From a random sample of size 18, a researcher states that (11.1, 15.7) inches is a 90% confidence interval for mu, the mean length of bass caught in a small lake. A normal distribution was assumed. Using - QAmmunity.org

From a random sample of size 18, a researcher states that (11.1, 15.7) inches is a 90% confidence interval for mu, the mean length of bass caught in a small lake. A normal distribution was assumed. Using

asked Mar 4, 2021 228k views

1 Answer

Complete Question

From a random sample of size 18, a researcher states that (11.1, 15.7) inches is a 90% confidence interval for mu, the mean length of bass caught in a small lake. A normal distribution was assumed. Using the 90% confidence interval obtain:

a. A point estimate of
$\mu$ and its 90% margin of error.

b. A 95% confidence interval for
$\mu$ .

Answer:

a

$\= x = 13.4$ .
E = 2.3

b

$10.7 <  \mu < 16.1$

Explanation:

From the question we are told that

The sample size is n = 18

The 90% confidence interval is (11.1, 15.7)

Generally the point estimate of
$\mu$ is mathematically evaluated as

$\= x = (11.1 + 15.7 )/(2)$

=>
$\= x = 13.4$

Generally the margin of error is mathematically evaluated as

E = (15.7 - 11.1)/(2 )

=>
E = 2.3

From the question we are told the confidence level is 90% , hence the level of significance is

$\alpha = (100 - 90 ) \%$

=>
$\alpha = 0.10$

Generally from the normal distribution table the critical value of
$(\alpha )/(2)$ is

$Z_{(\alpha )/(2) } =  1.645$

Generally the equation for the lower limit of the confidence interval is

$\= x - Z_{(\alpha )/(2) } * (s)/(√(18) ) = 11.1$

=>
13.4 - 0.3877 s = 11.1

=>
s = 5.932

From the question we are told the confidence level is 95% , hence the level of significance is

$\alpha = (100 - 95 ) \%$

=>
$\alpha = 0.05$

Generally from the normal distribution table the critical value of
$(\alpha )/(2)$ is

$Z_{(\alpha )/(2) } =  1.96$

Generally the margin of error is mathematically represented as

$E = Z_{(\alpha )/(2) } *  (\sigma )/(√(n) )$

=>
E = 1.96 * (5.932)/(√(18) )

=>
E = 2.7

Generally 95% confidence interval is mathematically represented as

$\= x -E <  \mu <  \=x  +E$

=>
$13.4  -  2.7  <  \mu < 13.4  +   2.7$

=>
$10.7 <  \mu < 16.1$

Ziad Gholmish answered Mar 10, 2021

by Ziad Gholmish

5.4k points

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