Answer:
Step-by-step explanation:
From the given information:
Mean [tex]\mu[/tex] = 3.2
Standard deviation [tex]\sigma[/tex] = 1.9
sample size n = 38
Mean of sampling distribution [tex]\mu _{\bar x} = \mu = 3.2[/tex]
Standard deviation of the sample mean is:
[tex]\sigma _{\bar x} = \dfrac{\sigma}{\sqrt{n}} \\ \\ = \dfrac{1.9}{\sqrt{38}} \\ \\ = 0.3082[/tex]
a)
To find P(x < 3.4)
[tex]= P\Big( \dfrac{(\bar x - \mu_{\bar x} ) }{\sigma_{\bar x}} < \dfrac{3.4 - 3.2}{0.3082}\ \Big)[/tex]
[tex]= P\Big( Z< \dfrac{0.2}{0.3082}\ \Big) \\ \\= P\Big( Z< 0.65 \ \Big)[/tex]
Using the standard normal table
[tex]P(z < 0.65) = 0.7422[/tex]
The Bell curved shape is attached in the diagram below.
b)
To find P(3 < x < 3.6)
[tex]= P\Big( \dfrac{3.0-3.2}{0.3082} < \dfrac{(\bar x - \mu_{\bar x} ) }{\sigma_{\bar x}} < \dfrac{3.6 - 3.2}{0.3082}\ \Big)[/tex]
[tex]= P\Big( \dfrac{-0.2}{0.3082} < \dfrac{(\bar x - \mu_{\bar x} ) }{\sigma_{\bar x}} < \dfrac{0.4}{0.3082}\ \Big)[/tex]
[tex]= P (-0.65 < Z < 1.30)[/tex]
[tex]= P(Z < 1.30) - P(Z < -0.65)[/tex]
Using the standard normal table
= 0.9032 -0.2578
=0.6454
