Question

Problem 9-8 (modified). The height of adult males in a particular city is normally distributed, with mean 69.5 in. and standard deviation 2.65 in. When a random sample is taken of size 20, find (1) the standard error of Xbar

Answer 1

Complete Question

Problem 9-8 (modified). The height of adult males in a particular city is normally distributed, with mean 69.5 in. and standard deviation 2.65 in.

When a random sample is taken of size  20, find (1) the standard error of Xbar

(2) the probability that Xbar falls within .5 in. of the true mean

Probability=

Answer:

1

$\sigma_{\= x} = 0.5926$

2

$P( 69 < \= X < 70) = 0.60134$

Step-by-step explanation:

From the question we are told that

  The true  mean is  $\mu = 69.5 \ in$

  The standard deviation is  $\sigma = 2.65 \ in$

    The sample size is n =  20

Generally the standard error of Xbar$(\= x)$ is mathematically represented as

     $\sigma_{\= x} = \frac{\sigma }{\sqrt{n} }$

=>  $\sigma_{\= x} = \frac{ 2.65 }{\sqrt{ 20 } }$

=>  $\sigma_{\= x} = 0.5926$

Generally for  $\= x$ to fall within 0.5 of the true mean the $\= x$ must be within

     $a = \mu + 0.5$

=>   $a = 69.5 + 0.5$

=>   $a = 70$

or  $b = \mu -0.5$

=>    $b = 69.5 -0.5$

=>    $b = 69$

Generally the probability that  $\= x$ fall with 0.5  of the true mean is mathematically represented as

     $P( 69 < \= X < 70) = P(\frac{69 - 69.5 }{ 0.5926} < \frac{\= X - \mu }{ \sigma_{\= x} } <\frac{70- 69.5 }{ 0.5926})$

=>  $P( 69 < \= X < 70) = P(-0.844 < Z < 0.844)$

=>  $P( 69 < \= X < 70) = P( Z < 0.844) - P( < -0.844)$

From the z table the probability of  (  Z    < 0.844)  and   (  Z    < -0.844)  is  

     $P( Z < 0.844) = 0.80067$

and

    $P( Z < - 0.844) = 0.19933$

So

   $P( 69 < \= X < 70) = 0.80067 - 0.19933$

=>  $P( 69 < \= X < 70) = 0.60134$

Answer 2

The standard error of the given data comes to be 0.593.

It is given that

mean $\mu$ = 69.5 inch

Standard deviation $\sigma$ = 2.65 inch

Sample size n =20

What is the standard error?

The standard error is a statistical term that measures the accuracy with which a sample distribution represents a population by using standard deviation.

The standard error is given by:

$SE =\frac{\sigma}{\sqrt{n} }$

$SE =\frac{2.65}{\sqrt{20} }$

$SE=0.593$

Hence, the standard error of the given data comes to be 0.593.

To get more about standard errors visit:

https://brainly.com/question/1191244