Inference for Two Proportions Two samples of categorical

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Inference for Two Proportions Two samples of categorical data 1

Inference for Two Proportions Two samples of categorical data 1

Introduction to Statistical Inference Methods � � Statistical Inference: Drawing conclusions about a population

Introduction to Statistical Inference Methods � � Statistical Inference: Drawing conclusions about a population from sample data. Methods Ø Point Estimation– Using a sample statistic to estimate a parameter Ø Confidence Intervals – supplements an estimate of a parameter with an indication of its variability Ø Hypothesis Tests - assesses evidence for a claim about a parameter by comparing it with observed data Parameter Measure Statistic Mean of a single population Proportion of a single population Mean difference of two dependent populations (MP) Difference in means of two independent populations Difference in proportions of two populations Variance of a single population Standard deviation of a single population S

The difference in two Proportions � The goal here is to conduct a hypothesis

The difference in two Proportions � The goal here is to conduct a hypothesis test comparing two in dependent samples of categorical data. � Our new parameter of interest will be: comparison of two populations proportions will be based on the statistic

SE Problem with Proportions � � � The population proportions used in the formula

SE Problem with Proportions � � � The population proportions used in the formula for standard error of the sampling distribution are unknown. Even if two population proportions are equal, the sample proportions drawn from these populations are usually different. Methods of estimating the population proportions differ depending on whether you are constructing a confidence interval or conducting a hypothesis test.

Hypothesis Test of the difference in proportions: Hypotheses �

Hypothesis Test of the difference in proportions: Hypotheses �

Pooling p � 9

Pooling p � 9

Hypothesis Test of the difference in proportions: Test Statistic �

Hypothesis Test of the difference in proportions: Test Statistic �

Hypothesis Test for the difference in two proportions � Data are from two independent

Hypothesis Test for the difference in two proportions � Data are from two independent random samples of male and female seniors on whether they expect to attend grad school. � Test Males Females Yes 18 33 51 No 30 19 49 48 52 100 to see whethere is a significant difference between genders at a = 0. 01

Set up � The question ask about a general “difference” H 0: p m

Set up � The question ask about a general “difference” H 0: p m = p f H a: p m ≠ p f � Use a = 0. 01 � Again it does not matter who we call group 1 and 2, as long as we’re consistent.

Check Conditions � First calculate the sample proportions � Conditions:

Check Conditions � First calculate the sample proportions � Conditions:

Test Statistic � Pooled � Test proportion: Statistic

Test Statistic � Pooled � Test proportion: Statistic

Critical Value � We need a Critical value with: ◦ a = 0. 01

Critical Value � We need a Critical value with: ◦ a = 0. 01 ◦ Two-Tailed � Table Value ◦ ± 2. 575 15

P-value � We need 2(Z < -2. 59) � Estimating ◦ 0. 0096 �

P-value � We need 2(Z < -2. 59) � Estimating ◦ 0. 0096 � By w/ table: Technology: ◦ 0. 0096

Conclusion � CV ◦ ◦ Method: Two-Tailed CV = ± 2. 575 TS =

Conclusion � CV ◦ ◦ Method: Two-Tailed CV = ± 2. 575 TS = -2. 59 TS is in Rejection Region � P-val Method: ◦ P-val= 0. 0096 < α = 0. 01 � Decision: ◦ Reject H 0, there is statistically significant evidence of a difference between genders’ interest in graduate school at α = 0. 01.

 � � In 2002, a Pew Poll based on a random sample of

� � In 2002, a Pew Poll based on a random sample of 1500 people suggested that 43% of Americans approved of stem cell research. In 2009 a new poll of a different sample of 1500 people found that 58% approved. Construct a 95% confidence interval for the difference between the proportion of people who support stem cell research.

Calculations �

Calculations �

Interpretation and Conclusion � Interval Interpretation Contains 0 The population proportions may be equal

Interpretation and Conclusion � Interval Interpretation Contains 0 The population proportions may be equal Both values are positive (+, +) The population proportions are most likely different and p 1 > p 2 Both values are negative (-, -) The population proportions are most likely different and p 1 < p 2

Link between HTs and CIs � 21

Link between HTs and CIs � 21