Question

The following data were randomly drawn from an approximately normal population 4, 7, 10, 16, 19, 25, 30 E) Send data to Excel data, find a 95% confidence interval for the population standard deviation. Then complete the table below. intermediate computations to at least three decimal places. Round your answers to at least two decimal places. Based on these (If necessary, consult a list of formulas.) what is the lower limit of the 95% confidence interval? What is the upper limit of the 95% confidence interval?

Answer 1

Solution:

Given that,

x	dx = a - x	dx2
4	-12	144
7	-9	81
10	-6	36
16	0	0
19	3	9
25	9	81
30	14	196
$\sum$x = 111	$\sum$dx = -1	$\sum$dx2 = 547

a ) The sample mean is $\bar x$

Mean $\bar x$   = ($\sum$x / n)

= ( 4 + 7 + 10 + 16 + 19 + 25 + 30 / 7)

= 111 / 7

= 15.857

$\bar x$ = 15.86

b ) The population standard is $\sigma$

$\sigma$ = $\sqrt{}$ ( $\sum$ dx² ) - (( $\sum$ dx )² / n ) / n

= $\sqrt{}$ ( 547 ( (- 1 )² / 7 ) / 7

   = $\sqrt{}$ ( 547 - 0.1429 ) 7

=$\sqrt{}$ (546.8571 / 7 )

= $\sqrt{}$ 78.1224

= 8.839

The population standard is $\sigma$ = 8.84

$\bar x$ = 15.86

$\sigma$ = 8.84

n = 7

A ) At 95% confidence level the z is ,

$\alpha$  = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

Z_$\alpha$/2 = Z_0.025 = 1.960

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

= 1.960 * (8.84 / $\sqrt$ 7 )

= 6.55

At 95% confidence interval estimate of the population mean is,

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

15.86 - 6.55< $\mu$ < 15.86 + 6.55

9.31 < $\mu$ < 22.41

The lower limit = 9.31

Upper limit = 22.41