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A consumer protection group randomly checks the volume of different beverages to ensure that companies are packaging the stated amount. Each individual volume is not exact, but a volume of iced tea beverages is supposed to average 300 mL with a standard deviation of 3 mL. The consumer protection group sampled 20 beverages and found the average to be 298.4 mL. Using the given table, which of the following is 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?

\begin{tabular}{|c|c|c|c|}
\hline \multicolumn{4}{|c|}{ Upper-Tail Values } \\
\hline a & [tex]$5 \%$[/tex] & [tex]$2.5 \%$[/tex] & [tex]$1 \%$[/tex] \\
\hline \begin{tabular}{c}
Critical \\
z-values
\end{tabular} & 1.65 & 1.96 & 2.58 \\
\hline \hline
\end{tabular}

A. [tex]$1 \%$[/tex]
B. [tex]$2.5 \%$[/tex]
C. [tex]$5 \%$[/tex]
D. [tex]$10 \%$[/tex]

Sagot :

GinnyAnswer
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]
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