STAT 3120 Statistical Methods I

1. STAT 3120Statistical Methods I Lecture 5 TtestsAbout Proportions

2. STAT3120 - Ttests The term “Ttest” comes from the application of the t-distribution to evaluate a hypothesis. As the number of obs increases, the t-distribution approaches normality. In practice, even hypothesis tests with sample sizes greater than 30, which utilize the normal distribution, are commonly referred to as “ttests”. Note: a “t-statistic” and a “z-score” are conceptually similar…page 501 shows that the formula for the two are the same.

3. STAT3120 – Ttests about Proportions • The testing formula for a one sample proportion is a simple z calculation: • Z = (sample estimate – Null value)/Null Standard Error • For a proportion, this would be: • Z=(p-po)/SQRT((po(1-po)/n)

4. STAT3120 – Ttests about Proportions If 30% of cars on a street are found to be speeding, the city will install “traffic calming” devices. John used his radar gun to measure the speeds of 400 cars on his street. He found that 32% were speeding. Will John get “traffic calming” devices on his street?

5. STAT3120 – Ttests about Proportions Answer the following: • Identify the Null and Alternative Hypotheses • Identify the Type I and Type II errors, including the implications • What is an appropriate alpha value? • What is the associated p-value? • What is your conclusion?

6. STAT3120 – Ttests about Proportions A recent news article from “VEG World” reported that at least 5% of all American Teenagers eat a vegetarian diet. A sample of 200 teenagers was interviewed about their diets. 8% indicated that they eat vegetarian. Does this prove Veg World’s claim?

7. STAT3120 – Ttests about Proportions • Answer the following: • Identify the Null and Alternative Hypotheses • Identify the Type I and Type II errors, including the implications • What is an appropriate alpha value? • What is the associated p-value? • What is your conclusion?

8. STAT3120 – Ttests about Proportions • 2. Two Sample Ttest - compares the proportion of the first sample minus the proportion of the second sample to a given number. It is of common interest to test of two population proportions are equal. • e.g. Is there a difference in the percentage of students who pass a standardized test between those who took a prep course and those who did not? • Formal Hypothesis Statement examples: • H0: pa - pb =0 H0: pa - pb <0 • H1: pa - pb  0 H1: pa - pb > 0

9. STAT3120 – Ttests about Proportions • Before you undertake a two sample ttest, there are few things to be determined: • The two samples must be independent • The number of individuals with each trait of interest and the number without the trait of interest must be at least 10 in each sample. • See page 529 for the formula.

10. STAT3120 – Ttests about Proportions Suppose that a random sample of 50 men and 50 women were asked if they make a list before they go to the grocery store. Of these 20 of the men and 30 of the women said “yes”. What do you conclude?

11. STAT3120 – Ttests about Proportions Answer the following: Identify the Null and Alternative Hypotheses Identify the Type I and Type II errors, including the implications What is an appropriate alpha value? Using the formula on page 529, determine the test statistic. What is the associated p-value? What is your conclusion?