Inference Confidence Interval for p

Confidence Interval Plausible values for the unknown population proportion, p. We have confidence in the process that produced this interval.

Inference – Using C.I. The population proportion, p, could be any of the values in the interval. Values outside the interval are not plausible values for p.

Inference – Hypothesis Test Propose a value for the population proportion, p. Does the sample data support this value?

Example A law firm will represent people in a class action lawsuit against a car manufacturer only if it is sure that more than 10% of the cars have a particular defect.

Example Population: Cars of a particular make and model. Parameter: Proportion of this make and model of car that have a particular defect.

Example Null Hypothesis Alternative Hypothesis H0: p = 0.10 HA: p > 0.10

Example The law firm contacts 100 car owners at random and finds out that 12 of them have cars that have the defect. Is this sufficient evidence for the law firm to proceed with the class action law suit?

Example How likely is it to get a sample proportion as extreme as the one we observe when taking a random sample of 100 from a population with p = 0.10?

Example Sampling distribution of Shape approximately normal because 10% condition and success/failure condition are satisfied. Mean: p = 0.10 Standard Deviation:

Draw a Picture

Standardize

Use Table Z z 0.05 0.06 0.07 0.5 0.6 0.7486 0.7

Interpretation Getting a sample proportion of 0.12 or more will happen about 25% (P-value = 0.25) of the time when taking a random sample of 100 from a population whose population proportion is p = 0.10.

Interpretation Getting a value of the sample proportion of 0.12 is consistent with random sampling from a population with population proportion p = 0.10. This sample result does not contradict the null hypothesis. The P-value is not small, therefore fail to reject H0.

Interpretation Even though the sample proportion, 0.12, is larger than the hypothesized population proportion, 0.10, it is not large enough for us to believe that the population proportion is greater than 0.10. There is not convincing evidence.

Conclusion Based on this sample, the law firm should not pursue the class action lawsuit because the population proportion of defective cars could be only 10%.

Test of Hypothesis Step 1: State your null and alternative hypotheses. H0: p = po HA: p > po

Test of Hypothesis Step 2: Check conditions Independence Random sampling condition 10% condition Success/Failure condition

Test of Hypothesis Step 3: Calculate the test statistic value and convert it into a P-value. Use Table Z.

Test of Hypothesis Step 4: Use the P-value to reach a decision. If the P-value is small, then reject Ho. If the P-value is not small, then fail to reject Ho.

Test of Hypothesis Step 5: State your conclusion in the context of the problem. What does rejecting, or failing to reject, Ho mean in the context of the problem.

Alternatives H0: p = po HA: p < po, P-value = Pr < z

Another Example According to the U.S. census, Story County has 9.7% of its population classified as non-white. Of 120 people called for jury duty in Story County only 3 are non-white. Is this convincing evidence of under-representation of non-whites?

Another Example Step 1: State your null and alternative hypotheses. H0: p = 0.097 HA: p < 0.097 p is the proportion of non-whites among all people in the jury pool for Story County.

Test of Hypothesis Step 2: Check conditions Independence Random sampling condition 10% condition Success/Failure condition

Test of Hypothesis Step 3: Calculate the test statistic value and convert it into a P-value.

Test of Hypothesis Step 4: Use the P-value to reach a decision. Because the P-value is small, we should reject Ho

Test of Hypothesis Step 5: State your conclusion in the context of the problem. This is convincing evidence that non-whites are under-represented in the jury pool.