Answer: (409.36, 434.64)
Step-by-step explanation:
When population standard deviation is unknown and sample is not so large , then the formula to find the confidence interval for population mean is given by :-
[tex]\overline{x}\pm t^*\dfrac{s}{\sqrt{n}}[/tex]
, where [tex]\overline{x}[/tex]= sample mean , n = sample size , s= sample population standard deviation, t*= two tailed critical value.
As , per given , [tex]\overline{x}=\$422[/tex], s=$57, n=130
For 99% confidence , [tex]\alpha=0.01[/tex]
By t-distribution table , t-value for [tex]\alpha/2=0.005[/tex] (two tailed) and df =129 [∵df=n-1] would be
t*=2.6145
Now , the 99% confidence interval for the mean amount of money spent by collage students on textbooks will be :
[tex]422\pm (2.6145)\dfrac{57}{\sqrt{139}}[/tex]
[tex]422\pm (2.6145)(4.834677)[/tex]
[tex]422\pm 12.640263[/tex]
[tex](422- 12.640263,\ 422+12.640263)\\\\=(409.359737,\ 434.640263)\approx(409.36,\ 434.64)[/tex]
Hence, a 99% confidence interval for the mean amount of money spent by collage students on textbooks will be (409.36, 434.64).
