Warm-Up

1. Warm-Up A poll of 300 randomly selected adults discovered that 185 had over $5000 in unsecured debt. • Are the conditions met assume the distribution is approximately normal? • Find a 98% confidence interval. • How large of a sample is needed to reduce the margin of error to 4%?

2. Hypothesis Tests for Proportions Idea • Create a hypothesis (null hypothesis) that a parameter is some value. • Take a sample and the compare the statistic from the sample to the null hypothesis. • If the statistic is consistent with hypothesis, then we will say that we “fail to reject the null hypothesis”. (We are not proving the null hypothesis is true, but instead finding no reason to reject it) • If the probability of the resulting statistic is very unlikely, we will “reject the null hypothesis”.

3. Definitions Null Hypothesis (Ho): the proposed value of the parameter in question (for now the parameter is the theoretical value of the proportion) Alternative Hypothesis (Ha): contains the value of the parameter we accept if we reject the null (not a precise value) P-Value: the probability that the statistic should occur if the null hypothesis is true Significance level (α): the level at which a decision will be made, usually 5%. In other words, the probability that which we “reject the null hypothesis” because the likelihood of a statistic happening is so small that it could not happen just by chance.

4. Thought Process Think of a defendant on trial. Our judicial system assumes a person is not guilty (null hypothesis). The burden is on the prosecution to prove the defendant is guilty (alternative hypothesis). If the prosecution fails to do this, the defendant is failed to be proven guilty. This does not mean he is innocent, just that he is not guilty. Therefore, the jury has failed to reject the null hypothesis.

5. Steps to a Complete Hypothesis Test • State the parameter you are testing in words. • Name the test (today, they will be a 1 proportion Z-test) • State the hypothesis • Show the conditions are met to use your model • Calculate your p-value (include a picture) • State your conclusions in context.

6. Ex: Past studies have shown 39% of graduating Placer County seniors attended a 4-year university the year following graduation. An SRS of 400 Placer County graduating seniors in 2007 found that 190 attended a 4-year university the following school year. Is this evidence that the 2007 graduating class had an unusually high percentage of students entering a 4-year university after graduation?

7. Confidence Intervals vs. Tests Sampling Error is different: C.I.’s – Tests – It is possible to use a Confidence interval to make a decision on whether to reject the null hypothesis or not. A 90% Confidence Interval is the same as a one-sided test at the 5% level. If the null hypothesis is in the Confidence Interval at the appropriate confidence level for α, then you will fail to reject the null hypothesis.

8. Ex 2: Write an appropriate null and alternative hypothesis for each situation. • A governor is concerned about his negatives – the percentage of state residents who express disapproval of his job performance. His political committee sets up a poll. The governor is hoping to keep the negatives below 30% • Is a coin fair? • Only about 20% of people who try to quit smoking succeed. Sellers of a motivational tape claim that listening to the recorded messages can help people quit.

9. Ex 3: Going back to the governor who is concerned about negatives. The poll finds the statistic is 32% show disapproval with his job performance and determines the p-value is 0.22. • What does this mean? • What conclusions can you make?

10. Ex. 4: The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996 31% of students reported that their mothers had graduated from college. In 2000, responses from 8368 students found that this figure had grown to 32%. Is this evidence of a change in education level among mothers?