Question

The mean of a population is 74 and the standard deviation is 16. The shape of the population is unknown. Determine the probability of each of the following occurring from this population. Appendix A Statistical Tables a. A random sample of size 36 yielding a sample mean of 75 or more b. A random sample of size 140 yielding a sample mean of between 72 and 75 c. A random sample of size 217 yielding a sample mean of less than 74.7

Answer 1

a

`P(X > 75)= 0.35402`

b

`P(72 < X < 75 ) = 0.2529`

c

`P( X < 74.7) = 0.74041`

Explanation:

From the question we are told that

The population mean is
`\mu = 74`

The population standard deviation is
`\sigma = 16`

Considering question a

The sample size is n = 36

Generally the standard error of mean is mathematically represented as

`\sigma_(x) = (\sigma )/(√(n) )`

=>
`\sigma_(x) = (16)/(√(36) )`

=>
`\sigma_(x) = 2.67`

Generally the probability that a random sample of size 36 yielding a sample mean of 75 or more is mathematically represented as

`P(X > 75) = P( (X - \mu )/( \sigma_(x)) > (75 - 74)/( 2.67 ) )`

`(X -\mu)/(\sigma )  =  Z (The  \ standardized \  value\  of  \ X )`

`P(X > 75) = P(Z > 0.3745 )`

From the z table the area under the normal curve representing 0.3745 to the right is

`P(Z > 0.3745 ) = 0.35402`

=>
`P(X > 75)= 0.35402`

Considering question b

The sample size is n = 104

Generally the standard error of mean is mathematically represented as

`\sigma_(x) = (\sigma )/(√(n) )`

=>
`\sigma_(x) = (16)/(√(104) )`

=>
`\sigma_(x) = 1.5689`

Generally the probability that a random sample of size 104 yielding a sample mean between 72 and 75 is mathematically represented as

`P(72 < X < 75 ) = P((72 - 74 )/(1.5689) < (X - \mu )/(\sigma_(x)) < (75 - 74 )/(1.5689) )`

=>
`P(72 < X < 75 ) = P(-1.275 < Z < 0.375 )`

=>
`P(72 < X < 75 ) = P(Z < 0.375 ) - P(Z < -1.275)`

From the z table the area under the normal curve representing -1.275 to to the left is

`P(Z < -1.275) =0.10115`

=>
`P(72 < X < 75 ) = 0.35402 - 0.10115`

=>
`P(72 < X < 75 ) = 0.2529`

Considering question c

The sample size is n = 217

Generally the standard error of mean is mathematically represented as

`\sigma_(x) = (\sigma )/(√(n) )`

=>
`\sigma_(x) = (16)/(√(217) )`

=>
`\sigma_(x) = 1.086`

Generally the probability that a random sample of size 217 yielding a sample mean of less than 74.7 is mathematically represented as

`P( X < 74.7) = P((X - \mu )/(\sigma_x) < ( 74.7 - 74 )/( 1.086 ))`

=>
`P( X < 74.7) = P(Z < 0.6446 )`

From the z table the area under the normal curve representing 0.6446 to to the left is

`P(Z < 0.6446 ) = 0.74041`

=>
`P( X < 74.7) = 0.74041`