Question 1206945: Workers at a certain soda drink factory collected data on the volumes (in ounces) of a simple random sample of 22 cans of the soda drink. Those volumes have a mean of 12.19 oz and a standard deviation of 0.09 oz, and they appear to be from a normally distributed population. If the workers want the filling process to work so that almost all cans have volumes between 11.96 oz and 12.56 oz, the range rule of thumb can be used to estimate that the standard deviation should be less than 0.15 oz. Use the sample data to test the claim that the population of volumes has a standard deviation less than 0.15 oz. Use a 0.025 significance level.
A) Compute the test statistic.
B) What is the P-value?
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **1. Set up the Hypotheses**
* **Null Hypothesis (H₀):** σ² ≥ 0.15²
* The population variance of the soda can volumes is greater than or equal to 0.15² (0.0225).
* **Alternative Hypothesis (Hₐ):** σ² < 0.15²
* The population variance of the soda can volumes is less than 0.15² (0.0225).
**2. Calculate the Test Statistic**
* The test statistic for this hypothesis test is the chi-square statistic:
χ² = (n - 1) * (s² / σ₀²)
where:
* n is the sample size (22)
* s is the sample standard deviation (0.09)
* σ₀ is the hypothesized population standard deviation (0.15)
* Calculate the test statistic:
χ² = (22 - 1) * (0.09² / 0.15²)
χ² = 21 * (0.0081 / 0.0225)
χ² = 7.56
**3. Determine the P-value**
* The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
* Since this is a left-tailed test (Hₐ: σ² < 0.15²), we need to find the area to the left of the calculated chi-square value in the chi-square distribution with (n - 1) degrees of freedom.
* **Using a chi-square table or statistical software:**
* Find the p-value associated with χ² = 7.56 and degrees of freedom (df) = n - 1 = 21.
* The p-value will be greater than 0.025 (as the test statistic falls in the right tail of the chi-square distribution).
**Therefore:**
* **A) Test Statistic:** χ² = 7.56
* **B) P-value:** p-value > 0.025
**Conclusion**
Since the p-value is greater than the significance level (0.025), we **fail to reject the null hypothesis**. There is not enough evidence to support the claim that the population standard deviation of the soda can volumes is less than 0.15 oz.
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