Significance Tests and Two Proportions

Presentation 9.5

Working with Two Proportions

• Be sure to define your proportions so you can keep track of them.

– For example: • 1: males • 2: females • Use proper notation: Population 1 Population 2 Sample size n n 1 2 Population proportion p 1 p 2 Sample proportion ˆ 1 

x n

1 1 ˆ 2 

x

2

n

2

Properties: Sampling Distribution of p 1 - p 2 If the two random samples are independent, the following properties hold:

mean of

ˆ 1  2 

p

1 

p

2  1  2  ˆ 1 ( 1  ˆ 1 ) 

n

1 2 ( 1  ˆ 2 )

n

2 If both n 1 10, n 2 p 2  and n 2 are large [n 1 p 1  10, n 2 (1- p 2 )  10, n 10], then p 1 1 (1- p and p 2 1 )  each have a sampling distribution that is approximately normal

Setting up the Two Proportion Test

• Check the conditions – Independent samples – Large enough samples • n 1 p 1  10, n 1 (1- p 1 )  10, n 2 p 2  10, n 2 (1- p 2 )  10 • Write Hypotheses – Null Hypothesis (the two proportions are equal) • H o : p 1 =p 2 or H o : p 1 -p 2 =0 – Alternate Hypothesis • Ha: p 1

p 2 • Ha: p 1 ≠p 2 or or Ha: p 1 -p 2 <0 Ha: p 1 -p 2 >0 Ha: p 1 -p 2 ≠ 0

Conducting the Two Proportion Test

• Calculations – Test Statistic z • We will use the calculator for these calculations – p-value

z

 ˆ 1  1 

n

1 ˆ 1 ˆ 1    2 ˆ 2  1 

n

2 • Calculate this as you always have using normalcdf 2 

Example #1: Big Brother

• A survey of 356 workers showed that 192 of them said that it was unethical for the company to monitor employee e-mail. When 106 senior-level bosses were surveyed, 38 said that it was seriously unethical for the company to monitor employee e-mail. Is there a significant difference between the workers’ opinion and the bosses’ opinion of monitored e-mail?

Example #1: Big Brother

• First, check conditions.

– These check out fine as the sample sizes are rather large • Set up hypotheses.

– H o : p 1 =p 2 – Ha: p 1 ≠p 2 Sample size Count (# opposed to monitoring) Sample proportion p-hat 356 192 0.5393

Population 1: Workers Population 2: Bosses 106 38 0.3585

Example #1: Big Brother

• Conduct calculations by using the 2-proportion z test • Enter in your statistics • Record your test statistic and p value (which can also be obtained using 2normalcdf(3.2687,99)

Example #1: Big Brother

• Conclusions – With such a small p-value, we reject the null – There is sufficient evidence to suggest that there is a difference between the proportion of workers who are opposed to e-mail monitoring and the proportion of bosses who are opposed to e-mail monitoring.

Example #2: Well Water

A major court case on the health effects of drinking contaminated water took place in the town of Woburn, Massachusetts. A town well was contaminated with industrial chemicals. During the period when the well was open, 16 birth defects out of 414 births. When this particular well was shut off from and water was supplied from other wells, 3 out of 228 birth defects were reported. The plaintiffs suing the firm responsible for contaminating the well claim that the rate of birth defects is higher when the contaminated well was in use. Conduct a significance test to determine if the plaintiffs have a case.

Example #2: Well Water

• Conduct this test on your own. Following are the results: – Test Statistic: z = 1.8238

– p-value: 0.0341

– At the 5% level, reject the null – At the 1% level, fail to reject the null – If you have questions about these answers, post them in the discussion board!

Significance Tests and Two Proportions

This concludes this presentation.