To find the value of the test statistic [tex]\( z \)[/tex] for the hypothesis test, we'll follow these steps:
1. Identify the given values:
- Sample size ([tex]\( n \)[/tex]) = 106
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 98.20°F
- Population mean ([tex]\( \mu \)[/tex]) = 98.6°F
- Population standard deviation ([tex]\( \sigma \)[/tex]) = 0.62°F
2. Write down the formula for the test statistic [tex]\( z \)[/tex]:
[tex]\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\][/tex]
3. Plug the given values into the formula:
- The numerator [tex]\( \bar{x} - \mu \)[/tex] is:
[tex]\[
98.20 - 98.6 = -0.4
\][/tex]
- The denominator [tex]\( \frac{\sigma}{\sqrt{n}} \)[/tex] is:
[tex]\[
\frac{0.62}{\sqrt{106}}
\][/tex]
4. Compute the denominator:
[tex]\[
\frac{0.62}{\sqrt{106}} \approx \frac{0.62}{10.2956} \approx 0.0602
\][/tex]
5. Compute the [tex]\( z \)[/tex]-value:
[tex]\[
z = \frac{-0.4}{0.0602} \approx -6.64
\][/tex]
So, the value of the test statistic [tex]\( z \)[/tex] is approximately -6.64.
Among the given options:
- [tex]\( -6.64 \)[/tex]
- [tex]\( 6.64 \)[/tex]
- [tex]\( -5.23 \)[/tex]
- [tex]\( 0.08 \)[/tex]
The correct value for the test statistic [tex]\( z \)[/tex] is:
[tex]\[
-6.64
\][/tex]
