8.2 Hypothesis Testing for a Proportion | STAT 200
Hypothesis Testing for a Proportion Printerfriendly version Here we will be using hypothesis tests to compare a proportion in one group to a specified population proportion.
Examples: Research Questions The following are research questions that could be answered using a hypothesis test for one proportion. In each case, we would test the hypothesis by comparing data from a sample to the hypothesized population parameter. Handedness. Are more than 80% of American’s right handed? Babies. Is the proportion of babies born male different from .50? Ice cream. Is the percentage of Creamery customers who prefer chocolate ice cream over vanilla less than 80%?
Recall from the last section the five step hypothesis testing procedure that we will be using in this course:
Five Step Hypothesis Testing Procedure 1. Check any necessary assumptions and write null and alternative hypotheses. The assumptions will vary depending on the test. The null and alternative hypotheses will also be written in terms of population parameters; the null hypothesis will always contain an equality (i.e., = =,≥ ≥, or ≤ ≤). 2. Calculate an appropriate test statistic. This will vary depending on the test, but it will typically be the difference observed in the sample divided by a standard error. In this class we will see z z, t t, χ 2 χ2 , and F Ftest statistics. 3. Determine a pvalue associated with the test statistic. This can be found using the tables in Appendix A or using Minitab Express. 4. Decide between the null and alternative hypotheses. If p ≤ α p≤α reject the null hypothesis. If p > α p>α fail to reject the null hypothesis. 5. State a "real world" conclusion. Based on your decision in step 4, write a conclusion in terms of the original research question. Some steps may vary depending on the test. Let's walk through these five steps for specifically for comparing the proportion of one group to a specified value.
1. Check Any Necessary Assumptions and Write Null and Alternative Hypotheses. As in previous lessons, the assumption is that both n × p ≥ 10 n×p≥10 and n × (1 − p) ≥ 10 n×(1−p)≥10 . Note that some textbooks use 15 instead of 10 believing that 10 is too liberal. We will continue to use 10 for our discussions. In terms of the hypotheses, the null hypothesis will always contain an equality, the alternative hypothesis will never contain an equality. Below is a table with the possible combinations of null and alternative https://onlinecourses.science.psu.edu/stat200/node/53
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8.2 Hypothesis Testing for a Proportion | STAT 200
hypotheses. p0 p0 is the hypothesized value of the population proportion.
Research Question
Is the proportion different from p0 p0 ?
Is the proportion greater than p0 p0 ?
Is the proportion less than p0 p0 ?
Null Hypothesis, H 0 H0
p = p0 p=p0
p ≤ p0 p≤p0
p ≥ p0 p≥p0
Alternative Hypothesis, H a Ha
p ≠ p0 p≠p0
p > p0 p>p0
p < p0 p .80 Ha:p>.80 This is a righttailed test. Babies. Is the proportion of babies born male different from .50?
H 0 : p = .50 H0:p=.50 H a : p ≠ .50 Ha:p≠.50 This is a twotailed test. Ice cream. Is the percentage of Creamery customers who prefer chocolate ice cream over vanilla less than 80%?
H 0 : p ≥ .80 H0:p≥.80 H a : p < .80 Ha:p α p>α, therefore we fail to reject the null hypothesis 5. State a "real world" conclusion. There is not sufficient evidence to state that the proportion of citizens of this city who are overweight is different from the national average of .690.
On Your Own Before continuing to the next session, take some time to work through the problems below on your own. Then click on the icon to the left to compare your answers with solutions.
1. According to the Center for Disease Control (CDC), 17.7% of children 6 11 years of age are obese (seehttp://www.cdc.gov/nchs/fastats/obesityoverweight.htm). The parentteacher association at one elementary school wants to find out if the obesity rate at their school is higher than the national average. They take a random sample from their school of 50 children between 611 years of age and found that 13 were obese. Use the fivestep hypothesis testing approach to answer the parentteacher association’s question.
8.2 Hypothesis Testing for a Proportion | STAT 200
2. According to the Center for Disease Control (CDC), 49.4% of American adults 18 years old and over meet their Physical Activity Guidelines for aerobic physical activity (see http://www.cdc.gov/nchs/fastats/exercise.htm). The owners of a gym want to know if the proportion of their members who meet these requirements is different from the national average. They took a random sample of 40 members and 28 met the CDC’s guidelines. Use the fivestep hypothesis testing approach to answer their question.
