Sure, I'd be happy to help with the detailed, step-by-step solution for finding the value of the z-test statistic.
Given data:
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 8.93 volts
- Population mean ([tex]\(\mu\)[/tex]) = 9 volts
- Standard deviation ([tex]\(\sigma\)[/tex]) = 0.2 volts
- Sample size ([tex]\(n\)[/tex]) = 64
The formula for the z-test statistic is:
[tex]\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\][/tex]
First, we will calculate the standard error of the mean, which is given by:
[tex]\[
\text{Standard error} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substitute the values:
[tex]\[
\text{Standard error} = \frac{0.2}{\sqrt{64}}
\][/tex]
[tex]\(\sqrt{64}\)[/tex] equals 8, so:
[tex]\[
\text{Standard error} = \frac{0.2}{8} = 0.025
\][/tex]
Now, we calculate the z-test statistic:
[tex]\[
z = \frac{8.93 - 9}{0.025}
\][/tex]
Subtract the population mean from the sample mean:
[tex]\[
8.93 - 9 = -0.07
\][/tex]
Now, divide this result by the standard error:
[tex]\[
z = \frac{-0.07}{0.025} = -2.8
\][/tex]
Thus, the value of the z-test statistic is:
[tex]\[
z = -2.8
\][/tex]
This concludes the calculation of the z-test statistic for testing whether the batch of batteries has a significantly different mean voltage from 9 volts.
