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Answer :
Final Answer:
The hypothesis test indicates that the production line should be stopped if the sample mean weight of soft water bottles deviates significantly from the target average of 12 ounces. The significance level is set at 2%.
Explanation:
In order to determine whether the production line should be stopped, a hypothesis test can be performed. The null hypothesis (H0) states that the mean weight of soft water bottles is equal to 12 ounces, while the alternative hypothesis (Ha) suggests that the mean weight is not equal to 12 ounces.
Hypotheses:
H0: μ = 12 ounces
Ha: μ ≠ 12 ounces
The sample size is 80 bottles, and the variance is given as 0.16 (16E-2). The significance level (α) is set at 0.02 (2%). The critical z-values for a two-tailed test at a 2% significance level are -2.33 and 2.33.
Calculation:
Standard Error (SE) = √(Variance / Sample Size) = √(0.16 / 80) ≈ 0.071
Z-score = (Sample Mean - Population Mean) / SE
Assuming the sample mean comes out to be 12.08 ounces, the calculated Z-score is:
Z = (12.08 - 12) / 0.071 ≈ 1.13
Since the calculated Z-score of 1.13 falls within the range of -2.33 to 2.33, we fail to reject the null hypothesis. This means that the sample mean is not significantly different from the population mean of 12 ounces, and the production line can continue operating.
The results of the hypothesis test indicate that the sample mean falls within an acceptable range of the target average. This implies that the soft water bottles' weight is not significantly different from the desired weight of 12 ounces. Therefore, there is no immediate need to stop the production line based on the sample mean.
The choice of a 2% significance level ensures that the probability of making a Type I error (rejecting a true null hypothesis) is limited to 2%. This level of significance helps maintain a balance between minimizing the chances of stopping production when it's unnecessary and ensuring that potential deviations from the desired weight are adequately addressed.
To summarize, the hypothesis test with a 2% significance level provides a reliable method for deciding whether the production line should be stopped based on the sample mean weight of soft water bottles. It ensures that the quality control process is appropriately stringent without causing unnecessary interruptions in the production process.
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