Respuesta :
Answer:
Hence, the confidence interval is [tex](63.517,74.483)[/tex] gallons.
Given :
Sample size[tex]=36[/tex]
Sample mean [tex]=69[/tex] gallons
Standard deviation [tex]=20[/tex] gallons
So, the standard error [tex]=\frac{20}{\sqrt{36}}=20/6=3.33[/tex]
Confidence level [tex]=95\%[/tex]
To find :
The [tex]95\%[/tex] confidence interval for the number of gallons of carbonated beverages consumed per year by Americans.
Explanation :
[tex]\because[/tex] Confidence interval [tex]=(\text{Sample mean}-\text{margin error},\text{Sample mean}+\text{margin error})[/tex]
Margin error [tex]=Z_{\frac{\alpha}{2}}\times\frac{\sigma}{\sqrt{n}}[/tex]
Here, [tex]\alpha=1-95\%=1-0.95[/tex]
[tex]\Rightarrow \alpha=0.05\Rightarrow \frac{\alpha}{2}=0.025[/tex]
[tex]Z_{\frac{\alpha}{2}}=Z_{0.025}=1.645[/tex] from the [tex]z-[/tex]table.
So, Margin error [tex]=Z_{\frac{\alpha}{2}}\times\frac{\sigma}{\sqrt{n}}[/tex]
[tex]=1.645\times\frac{20}{6}=1.645\times3.33[/tex]
[tex]=5.483[/tex]
Therefore, the confidence interval is:
[tex](69-5.483,69+5.483)[/tex]
[tex]=(63.517,74.483)[/tex] gallons.
