Question

A company manufactures and markets a traditional type of disposable coffee cup that is used in many fast food restaurants. The company has created a new cup that it believes insulates better than the traditional cup. To investigate whether the new cup insulates better, the company plans to conduct a study. In the study, a random sample of cups for each of the two types will be selected. In each sample, each cup will be filled with the same amount of coffee that has been heated to 150 degrees Fahrenheit (ºF). The amount of time (in minutes) it takes for the coffee to cool to 100º F will be measured for each cup.

The hypotheses that the company will test are shown below, where µ_N is the true mean time it takes coffee to cool from 150º F to 100º F in the new cup and µ_T is the true mean time it takes coffee to cool from 150º F to 100º F in the traditional cup.

H₀: μ_N = μ_T

H_a: μ_N > μ_T

1. Describe a Type II error in the context of the study.

2. The company is concerned about the probability of a Type II error. Which test procedure, one that uses a significance level of ? = 0.10 or one that uses a significance level of ? = 0.01, would result in a smaller probability of a Type II error? Explain.

3. The marketing department in the company has suggested that a 2-minute increase in the time it takes the coffee to cool from 150º F to 100º F would be a noticeable improvement to customers. Suppose the company statistician estimates that the power of the appropriate significance test is 0.88 when the true mean cooling time for the new cups is 2 minutes greater than the true mean cooling time for the traditional cups. Interpret the value of 0.88 in the context of the study.

Answer 1

In the above context of manufacturing a new insulating coffee cup, in the study of comparing with a traditional coffee cup the type two error can be defined as -

Type two error

Type two error= Probability (accepting H0 when H0 is not true)

under the null hypothesis that H₀: μ_N = μ_T

in other words, it can be defined as

Type two error= Probability (declaring no difference between new sup and traditional cup when there is actually there is a significant difference)

Now which significance level to used to minimize type two error?

Since the type one error and type two error are inversely related to each other. If one is set to minimize others will have increasing nature we can not set to minimize both errors simultaneously. So a smart strategy is to fix one of the errors at a particular level and minimize the other one.

If coming to the point if we select the significance level at a relaxing level then we can have a smaller type two error. So at alpha = 0.1 the type two error would we at maximum.

Interpretation: If the power of the appropriate significance test is 0.88 when the true mean cooling time for the new cups is 2 minutes greater than the true mean cooling time for the traditional cups then its simply means that there is 88% chance that this difference detected by the statistician is actually the difference between both the cups. Or out of a hundred times 88 times this test is able to detect the true difference between both the cups.