Question

You are interested in finding a 90% confidence interval for the average commute that non-residential students have to their college. The data below show the number of commute miles for 11 randomly selected non-residential college students. Round answers to 3 decimal places where possible.
25 21 26 6 25 14 26 24 7 10 14
a. To compute the confidence interval use a distribution.
b. With 90% confidence the population mean commute for non-residential college students is between and miles.

Answer 1

Answer:

a)  90 % of confidence interval is determined by

$(x^{-} -t_{\frac{\alpha }{2} } \frac{S}{\sqrt{n} } , x^{-} +t_{\frac{\alpha }{2} } \frac{S}{\sqrt{n} } )$

b) The 90% confidence intervals for the population mean

(13.6572 , 22.3428)

Step-by-step explanation:

Step(i):-

Given data

Non residential  college students  25 21 26 6 25 14 26 24 7 10 14

Mean of   Non residential  college students

    x⁻  = ∑x/n

          = $\frac{25+21+26+6+25+14+26+24+7+10+14}{11}$

   x⁻ = 18

now

Non residential

college students 'x'  : 25    21     26   6      25    14    26   24     7    10    14

       x - x⁻                  :   7      3       8   -12      7     -4     8      6     -11   -8    -4

     (x-x⁻)²                  :    49   9      64  144    49    16    64   36   121   64    16

$s^{2} = \frac{49+9+64+144+49+16+64+36+121+64+16 }{11-1}$

S² = 63.2

S = √63.2 = 7.949

Step(ii):-

The 90% confidence the population mean commute for non-residential college students is between and miles.

$(x^{-} -t_{\frac{\alpha }{2} } \frac{S}{\sqrt{n} } , x^{-} +t_{\frac{\alpha }{2} } \frac{S}{\sqrt{n} } )$

Degrees of freedom

ν =n-1 =11-1 = 10

t $t_{\frac{0.10}{2} } = t_{0.05} = 1.812$

Step(iii):-

The 90% confidence the population mean commute for non-residential college students is between and miles.

$(x^{-} -t_{\frac{\alpha }{2} } \frac{S}{\sqrt{n} } , x^{-} +t_{\frac{\alpha }{2} } \frac{S}{\sqrt{n} } )$

$((18- 1.812 \frac{7.949}{\sqrt{11} } , (18-+1.812 \frac{7.949}{\sqrt{11} })$

(18 - 4.3428 , 18 + 4.3428)

(13.6572 , 22.3428)

Conclusion:-

The 90% confidence the population mean commute for non-residential college students is between and miles.

(13.6572 , 22.3428)