Question

Construct a confidence interval for u based on the sample data given here: 12.3 11.6 12.8 11.9 11.7 12.1 13.1 11.4 12 11.8 Use 97% as the confidence level. Round the final values to two digits after the decimal point. The lower limit in the interval is: Number and the upper limit in the interval is: Number

Answer 1

Solution:

x	x2
12.3	151.29
11.6	134.56
11.9	141.61
12.8	163.84
12.1	146.41
11.4	129.96
12	144
11.7	136.89
11.8	139.24
13.1	171.61
∑x=120.7	∑x2=1459.41

Mean ˉx=∑xn

=12.3+11.6+11.9+12.8+12.1+11.4+12+11.7+11.8+13.1/10

=120.7/10

=12.07

Sample Standard deviation S=√∑x2-(∑x)2nn-1

=√1459.41-(120.7)210/9

=√1459.41-1456.849/9

=√2.561/9

=√0.2846

=0.5334

Degrees of freedom = df = n - 1 = 10 - 1 =9

At 97% confidence level the t is ,

$\alpha$ = 1 - 97% = 1 - 0.97 = 0.03

$\alpha$ / 2 = 0.03 / 2 = 0.015

t_{$\alpha$ /2,df} = t_0.015,9=2.574

Margin of error = E = t_{$\alpha$/2,df} * (s /$\sqrt$n)

= 2.574 * (0.53 / $\sqrt$ 10)

= 0.43

Margin of error = 0.43

The 97% confidence interval estimate of the population mean is,

$\bar x$ - E <$\mu$  < $\bar x$ + E

12.07 - 0.43 < $\mu$ < 12.07 + 0.43

11.64 < $\mu$ < 12.50

(11.64, 12.50 )