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Sagot :
To determine the most restrictive level of significance that indicates the company is packaging an average amount of iced tea beverages less than the required average of [tex]\(300 \, \text{mL}\)[/tex], we need to perform a hypothesis test.
Here’s a step-by-step solution:
### Step 1: Formulate the Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): The true mean volume is [tex]\(300 \, \text{mL}\)[/tex], i.e., [tex]\( \mu = 300 \, \text{mL}\)[/tex].
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The true mean volume is less than [tex]\(300 \, \text{mL}\)[/tex], i.e., [tex]\( \mu < 300 \, \text{mL}\)[/tex].
### Step 2: Calculate the Standard Error
The standard error of the mean (SE) is computed using the population standard deviation ([tex]\(\sigma\)[/tex]) and the sample size ([tex]\(n\)[/tex]):
[tex]\[ \text{SE} = \frac{\sigma}{\sqrt{n}} \][/tex]
Given:
- Population standard deviation ([tex]\(\sigma\)[/tex]) = [tex]\(3 \, \text{mL}\)[/tex]
- Sample size ([tex]\(n\)[/tex]) = 20
Plugging in these values:
[tex]\[ \text{SE} = \frac{3}{\sqrt{20}} \approx 0.6708 \][/tex]
### Step 3: Calculate the Test Statistic (Z-score)
The Z-score is calculated with the formula:
[tex]\[ Z = \frac{\bar{X} - \mu}{\text{SE}} \][/tex]
Where:
- [tex]\(\bar{X}\)[/tex] is the sample mean = [tex]\(298.4 \, \text{mL}\)[/tex]
- [tex]\(\mu\)[/tex] is the population mean = [tex]\(300 \, \text{mL}\)[/tex]
Plugging in the values:
[tex]\[ Z = \frac{298.4 - 300}{0.6708} \approx -2.385 \][/tex]
### Step 4: Compare the Test Statistic to Critical Z-values
We compare the calculated Z-score to the critical Z-values from the table for different levels of significance.
Given critical Z-values for the upper tail:
- [tex]\(5\%\)[/tex] significance level: [tex]\(Z = 1.65\)[/tex]
- [tex]\(2.5\%\)[/tex] significance level: [tex]\(Z = 1.96\)[/tex]
- [tex]\(1\%\)[/tex] significance level: [tex]\(Z = 2.58\)[/tex]
Since our test is one-tailed (left-tailed), we compare the negative of these critical values with our calculated Z-score:
Critical Z-values (left-tailed):
- [tex]\(5\%\)[/tex]: [tex]\(-1.65\)[/tex]
- [tex]\(2.5\%\)[/tex]: [tex]\(-1.96\)[/tex]
- [tex]\(1\%\)[/tex]: [tex]\(-2.58\)[/tex]
The test statistic [tex]\(Z \approx -2.385\)[/tex] is less than [tex]\(-1.96\)[/tex] but greater than [tex]\(-2.58\)[/tex].
### Conclusion:
The most restrictive level of significance at which we can reject the null hypothesis and conclude that the company is packaging less than the required average [tex]\(300 \, \text{mL}\)[/tex] is [tex]\(2.5\%\)[/tex].
Therefore, the answer is [tex]\( \boxed{2.5\%} \)[/tex].
Here’s a step-by-step solution:
### Step 1: Formulate the Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): The true mean volume is [tex]\(300 \, \text{mL}\)[/tex], i.e., [tex]\( \mu = 300 \, \text{mL}\)[/tex].
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The true mean volume is less than [tex]\(300 \, \text{mL}\)[/tex], i.e., [tex]\( \mu < 300 \, \text{mL}\)[/tex].
### Step 2: Calculate the Standard Error
The standard error of the mean (SE) is computed using the population standard deviation ([tex]\(\sigma\)[/tex]) and the sample size ([tex]\(n\)[/tex]):
[tex]\[ \text{SE} = \frac{\sigma}{\sqrt{n}} \][/tex]
Given:
- Population standard deviation ([tex]\(\sigma\)[/tex]) = [tex]\(3 \, \text{mL}\)[/tex]
- Sample size ([tex]\(n\)[/tex]) = 20
Plugging in these values:
[tex]\[ \text{SE} = \frac{3}{\sqrt{20}} \approx 0.6708 \][/tex]
### Step 3: Calculate the Test Statistic (Z-score)
The Z-score is calculated with the formula:
[tex]\[ Z = \frac{\bar{X} - \mu}{\text{SE}} \][/tex]
Where:
- [tex]\(\bar{X}\)[/tex] is the sample mean = [tex]\(298.4 \, \text{mL}\)[/tex]
- [tex]\(\mu\)[/tex] is the population mean = [tex]\(300 \, \text{mL}\)[/tex]
Plugging in the values:
[tex]\[ Z = \frac{298.4 - 300}{0.6708} \approx -2.385 \][/tex]
### Step 4: Compare the Test Statistic to Critical Z-values
We compare the calculated Z-score to the critical Z-values from the table for different levels of significance.
Given critical Z-values for the upper tail:
- [tex]\(5\%\)[/tex] significance level: [tex]\(Z = 1.65\)[/tex]
- [tex]\(2.5\%\)[/tex] significance level: [tex]\(Z = 1.96\)[/tex]
- [tex]\(1\%\)[/tex] significance level: [tex]\(Z = 2.58\)[/tex]
Since our test is one-tailed (left-tailed), we compare the negative of these critical values with our calculated Z-score:
Critical Z-values (left-tailed):
- [tex]\(5\%\)[/tex]: [tex]\(-1.65\)[/tex]
- [tex]\(2.5\%\)[/tex]: [tex]\(-1.96\)[/tex]
- [tex]\(1\%\)[/tex]: [tex]\(-2.58\)[/tex]
The test statistic [tex]\(Z \approx -2.385\)[/tex] is less than [tex]\(-1.96\)[/tex] but greater than [tex]\(-2.58\)[/tex].
### Conclusion:
The most restrictive level of significance at which we can reject the null hypothesis and conclude that the company is packaging less than the required average [tex]\(300 \, \text{mL}\)[/tex] is [tex]\(2.5\%\)[/tex].
Therefore, the answer is [tex]\( \boxed{2.5\%} \)[/tex].
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