91Ó°ÊÓ

A can of soda is labeled as containing 12 fluid ounces. The quality control manager wants to verify that the filling machine is neither over-filling nor under-filling the cans. (a) Determine the null and alternative hypotheses that would be used to determine if the filling machine is calibrated correctly. (b) The quality control manager obtains a sample of 75 cans and measures the contents. The sample evidence leads the manager to reject the null hypothesis. Write a conclusion for this hypothesis test. (c) Suppose, in fact, the machine is not out of calibration. Has a Type I or Type II error been made? (d) Management has informed the quality control department that it does not want to shut down the filling machine unless the evidence is overwhelming that the machine is out of calibration. What level of significance would you recommend the quality control manager use? Explain.

Step by step solution

01

Formulate null and alternative hypotheses

Define the null hypothesis (H_0) and the alternative hypothesis (H_1). The null hypothesis should state that the filling machine is calibrated correctly (i.e., the average fill is 12 fluid ounces), and the alternative hypothesis should state that the fill is not equal to 12 fluid ounces.\(H_0: \mu = 12\) fl oz\(H_1: \mu eq 12\) fl oz

02

Conclusion from hypothesis test

Since the quality control manager rejected the null hypothesis, it means that the sample evidence is strong enough to conclude that the average fill is not 12 fluid ounces, suggesting the machine might be out of calibration.

03

Identify type of error

Type I error occurs if the null hypothesis is true but is rejected. Type II error occurs if the null hypothesis is false but is not rejected. Since the null hypothesis was rejected but it is actually true (the machine is not out of calibration), a Type I error has been made.

04

Determine level of significance

To avoid unnecessarily shutting down the machine, the level of significance (\(\alpha\)) should be very low. A smaller \(\alpha\), such as 0.01 or 0.001, would mean that there must be overwhelming evidence to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In hypothesis testing, the null hypothesis, denoted as \(H_0\), represents a statement of no effect or no difference. It is the hypothesis that there is nothing new happening. For the soda filling example, the null hypothesis is that the filling machine is calibrated correctly. This means the average content of the soda cans is exactly 12 fluid ounces, that is \(H_0: \mu = 12\). The null hypothesis assumes that any observed variation in the sample is due to random chance.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_1\), is what you want to prove. It contradicts the null hypothesis and represents a statement of effect or difference. For the soda filling machine, the alternative hypothesis states that the machine is not calibrated correctly, i.e., the average content of the soda cans is not equal to 12 fluid ounces, which is represented as \(H_1: \mu \eq 12\). The alternative hypothesis aims to show that the observed effect (filling not being 12 fluid ounces) is due to some cause other than random chance.

Type I Error

A Type I error occurs when the null hypothesis \(H_0\) is true, but you incorrectly reject it. This is also known as a 'false positive.' For example, in the soda filling machine scenario, a Type I error would happen if the machine is actually filling the cans correctly at 12 fluid ounces, but the quality control manager concludes, based on sample evidence, that it is not filling them correctly and therefore takes corrective action that isn't necessary. The probability of making a Type I error is denoted by the significance level \(\alpha\).

Type II Error

A Type II error occurs when the null hypothesis \(H_0\) is false, but you fail to reject it. This is also known as a 'false negative.' In the context of the soda filling machine, a Type II error would happen if the machine is not filling the cans correctly (not averaging 12 fluid ounces), but the quality control manager concludes that it is filling them correctly. This error means a false acceptance of the null hypothesis \(H_0\). The probability of making a Type II error is denoted by \(\beta\).

Level of Significance

The level of significance, denoted by \(\alpha\), is the threshold we set for deciding when to reject the null hypothesis. It represents the probability of committing a Type I error. A common \(\alpha\)-level used is 0.05, but it can be set to other values like 0.01 or 0.001 depending on how cautious you want to be about making a Type I error. For example, in the soda filling machine exercise, using an \(\alpha \) level of 0.01 means that you would only reject the null hypothesis if the evidence was strong enough that there is less than a 1% chance that you are making a Type I error. This more stringent \(\alpha\) ensures that the quality control manager only shuts down the filling machine when there is overwhelming evidence that it is not filling the cans correctly.