Question 1178600: A manufacturer of salad dressings uses machines to dispense liquid ingredients into bottles that move along a filling line. The machine that dispenses salad dressings is working properly when 10ml are dispensed. Suppose that the average amount dispensed in a particular sample of 35 bottles is 8.92ml with a variance of
0.24 ml2 .Is there enough evidence that the machine should be stopped and
production wait for repairs at a=1%.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to approach this hypothesis test:
1. Define the Hypotheses:
Null Hypothesis (H0): The machine is working properly (the mean dispensed is 10 ml). μ = 10
Alternative Hypothesis (H1): The machine is not working properly (the mean dispensed is less than 10 ml). μ < 10 (This is a left-tailed test since we're concerned if the machine is underfilling).
2. Determine the Test Statistic:
We'll use a t-test because we have the sample variance and not the population variance.
Calculate the sample standard deviation (s): s = √variance = √0.24 ≈ 0.4899 ml
Calculate the t-statistic: t = (x̄ - μ) / (s / √n)
x̄ = 8.92 ml (sample mean)
μ = 10 ml (population mean)
s = 0.4899 ml (sample standard deviation)
n = 35 (sample size)
t = (8.92 - 10) / (0.4899 / √35)
t = -1.08 / (0.4899 / 5.916)
t = -1.08 / 0.0828
t = -13.04
3. Determine the Critical Value:
Significance level (α) = 0.01
Degrees of freedom (df) = n - 1 = 35 - 1 = 34
Using a t-table or calculator, find the critical t-value for a one-tailed test with α = 0.01 and df = 34. The Critical t-value is approximately -2.441.
4. Make a Decision:
Compare the calculated t-statistic (-13.04) with the critical t-value (-2.441).
Since -13.04 < -2.441, the calculated t-statistic falls in the rejection region.
5. Conclusion:
Reject the null hypothesis. There is sufficient evidence to conclude that the machine is not working properly and the average amount dispensed is less than 10 ml at a 1% significance level. Therefore, the machine should be stopped and production should wait for repairs.
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