Business Research Methods: Inferential Statistics and Effect Sizes

Part I: t-tests Comparing Means

Problems 1-3 are drawn from Stat Trek. After taking a crack at the problems on your own, you can check your answers and read through the thorough explanation offered on the web site.

1. An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. Suppose a simple random sample of 50 engines is tested. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. Test the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run times for the population of engines are normally distributed.)

2. Bon Air Elementary School has 300 students. The principal of the school thinks that the average IQ of students at Bon Air is at least 110. To prove her point, she administers an IQ test to 20 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 10. Based on these results, should the principal accept or reject her original hypothesis? Assume a significance level of 0.01.

3. Within a school district, students were randomly assigned to one of two Math teachers - Mrs. Smith and Mrs. Jones. After the assignment, Mrs. Smith had 30 students, and Mrs. Jones had 25 students. At the end of the year, each class took the same standardized test. Mrs. Smith's students had an average test score of 78, with a standard deviation of 10; and Mrs. Jones' students had an average test score of 85, with a standard deviation of 15. Test the hypothesis that Mrs. Smith and Mrs. Jones are equally effective teachers. Use a 0.10 level of significance. (Assume that student ability is approximately equal across classes.)

4. The Acme Widget Company claims that their widgets last 5 years, with a standard deviation of 1 year. Assume that their claims are true. If you test a random sample of 9 Acme widgets, what is the probability that the standard deviation in your sample will be less than 0.95 years? (Hint: Compute the χ² statistic).

Problem 4 is also drawn from Stat Trek.

Part II: Effect Sizes Comparing Means

One of the common uses of central tendency and standard deviation statistics is to make comparisons between different samples. In some situations, a simple comparison of the difference between the mean (average) of the two samples is sufficient to meaningfully interpret the difference. In others, the properties of the variable itself may not be inherently meaningful and scaling of the variable is useful in helping determine how different two samples may be on some variable. In these situations, a standardized difference, like Cohen's d is useful.

For each of the following exercises, calculate the standardized difference (Cohen's d) suggested in each scenario. For your convenience, here is the population and sample formulas for Cohen's d, when the SDs are to be pooled (i.e., when they come from the same population).

Cohen's d (in the population) = (μ₁ - μ₂)/σ_pooled        σ_pooled = √((σ₁²+σ₂²)/(n₁ + n₂))

Cohen's d (in samples) = (x ¯₁ - x ¯₂)/SD_pooled            SD_pooled = √(((n₁-1)SD₁² + (n₂-1)SD₂²)/(n₁+n₂ - 2))

5. An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. Suppose a simple random sample of 50 engines is tested. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. How different is the run time for the sample of engines from the expected value?

6. Bon Air Elementary School has 300 students. The principal of the school thinks that the average IQ of students at Bon Air is at least 110. To prove her point, she administers an IQ test to 20 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 10. How different is the sample of IQ scores for the tested students from the expected value of 110?

7. Within a school district, students were randomly assigned to one of two Math teachers - Mrs. Smith and Mrs. Jones. After the assignment, Mrs. Smith had 30 students, and Mrs. Jones had 25 students. At the end of the year, each class took the same standardized test. Mrs. Smith's students had an average test score of 78, with a standard deviation of 10; and Mrs. Jones' students had an average test score of 85, with a standard deviation of 15. How different are the test scores of the two teachers' students?

Part III: Short Answer Questions for review and preparation for next week

8. Discuss the difference between estimates of effect sizes and inferential statistics.

9. What is a standard error? How is it influenced by sample size?

10. Discuss the differences between one-tailed and two-tailed tests of statistical significance. When would you choose each?

11. Discuss Type I and Type II errors. Discuss how the potential risks associated with type I and type II error might differ across studies. How can researchers influence the potential for type I and type II errors?

12. What is statistical power? How can a researcher influence the statistical power of a research design?

13. What is the difference between statistical significance and substantive significace (or practical importance)? Under what circumstances might interpretations of research findings using statistical significance and substantive significance reach different conclusions?

Attachment:- Assignment File.rar