Answer:
a) P(X < 6) = 0.0838
b) P(8<=x<=10) = 0.3682
c) a = 7.1244
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 8.88, \sigma = 2.09[/tex]
a. Find P(x < 6)
This is the pvalue of Z when X = 6. So:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{6 - 8.88}{2.09}[/tex]
[tex]Z = -1.38[/tex]
[tex]Z = -1.38[/tex] has a pvalue of 0.0838.
So P(X < 6) = 0.0838
b. Find P(8<=x<=10)
This is the pvalue of Z when X = 10 subtracted by the pvalue of Z when X = 8. So:
X = 10
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{10 - 8.88}{2.09}[/tex]
[tex]Z = 0.54[/tex]
[tex]Z = 0.54[/tex] has a pvalue of 0.7054.
X = 8
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{8 - 8.88}{2.09}[/tex]
[tex]Z = -0.42[/tex]
[tex]Z = -0.42[/tex] has a pvalue of 0.3372.
So P(8<=x<=10) = 0.7054 - 0.3372 = 0.3682
c. Find the value for which P(x < a) = 0.2
This is X = a when Z has a pvalue of 0.2. So [tex]Z = -0.84[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.84 = \frac{a - 8.88}{2.09}[/tex]
[tex]a - 8.88 = 2.09*(-0.84)[/tex]
[tex]a = 7.1244[/tex]
For the normal distribution, P(x < 6) is 8.38% while P(8<=x<=10) is 36.82%
The z score is used to determine by how many standard deviations the raw score is above or below the mean. It is given by:
z = (x - μ)/σ
where x is raw score, σ is standard deviation and μ is mean
From the question,
μ = 8.88, σ = 2.09
For x < 6:
z = (6 - 8.88)/2.09 = -1.38
P(z < -1.38) = 0.0838
For x > 8:
z = (8 - 8.88)/2.09 = -0.42
For x < 10:
z = (10 - 8.88)/2.09 = 0.54
P(-0.42< z < 0.54) = P(z < 0.54) - P(z < -0.42) = 0.7054 - 0.3372 = 0.3682
P(x < 6) is 8.38% while P(8<=x<=10) is 36.82%
Find out more on Z score at: https://brainly.com/question/25638875
