -0.57) = 0.28434
b) proportion of healthy children have left atrial diameters between 25 and 30 mm
P(25 < x < 30)
We first normalize/standardize 25 mm and 30 mm
For 25 mm
z = (x - μ)/σ = (25 - 26.4)/4.2 = -0.33
For 30 mm
z = (x - μ)/σ = (30 - 26.4)/4.2 = 0.86
The required probability
P(25 < x < 30) = P(-0.33 < z < 0.86)
We'll use data from the normal probability table for these probabilities
P(25 < x < 30) = P(-0.33 < z < 0.86)
= P(z < 0.86) - P(z < -0.33)
= 0.80511 - 0.37070 = 0.43441
c) For healthy children, what is the value for which only about 20% have a larger left atrial diameter.
Let the value be x' and its z-score be z'
P(x > x') = P(z > z') = 20% = 0.20
P(z > z') = 1 - P(z ≤ z') = 0.20
P(z ≤ z') = 0.80
Using normal distribution tables
z' = 0.842
z' = (x' - μ)/σ
0.842 = (x' - 26.4)/4.2
x' = 29.9364 = 29.9 mm
Part 2
Population mean = μ = 65 mm
Population Standard deviation = σ = 5 mm
The central limit theory explains that the sampling distribution extracted from this distribution will approximate a normal distribution with
Sample mean = Population mean
¯x = μₓ = μ = 65 mm
Standard deviation of the distribution of sample means = σₓ = (σ/√n)
where n = Sample size = 100
σₓ = (5/√100) = 0.5 mm
So, probability that the sample mean distance ¯x for these 100 will be between 64 and 67 mm = P(64 < x < 67)
We first normalize/standardize 64 mm and 67 mm
For 64 mm
z = (x - μ)/σ = (64 - 65)/0.5 = -2.00
For 67 mm
z = (x - μ)/σ = (67 - 65)/0.5 = 4.00
The required probability
P(64 < x < 67) = P(-2.00 < z < 4.00)
We'll use data from the normal probability table for these probabilities
P(64 < x < 67) = P(-2.00 < z < 4.00)
= P(z < 4.00) - P(z < -2.00)
= 0.99997 - 0.02275 = 0.97722
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