Business Decision Making Project Part 2

Hypothesis Testing Definitions: A statistical hypothesis is a guess about a population parameter. The guess may or not be true. The null hypothesis, written H0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters. The alternative hypothesis, written H1 or HA, is a statistical hypothesis that specifies a specific difference between a parameter and a specific value, or that there is a difference between two parameters. Example 1: A medical researcher is interested in finding out whether a new medication will have undesirable side effects. She is particularly concerned with the pulse rate of patients who take the medication. The research question is, will the pulse rate increase, decrease, or remain the same after a patient takes the medication? Since the researcher knows that the mean pulse rate for the population under study is 82 beats per minute, the hypotheses for this study are:

H0: µ = 82 HA: µ ≠ 82

The null hypothesis specifies that the mean will remain unchanged and the alternative hypothesis states that it will be different. This test is called a two-tailed test since the possible side effects could be to raise or lower the pulse rate. Notice that this is a non directional hypothesis. The rejection region lies in both tails. We divide the alpha in two and place half in each tail. Example 2: An entrepreneur invents an additive to increase the life of an automobile battery. If the mean lifetime of the automobile battery is 36 months, then his hypotheses are:

H0: µ ≤ 36 HA: µ > 36

Here, the entrepreneur is only interested in increasing the lifetime of the batteries, so his alternative hypothesis is that the mean is greater than 36 months. The null hypothesis is that the mean is less than or equal to 36 months. This test is one-tailed since the interest is only in an increased lifetime. Notice that the direction of the inequality in the alternate hypothesis points to the right, same as the area of the curve that forms the rejection region.

Example 3: A landlord who wants to lower heating bills in a large apartment complex is considering using a new type of insulation. If the current average of the monthly heating bills is $78, his hypotheses about heating costs with the new insulation are:

H0: µ ≥ 78 HA: µ < 78

This test is also a one-tailed test since the landlord is interested only in lowering heating costs. Notice that the direction of the inequality in the alternate hypothesis points to the left, same as the area of the curve that forms the rejection region. Study Design: After stating the hypotheses, the researcher’s next step is to design the study. In designing the study, the researcher selects an appropriate statistical test, chooses a level of significance, and formulates a plan for conducting the study. In the first example, the medical researcher will select a sample of sample of patients who will be given the drug. After the drug takes effect, the researcher will measure each person’s pulse rate. Remember that the sample means vary about the population mean. Thus the mean of the sample will in most cases, not be exactly equal to 82 (the mean of the population). This raises the question, if the mean of the sample is not exactly equal to the population mean, how do we know that the medication really does affect the pulse rate? In other words, is the difference due to chance or is it due to the effects of the medication? If the mean pulse rate of the sample is 83, the researcher would probably not reject the null hypothesis. However, if the mean pulse rate is 110, the researcher would probably conclude that there really is a difference. The question is, where does the researcher draw the line between chance variation and a real difference? The difference is not made on feelings or intuition, but is made statistically. More Definitions: A statistical test uses the data obtained from a sample to make a decision about whether or not the null hypothesis can be rejected. The numerical value obtained from a statistical test is called the test statistic or test value. The critical value separates the critical region from the non-critical region. The critical region is the region of values that indicates that there is a significant difference and that the null hypothesis should be rejected. A level of significance is the maximum probability of committing a Type I error. In a statistical test, the mean is computed for the data obtained from the sample and is compared with the population mean. Then, a decision is made to reject or not reject the null hypothesis on the basis of the value obtained from the statistical test. If the difference is significant, the null is rejected. If it is not significant, the null is not rejected.

There are 4 possible outcomes:

H0 True H0 False Reject H0 Type I error Correct decision

Accept H0 Correct decision Type II error

A Type I error occurs when the null hypothesis is rejected when it is true. A Type II error occurs when the null hypothesis is not rejected when it is false. A decision to reject or not reject the null does not really prove anything. The only way to prove anything statistically is to use the entire population . . . in most cases, this is not possible. Instead, we use probabilities to make a decision. When there is a large enough difference between the sample mean and the hypothesized mean, the null is probably not true. How large does this difference need to be? First, we need to set a level of significance. In most cases, this level will be either .05 or .01. That means that if the null is rejected, the probability of a Type I error will either be 5% or 1% and the probability of a correct decision is either 95% or 99%. The probability of a Type II error is symbolized by β (Greek letter beta). The power of a test is its sensitivity, or the probability that a statistical test will be able to detect a true difference. Power is equal to 1-β. Therefore, if β.= .05, power = .95. Power is directly related to both sample size and effect size; power increases and sample size or effect size increases. After a significance level is chosen, a critical value is chosen from a table for the appropriate test. If a z-test is used, the z-table (Appendix D) is consulted to find the critical value. The critical value determines the critical (rejection) and noncritical(acceptance) regions. For a one-tailed test, the critical value can be on the right or left side of the mean. Here, the entire alpha is placed in the appropriate tail. For a two tailed test, the critical region is split into two equal parts and the alpha is split between the tails.

Five step procedure for hypothesis testing: 1. State the null & alternative hypotheses

Tip: It’s often easier to state the alternative hypothesis first.

2. Design the study and select a level of significance (alpha)

Tip: Your alpha level should reflect your level of comfort with either a Type I or Type II error.

3. Identify the test statistic

Tip: List all relevant info from your data first, then substitute into the appropriate formula. This can include n, s or σ, x-bar or p, µ or π. It’s also helpful to draw the curve and rejection area.

4. Formulate a decision rule

Tip: Use the Z-table to find the appropriate table value for your chosen alpha level, then decide under what conditions you’ll reject the null hypothesis.

5. Take a sample and arrive at a decision to reject or not reject the null.

Tip: Explain your decision in words rather than reporting “reject” or “accept”. Non- statisticians will thank you for it.

Discussion questions:

Your learning team has just been asked to perform some data analysis for the same entrepreneur who invented the battery additive. His newest product line is a gas additive that he thinks will significantly improve gas mileage in older cars. He's

ready to take it to market with a splashy ad campaign but thinks that it's a good idea to back up his claims with statistics.

1. What type of sample data will you need?

2. How will you choose and collect your sample data?

3. Set up your null and alternate hypotheses.

4. Did you use a one or two tailed test? Why?

5. Select a level of significance. What level did you use? Why?

6. How likely is it that you'll experience a Type I error? How can you reduce this likelihood?