Question

Let Y be a random variable. In a​ population, mu Subscript Upper Y Baseline equals 65μY=65 and sigma Subscript Upper Y Superscript 2 Baseline equals 49σ2Y=49. Use the central limit theorem to answer the following questions. ​ (Note​: any intermediate results should be rounded to four decimal​ places)

In a random sample of size n​ = 69​, find Pr(Y <68) =
In a random sample of size n​ = 124​, find Pr (68< Y <69)=
In a random sample of size n​ = 196​, find Pr (Y >66)=

Answer 1

a.
`\mathbf{P(\overline x < 68) = 0.9998}`

b.
`\mathbf{P(68 < \overline x < 69 ) =0}`

c.
`\mathbf{P ( \overline x > 66 ) =0.02275}`

Explanation:

Given that ;

Let Y be a random variable In a​ population, where:

mean
`\mu_y` = 65

`\sigma^2_y` = 49

standard deviation σ =
`√(49)` = 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

`P(\overline x < 68) = \begin {pmatrix} (\overline x - \mu )/((\sigma)/(√(n))) < (68 - \mu )/((\sigma)/(√(n))) } \end {pmatrix}`

`P(\overline x < 68) = \begin {pmatrix}Z < (68 - 65 )/((7)/(√(69))) } \end {pmatrix}`

`P(\overline x < 68) = \begin {pmatrix}Z < (3 )/((7)/(8.3066)) } \end {pmatrix}`

`P(\overline x < 68) = (Z < 3.5599 )`

From the z tables:

`\mathbf{P(\overline x < 68) = 0.9998}`

In a random sample of size n​ = 124​, find Pr (68< Y <69)=

`P(68 < \overline x < 69 ) = P \begin {pmatrix} (68- \mu)/((\sigma)/(√(n))) < (\overline x - \mu)/((\sigma)/(√(n))) < ( 69 - \mu)/((\sigma)/(√(n))) \end {pmatrix}`

`P(68 < \overline x < 69 ) = P \begin {pmatrix} (68- 65)/((7)/(√(124))) < Z < ( 69 - 65)/((7)/(√(124))) \end {pmatrix}`

`P(68 < \overline x < 69 ) = P \begin {pmatrix} (3)/((7)/(11.1355)) < Z < ( 4)/((7)/(11.1355)) \end {pmatrix}`

`P(68 < \overline x < 69 ) = P \begin {pmatrix} 4.7724 < Z < 6.3631 \end {pmatrix}`

`P(68 < \overline x < 69 ) = P( Z < 6.3631 ) - P ( Z < 4.7724 )`

From z tables

`P(68 < \overline x < 69 ) = 0.9999 - 0.9999`

`\mathbf{P(68 < \overline x < 69 ) =0}`

In a random sample of size n​ = 196​, find Pr (Y >66)=

`P ( \overline x > 66 ) = P ( (\overline x -\mu )/((\sigma)/(√(n))) > (66 -\mu )/((\sigma)/(√(n))))`

`P ( \overline x > 66 ) = P ( Z> (66 - 65 )/((7)/(√(196))))`

`P ( \overline x > 66 ) = P ( Z> (1 )/((7)/(14)))`

`P ( \overline x > 66 ) = P ( Z> (14 )/(7))`

`P ( \overline x > 66 ) = P ( Z>2)`

`P ( \overline x > 66 ) = 1 - P ( Z<2)`

from z tables

`P ( \overline x > 66 ) = 1 - 0.9773`

`\mathbf{P ( \overline x > 66 ) =0.02275}`