Question

E Question Help 21.8 A simple randorm sample of 20 new automobile models yielded the data shown to the right on fuel tank capacity, in gallons. 14.5 17.6 15.3 17.1 18.3 19.5 21.5 12.7 19.7 16.5 13.6 20 2 21.6 17.3 21.8 18.1 17.3 22.3 21.1 a Find a point estimate for the mean fuel tank capacity for all new automobile models. (Note: 2xi = 367 8) A point estimate is gallons. (Type an integer or a decimal. Do not round) b. Determine a 95% confidence interval for the mean fuel tank capacity of all new automobile models. Assume o = 3.30 gallons. The 95% confidence interval is from O gallons to gallons. (Do not round until the final answer. Then round to two decimal places as needed.)

Answer 1

Solution:

a) The point estimate is sample mean $\bar x$

$\bar x$ = Summation X_i / n = 367.8/20 = 18.39

The point estimate is 18.39 gallons

b)Given , $\sigma$ = 3.30

c = 95% = 0.95

= 0.95

$\therefore$$\alpha$ = 1- c = 1- 0.95 = 0.05

$\therefore$  $\alpha$/2 = 0.05 $\slash$ 2 = 0.025 and 1- $\alpha$ /2 = 0.975

Search the probability 0.975 in the Z table and see corresponding z value

$\therefore$$Z_{\alpha/2}$ = 1.96   

The margin of error is given by

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

= 1.96 * (3.30 / $\sqrt{}$ 20 )

= 1.45

Now , confidence interval for mean($\mu$) is given by:

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

(18.39 - 1.45 )   <  $\mu$ <  (18.39 + 1.45)

16.94 <  $\mu$ < 19.84

The 95% confidence interval is from 16.94 to 19.84.