Fowle Marketing Research, Inc., bases charges to a client on...

Fowle Marketing Research, Inc., bases charges to a client on the assumption that telephone surveys can be completed within 15 minutes or less. If more time is required, a premium rate is charged. With a sample of 35 surveys, a population standard deviation of 4 minutes, and a level of significance of .01, the sample mean will be used to test the null bypothesis $H_0: \mu \subseteq 15$. a. What is your interpeetation of the Type Il error for this problem? What is its impact on the firm? b. What is the probability of making a Type II error when the actual mean time is $\mu=17$ minutes? c. What is the probability of making a Type II error when the actual mean time is $\mu=18$ minutes? d. Sketch the general shape of the power curve for this test.

Fowle Marketing Research, Inc., bases charges to a client on the assumption that telephone surveys can be completed within 15 minutes or less. If more time is required, a premium rate is charged. With a sample of 35 surveys, a population standard deviation of 4 minutes, and a level of significance of .01, the sample mean will be used to test the null bypothesis $H_0: \mu \subseteq 15$.
a. What is your interpeetation of the Type Il error for this problem? What is its impact on the firm?
b. What is the probability of making a Type II error when the actual mean time is $\mu=17$ minutes?
c. What is the probability of making a Type II error when the actual mean time is $\mu=18$ minutes?
d. Sketch the general shape of the power curve for this test.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or decisions about a population parameter based on sample data. It involves formulating a null hypothesis (typically a statement of no effect or no difference) and an alternative hypothesis, then using a test statistic along with a predetermined significance level to decide whether to reject the null hypothesis in favor of the alternative.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected even though it is true. In the context of quality control or service benchmarks, this means concluding that the process or service exceeds a threshold when it actually meets the standard, potentially leading to unnecessary costs or unjustified changes in procedure.

Type II Error

A Type II error happens when the test fails to reject the null hypothesis when the alternative hypothesis is true. This error reflects a missed opportunity to detect a significant deviation from the claimed standard. It is crucial in applications like service benchmarks because it can lead to continued underperformance or mispricing, as true deviations are not addressed.

Statistical Power

Statistical power is the probability that a test correctly rejects the null hypothesis when the alternative hypothesis is true. It is calculated as one minus the probability of a Type II error (1 - ?). High power is desirable because it indicates a high likelihood of detecting an actual effect or deviation from the standard, thus ensuring that real issues are identified and rectified.

Sampling Distribution of the Mean

The sampling distribution of the mean describes the distribution of sample means over many samples drawn from the same population. When the population standard deviation is known and the sample size is sufficiently large, the Central Limit Theorem ensures that the sample means will be approximately normally distributed. This distribution is key to calculating probabilities for hypothesis tests involving the mean.

Significance Level and Critical Values

The significance level in hypothesis testing, often denoted as alpha, is the threshold probability for committing a Type I error. It determines the critical values or rejection regions of the test statistic under the null hypothesis. By choosing a significance level (e.g., 0.01), researchers set a strict criterion for evidence against the null hypothesis, balancing the risk of Type I and Type II errors.

Transcript

00:01 So in this test, we would be assuming that the mean is less than or equal to 15 minutes, and alternately it is higher than 15 minutes.

00:10 And so we are also told that the population standard deviation is 4 minutes, and we're supposed to use an alpha level of 0 .01.

00:20 And for this scenario, in part a, we want to look at what does it mean to have a type 2 error, and what would be the implications of that? and a type 2 error is, let's draw a picture here.

00:34 In this picture, we're doing a one -tail test.

00:38 We're going to assume that the mean is 15 minutes.

00:41 And if it's long as it's under 15 minutes, they have a certain fee that they charge.

00:46 And if it's higher than 15 minutes, then they charge some type of a premium fee is added on.

00:53 So they're assuming that these calls are less than or equal to 15 minutes.

00:59 And if they're higher than that, then they should actually be charging more money.

01:04 So at a 1 % significance level, that means we would have that all of that 0 .01 would be in this value.

01:12 And if a test statistic, wherever 2 .326, if it were higher than this, if the z value is higher than 2 .326, that would cause us to reject the null.

01:26 And then conclude that the mean is actually higher than 15 minutes, and then they would start charging that premium and thinking that they need to charge a premium rate.

01:35 And alternately, if it is less than are equal to that 2 .36, they would fail to reject the null.

01:48 Or some will say accept, but really it's better to say fail to reject the null.

01:54 So what is a type 2 error? a type 2 air is where we fail to reject them all.

02:03 However, we should have rejected because then we should have rejected, because the mean is actually higher.

02:16 The mean is actually higher than 15 minutes, which means you should be charging some type of a premium, an extra premium for that because of that taking longer periods of time.

02:26 So the type 2 air, you don't think that you have a situation of the alternative, and if it were the alternative, you'd be charging more.

02:34 Well, you're not going to charge more, but you should have.

02:37 So that's what the impact is.

02:39 Now we want to find what's the likelihood of a type 2 air if the actual mean is 17 minutes.

02:50 So it's actually two minutes higher than what we posted.

02:53 So we need to find what type of mean will cause us to fail to reject.

02:58 So let's turn this into a mean.

03:00 If we take 15 plus 2 .326 times the size of a standard deviation divided by the square root of n, which was 35, that tells us that a value of 16 .57.

03:15 If we get an x bar that is less than or equal to this value, that will cause us to fail to reject.

03:22 So we want to know what's the likelihood of getting an x bar that is less than or equal to this 16 .57...

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