Answer:
a. [tex]\mathbf{P(\overline x < 68) = 0.9998}[/tex]
b. [tex]\mathbf{P(68 < \overline x < 69 ) =0}[/tex]
c. [tex]\mathbf{P ( \overline x > 66 ) =0.02275}[/tex]
Step-by-step explanation:
Given that ;
Let Y be a random variable In a population, where:
mean [tex]\mu_y[/tex] = 65
[tex]\sigma^2_y[/tex] = 49
standard deviation σ = [tex]\sqrt{49}[/tex] = 7
The objective is to determine the following :
In a random sample of size n = 69, find Pr(Y <68) =
Using the Central limit theorem
[tex]P(\overline x < 68) = \begin {pmatrix} \dfrac{\overline x - \mu }{\dfrac{\sigma}{\sqrt{n}}} < \dfrac{68 - \mu }{\dfrac{\sigma}{\sqrt{n}}} } \end {pmatrix}[/tex]
[tex]P(\overline x < 68) = \begin {pmatrix}Z < \dfrac{68 - 65 }{\dfrac{7}{\sqrt{69}}} } \end {pmatrix}[/tex]
[tex]P(\overline x < 68) = \begin {pmatrix}Z < \dfrac{3 }{\dfrac{7}{8.3066}} } \end {pmatrix}[/tex]
[tex]P(\overline x < 68) = (Z < 3.5599 )[/tex]
From the z tables:
[tex]\mathbf{P(\overline x < 68) = 0.9998}[/tex]
In a random sample of size n = 124, find Pr (68< Y <69)=
[tex]P(68 < \overline x < 69 ) = P \begin {pmatrix} \dfrac{68- \mu}{\dfrac{\sigma}{\sqrt{n}}} < \dfrac{\overline x - \mu}{\dfrac{\sigma}{\sqrt{n}}} < \dfrac{ 69 - \mu}{\dfrac{\sigma}{\sqrt{n}}} \end {pmatrix}[/tex]
[tex]P(68 < \overline x < 69 ) = P \begin {pmatrix} \dfrac{68- 65}{\dfrac{7}{\sqrt{124}}} < Z < \dfrac{ 69 - 65}{\dfrac{7}{\sqrt{124}}} \end {pmatrix}[/tex]
[tex]P(68 < \overline x < 69 ) = P \begin {pmatrix} \dfrac{3}{\dfrac{7}{11.1355}} < Z < \dfrac{ 4}{\dfrac{7}{11.1355}} \end {pmatrix}[/tex]
[tex]P(68 < \overline x < 69 ) = P \begin {pmatrix} 4.7724 < Z < 6.3631 \end {pmatrix}[/tex]
[tex]P(68 < \overline x < 69 ) = P( Z < 6.3631 ) - P ( Z < 4.7724 )[/tex]
From z tables
[tex]P(68 < \overline x < 69 ) = 0.9999 - 0.9999[/tex]
[tex]\mathbf{P(68 < \overline x < 69 ) =0}[/tex]
In a random sample of size n = 196, find Pr (Y >66)=
[tex]P ( \overline x > 66 ) = P ( \dfrac{\overline x -\mu }{\dfrac{\sigma}{\sqrt{n}}} > \dfrac{66 -\mu }{\dfrac{\sigma}{\sqrt{n}}})[/tex]
[tex]P ( \overline x > 66 ) = P ( Z> \dfrac{66 - 65 }{\dfrac{7}{\sqrt{196}}})[/tex]
[tex]P ( \overline x > 66 ) = P ( Z> \dfrac{1 }{\dfrac{7}{14}})[/tex]
[tex]P ( \overline x > 66 ) = P ( Z> \dfrac{14 }{7})[/tex]
[tex]P ( \overline x > 66 ) = P ( Z>2)[/tex]
[tex]P ( \overline x > 66 ) = 1 - P ( Z<2)[/tex]
from z tables
[tex]P ( \overline x > 66 ) = 1 - 0.9773[/tex]
[tex]\mathbf{P ( \overline x > 66 ) =0.02275}[/tex]
