To determine the most restrictive level of significance on a hypothesis test that would indicate the company is packaging less than the required average of 300 mL, we need to follow these steps:
1. State the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population mean volume ([tex]\(\mu\)[/tex]) is 300 mL.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The population mean volume ([tex]\(\mu\)[/tex]) is less than 300 mL.
2. Collect the Known Values:
- Population mean ([tex]\(\mu\)[/tex]): 300 mL
- Sample mean ([tex]\(\bar{x}\)[/tex]): 298.4 mL
- Population standard deviation ([tex]\(\sigma\)[/tex]): 3 mL
- Sample size ([tex]\(n\)[/tex]): 20
3. Calculate the Z-score:
The Z-score formula for a sample mean is given by:
[tex]\[
Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\][/tex]
Plugging in the values:
[tex]\[
Z = \frac{298.4 - 300}{\frac{3}{\sqrt{20}}}
\][/tex]
[tex]\[
Z \approx -2.39
\][/tex]
4. Compare the Z-score with Critical Values:
The given critical values from the table are:
- For 5% significance level: 1.65
- For 2.5% significance level: 1.96
- For 1% significance level: 2.58
Since this is a lower-tailed test (testing if the mean is less than 300 mL), we take the negative of these critical values:
- For 5% significance level: -1.65
- For 2.5% significance level: -1.96
- For 1% significance level: -2.58
5. Determine the Most Restrictive Level of Significance:
We compare our Z-score (approximately -2.39) to these critical values:
- [tex]\( -2.39 \)[/tex] is less than [tex]\( -1.96 \)[/tex] (critical value for 2.5% significance level).
- [tex]\( -2.39 \)[/tex] is greater than [tex]\( -2.58 \)[/tex] (critical value for 1% significance level).
Therefore, the most restrictive level of significance that still rejects the null hypothesis is 2.5%.
So, the most restrictive level of significance indicating the company is packaging less than the required average 300 mL is 2.5%.
