Two events are independent when the occurrence of either one does not change the probability that the other will occur. In mathematical terms, this is defined by saying that event A and event B are independent if p(AB) p(A). In this case, the multiplication rule reduces to:

p(A B) p(A) * p(B) [3.15]

It is important to understand that independent events are not mutually exclusive events; they can co-occur but do not depend on one another. Mutually exclusive events are not independent insofar as the occurrence of one depends on the other one’s not occurring

1. An example of two independent events is rolling a 3 on a die and flipping a tail on a coin. Rolling a 3 does not affect the probability of flipping the tail. If events are independent, then the joint probability that both will occur is simply the product of the probabilities that each will occur:

p(A and B) p(A) * p(B) [3.16]

For example, the probability of rolling a 3 on a die is .17, and the probability of flipping a tail on a coin is .5. Then the probability of both rolling a 3 on a die and flipping a tail is .17 * .5 .085.

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2. Common -levels used are .10, .05, and .01. An -level of .10 means that for a result to be significant, it cannot occur more than 10% of the time by chance. Similarly, an -level of .05 means that the result cannot occur more than 5% of the time by chance, and an -level of .01 means that it cannot occur more than 1% of the time by chance.

3. For example, a recent evaluation of a school-based obesity prevention program sought to compare the effects of three groups who experienced three separate interventions concerning knowledge about nutrition with a control group over a 20-week period (Warren, Henry, Lightowler, Bradshaw, and Perwaiz, 2003). The researchers set their -level at .05. In doing any study, it is important to know if the groups are similar at the beginning of the study. Thus, they compared the body mass index (BMI), a measure of obesity, at baseline of the four groups using a one-way ANOVA, which is used to test if the means of different groups are significantly different from one another. The p-value of the computed ANOVA statistic was greater than .05, so the researchers accepted the null hypothesis. Thus, they concluded that the average BMI did not differ significantly between the four groups.

A one-sample z-test is used to compare the mean value of a variable obtained from a sample to the population mean of that variable to see if it is statistically significantly different. The steps for hypothesis testing are illustrated below by conducting a one-sample z-test to see if the BMIs of women attending a local church based health fair differ from the average BMIs of U.S. women. The church, which serves a low-income, African American population, has an active health ministry. Because obesity is increasingly a threat to the health of the U.S. population, especially among low-income African Americans (Truong and Sturm, 2005), the researchers wanted to know how the BMIs of the women in the sample compared with those of U.S. women in general. A BMI of 25 to 29.99 indicates overweight, and a BMI of 30.00 or higher indicates obesity. The mean BMI of the 48 women in the sample was 29.2, with a standard deviation of 3.4. In 1999 to 2000, the mean BMI for U.S. women was 27.9, with a standard deviation of 5.4 (Okusun, Chandra, Boev, et al., 2004).