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Statistics: Hypothesis and Error Types

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In a specific company, there are three levels of workers: skilled, unskilled, and manager. The owner of this company wants to know if the level of job satisfaction differs amongst the three levels of workers.

Data: 30 workers in each level are randomly selected to take the job satisfaction survey. The survey asks each worker on a scale of 1-10 how satisfied he or she is with his or her job. The data will consist of 90 workers total (30 belonging to skilled, 30 belonging to unskilled, and 30 belonging to manager) and their satisfaction rating.

Null Hypothesis: The population means of satisfaction ratings of each of level of worker are equal to each other.
Alternative Hypothesis: The population means of satisfaction ratings of each of level of worker are not all equal to each other.

Type 1 Error: We incorrectly reject the null hypothesis that the population means of satisfaction ratings of each of level of worker are equal to each other.
Type 2 Error: We fail to reject the null hypothesis that the population means of satisfaction ratings of each level of worker are equal to each other even though the null hypothesis if false.

If we obtained a p-value a .2 from a test statistic, that means we have a 20 percent probability of obtaining a test statistic at least as extreme as our test statistic assuming the null hypothesis is true. At a .05 level of significance, we would fail to reject the null hypothesis.

If we obtained a p-value a .01 from a test statistic, that means we have a 1 percent probability of obtaining a test statistic at least as extreme as our test statistic assuming the null hypothesis is true. At a .05 level of significance, we would reject the null hypothesis.

a. Reformulate your hypothesis test to incorporate a 2-sample hypothesis test. What would be your data? What is your null hypothesis? What is your alternate hypothesis? What would be your Type 1 and Type 2 errors relative to your decision? Suppose you have a p-value of 0.01, what does this mean relative to your problem and decision? Suppose the p-value is 0.20, what does this mean relative to your problem and decision?

b.If you reformulated your design for 3 or more samples, what would be the implications of interaction? When would you use Tukey-Kramer test?

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Solution Preview

a. Reformulate your hypothesis test to incorporate a 2-sample hypothesis test. What would be your data? What is your null hypothesis? What is your alternate hypothesis? What would be your Type 1 and Type 2 errors relative to your decision? Suppose you have a p-value of 0.01, what does this mean relative to your problem and decision? Suppose the p-value is 0.20, what does this mean relative to your problem and decision?

Say we combine the data from the skilled and unskilled workers into one group. We would have two different samples: the first sample representing the skilled and unskilled workers and the second sample representing the managers.

Null Hypothesis: The population mean of the satisfaction ratings of the first group of workers (containing skilled and unskilled workers) is equal to the population mean of the satisfaction ...

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Statistics - Hypothesis Testing and Type 2 Error - A corporation owns a factory that produces sulfuric acid. Because of changing conditions, the plant's output is quite ...

A corporation owns a factory that produces sulfuric acid. Because of changing
conditions, the plant's output is quite variable. The company's president has observed that the output is normally distributed with a mean of 8,200 pieces per hour. He recently has been informed that the government is considering a new law that would require him to alter the way in which the acid is produced. He reorganizes production facilities to comply and observes the hourly output for two working days (16 hours). At a 5% significance level do these data indicate that the new guidelines will reduce output? The
data are shown below:

8,283
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An employee at the company believes that production has really gone down to 8000 Liters per hour, which represents a true reduction of output and is surely a possible value supporting the alternative hypothesis. Find the probability that the statistician makes an error of Type 2 - ie that the true mean is really 8000 L per hour but yet we accept the null hypothesis.

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