Answer: (0.465, 5.535)
Step-by-step explanation:
Formula to calculate the confidence interval (when population standard deviation is unknown) is given by :-
[tex]\overline{x}\pm t_{\alpha/2} \dfrac{s}{\sqrt{n}}[/tex]
, where [tex]\overline{x}[/tex] = sample mean.
s= sample standard deviation.
n= Sample size.
[tex]t_{\alpha/2}[/tex] = critical value
By considering the given information , we have
[tex]\overline{x}=6[/tex]
s=0.75
n= 9
Significance level = [tex]\alpha=0.1[/tex] [1-0.90=0.1]
By using students' t distribution -table , the critical value for 95% confidence level :
[tex]t_{\alpha/2 , n-1}=t_{0.1/2,\ 8}=1.86[/tex]
[Note: degree of freedom = n-1]
Now, the 90% confidence interval for the true mean weight of these Southern California avocados will be :
[tex]6\pm (1.86) \dfrac{0.75}{\sqrt{9}}[/tex]
[tex]6\pm (1.86) \dfrac{0.75}{3}[/tex]
[tex]6\pm (1.86) 0.25[/tex]
[tex]6\pm 0.465=(6-0.465,\ 6+0.465)=(0.465,\ 5.535)[/tex]
Hence, the required confidence interval =(0.465, 5.535)
