An environmental group at a local college is conducting independent tests to determine the distance a particular make of automobile will travel while consuming only 1 gallon of gas.They test a sample of five cars and obtain a mean of 28.2 miles.Assuming that the standard deviation is 2.7 miles,find the 95 percent confidence interval for the mean distance traveled by all such cars using 1 gallon of gas.
A) $\left[ \begin{array} { l l }
26.16 & 30.24
\end{array} \right]$
B) $[ 20.70$$ \quad$$ \quad$$35.70 ]$
C) $\left[ \begin{array} { l l }
24.85 & 31.55
\end{array} \right]$
D) $[ 26.70$$ \quad$$ \quad$$29.70 ]$
E) $\left[ \begin{array} { l l }
25.83 & 30.57
\end{array} \right.$

Question

An environmental group at a local college is conducting independent tests to determine the distance a particular make of automobile will travel while consuming only 1 gallon of gas.They test a sample of five cars and obtain a mean of 28.2 miles.Assuming that the standard deviation is 2.7 miles,find the 95 percent confidence interval for the mean distance traveled by all such cars using 1 gallon of gas.
A) $\left[ \begin{array} { l l } 
26.16 & 30.24
\end{array} \right]$
B) $[ 20.70$$ \quad$$ \quad$$35.70 ]$
C) $\left[ \begin{array} { l l } 
24.85 & 31.55
\end{array} \right]$
D) $[ 26.70$$ \quad$$ \quad$$29.70 ]$
E) $\left[ \begin{array} { l l } 
25.83 & 30.57
\end{array} \right.$

Quizplus · Accepted Answer

We can use the formula for the confidence interval of a mean with known standard deviation:

$$	ext{CI}=\left[\bar{x}-z_{\alpha/2}\frac{\sigma}{\sqrt{n}},\bar{x}+z_{\alpha/2}\frac{\sigma}{\sqrt{n}}ight]$$

where $\bar{x}$ is the sample mean, $\sigma$ is the population standard deviation, $n$ is the sample size, and $z_{\alpha/2}$ is the critical value from the standard normal distribution corresponding to the desired level of confidence.

For a 95% confidence interval, $\alpha=0.05$ and $z_{\alpha/2}=1.96$. Plugging in the given values, we get:

$$	ext{CI}=\left[28.2-1.96\frac{2.7}{\sqrt{5}},28.2+1.96\frac{2.7}{\sqrt{5}}ight]\approx\left[24.85,31.55ight]$$

Therefore, the best choice is C.

An Environmental Group at a Local College Is Conducting Independent [26.1630.24]\left[ \begin{array} { l l } 26.16 & 30.24\end{array} \right][26.16​30.24​]

An Environmental Group at a Local College Is Conducting Independent $\left[ \begin{array} { l l } 26.16 & 30.24\end{array} \right]$