Question

A sample of 45 people were randomly selected from among the workers in a shoe factory. The time taken for each person to polish a finished shoe was measured. The sample mean was 2.3 minutes. Assume that o=0.23 minutes. Construct a 90% confidence interval for the true mean time, u, to polish a shoe. 02.24 to 2.36 minutes 2.21 to 2.39 minutes 2.22 to 2.38 minutes 0 2.23 to 2.37 minutes 0 2.29 to 2.31 minutes O None of the above

Answer 1

Solution :

Given that,

Point estimate = sample mean = $\bar x$= 2.3

Population standard deviation =   $\sigma$ = 0.23
Sample size = n =45

At 90% confidence level the z is

$\alpha$ = 1 - 90% = 1 - 0.90 = 0.1

$\alpha$ / 2 = 0.1 / 2 = 0.05

Z$\alpha$/2 = Z0.05 = 1.645 ( Using z table )

Margin of error = E = Z$\alpha$/2    * ( $\sigma$ /$\sqrt$n)

=1.645 * ( 0.23 /  $\sqrt$45 )

= 0.0564
At 90% confidence interval estimate of the population mean
is,

$\bar x$ - E < $\mu$ < $\bar x$ + E

2.3 - 0.0564 <  $\mu$ < 2.3 + 0.0564

2.24 <  $\mu$ < 2.36

( 2.24 to 2.36 )