AP Statistics Chapter 22 Notes “Comparing Two Proportions”

Introduction  Comparisons between two percentages are much more common in the real-world setting than questions about isolated percentages  We often want to know how two groups differ, whether a treatment works better than a placebo, or whether this year’s results are better than last year’s

Null and Alternate Hypotheses  The null hypothesis states that there is no difference between the percentages of the first group and the second group.  The alternate hypothesis states that: – the first group has a lower percentage than the second group (a negative difference) –the first group has a higher percentage than the second group (a positive difference) –or there is a difference between the first and second groups’ percentages (either positive or negative difference) or

Mean and Standard Deviation – Comparing 2 Proportions  The mean is the null hypothesis (no difference between the first group’s and the second group’s percentages)  Mean :  The standard deviation of the difference between two sample proportions (or standard error if working with p-hat instead of p) is:  SD(p 1 – p 2 ) :

Example  Do snoring rates differ for those under and over 30 years old? The National Sleep Foundation asked a random sample of 995 U.S. adults about their sleep habits. Of the questions asked about snoring, 26.1% of the 184 people under 30 snored, compared with 39.2% of the 811 people in the over 30 age group. Is this difference of 13% sufficient evidence to suggest that older people snore more than younger people?

The Step-by-Step Solution Plan  Step 1: State the hypotheses – The hypotheses are about the DIFFERENCE in the proportions. –H o : p 1 = p 2 or p 1 – p 2 = 0 (AKA no difference in snoring rates) –H a : p 1 < p 2 or p 1 – p 2 < 0 –(AKA there is a negative difference in snoring rates because the first group snores less than the second group)

The Step-by-Step Solution Plan  Step 2: Check the conditions and assumptions. – patients randomly selected –the groups are independent of each other –we have at least 9,950 people in the U.S. –we have at least 10 people that snore and 10 people that don’t snore within each age group

The Step-by-Step Solution Plan  Step 3 – Calculations –STD(p 1 – p 2 ) = Perform a 2-proportion z-test (Stat, Tests, #6) –Group 1 (under 30)Group 2 (over 30) –X 1 = x 2 = –n 1 = n 2 = –P-value =

The Step-by-Step Solution Plan  Step 4: State your conclusion Reject the null hypothesis: Based on this sample, there is enough evidence to suggest that older people snore more than younger people. Retain the null hypothesis: Based on this sample, there is not enough evidence to suggest that older people snore more than younger people. OR

Using Confidence Intervals  Find a 95% confidence interval. (Stat, Tests, #B – 2 PropZInt)  Explain what it means in the context of this problem.  Does it support your conclusion?