To determine the value of the test statistic [tex]\( z \)[/tex] using the provided information, follow these steps:
1. Identify the given information:
- Sample mean ([tex]\(\overline{x}\)[/tex]) = 6.7 ounces
- Population mean ([tex]\(\mu\)[/tex]) = 6.5 ounces
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 0.22 ounces
- Sample size ([tex]\(n\)[/tex]) = 40
- Significance level ([tex]\(\alpha\)[/tex]) = 0.10
2. Use the formula for calculating the test statistic [tex]\( z \)[/tex]:
[tex]\[
z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\][/tex]
3. Substitute the known values into the formula:
[tex]\[
z = \frac{6.7 - 6.5}{\frac{0.22}{\sqrt{40}}}
\][/tex]
4. Simplify the expression:
- Calculate the denominator: [tex]\(\frac{0.22}{\sqrt{40}}\)[/tex]
Let's first find [tex]\(\sqrt{40}\)[/tex]:
[tex]\[
\sqrt{40} \approx 6.3246
\][/tex]
Now, divide the population standard deviation by this value:
[tex]\[
\frac{0.22}{6.3246} \approx 0.03478
\][/tex]
So the expression becomes:
[tex]\[
z = \frac{6.7 - 6.5}{0.03478}
\][/tex]
5. Calculate the numerator:
[tex]\[
6.7 - 6.5 = 0.2
\][/tex]
6. Divide the numerator by the denominator:
[tex]\[
z = \frac{0.2}{0.03478} \approx 5.75
\][/tex]
Hence, the value of the test statistic [tex]\( z \)[/tex] is approximately [tex]\( 5.75 \)[/tex].
Therefore, the correct answer from the given options is:
[tex]\[
5.75
\][/tex]
