To determine the most restrictive level of significance for the hypothesis test, we need to conduct a one-sample z-test comparing the sample mean to the population mean. Here are the detailed steps to arrive at the solution:
1. State the Hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): The true mean volume is 300 mL.
- Alternative hypothesis ([tex]\(H_1\)[/tex]): The true mean volume is less than 300 mL.
2. Given Data:
- Sample size ([tex]\(n\)[/tex]): 20
- Sample mean ([tex]\(\bar{x}\)[/tex]): 298.4 mL
- Population mean ([tex]\(\mu\)[/tex]): 300 mL
- Population standard deviation ([tex]\(\sigma\)[/tex]): 3 mL
3. Calculate the z-score:
The z-score formula for a sample mean in hypothesis testing is:
[tex]\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\][/tex]
Substituting the given values:
[tex]\[
z = \frac{298.4 - 300}{\frac{3}{\sqrt{20}}} \approx -2.3851
\][/tex]
4. Determine the Critical z-values for Given Significance Levels:
The table provides the following critical z-values for upper-tail (one-tailed) tests:
- [tex]\(5\%\)[/tex] significance level: [tex]\(1.65\)[/tex]
- [tex]\(2.5\%\)[/tex] significance level: [tex]\(1.96\)[/tex]
- [tex]\(1\%\)[/tex] significance level: [tex]\(2.58\)[/tex]
5. Compare the Calculated z-score to the Critical z-values:
Since we are performing a left-tail test (sample mean < population mean), we consider the negative critical z-values:
- For [tex]\(5\%\)[/tex], the critical z-value is [tex]\(-1.65\)[/tex]
- For [tex]\(2.5\%\)[/tex], the critical z-value is [tex]\(-1.96\)[/tex]
- For [tex]\(1\%\)[/tex], the critical z-value is [tex]\(-2.58\)[/tex]
6. Determine the Most Restrictive Significance Level:
- The calculated z-score is approximately [tex]\(-2.3851\)[/tex].
- This z-score is lower than [tex]\(-1.96\)[/tex] but higher than [tex]\(-2.58\)[/tex]. This means it falls between the critical z-values for [tex]\(2.5\%\)[/tex] and [tex]\(1\%\)[/tex] significance levels.
Therefore, the most restrictive level of significance where the null hypothesis is rejected (indicating the company is packaging less than 300 mL on average) is [tex]\(2.5\%\)[/tex].
Hence, the answer is:
[tex]\[ \boxed{2.5\%} \][/tex]
