Understanding Inferences About Population Proportions Using the Z Statistic in Binomial Experiments

1. Chapter 11 Inferences about population proportions using the z statistic

2. The Binomial Experiment • Situations that conform to a binomial experiment include: • There are n observations • Each observation can be classified into 1 of 2 mutually exclusive and exhaustive outcomes • Observations come from independent random sampling • Proportion is the parameter of interest

3. Mutually Exclusive and Exhaustive • When an observation is measured, the outcome can be classified into one of two category • Exclusive – the categories do not overlap • An observation can not be part of both categories • Exhaustive – all observations can be put into the two categories

4. Exclusive and Exhaustive • For convenience, statisticians call the two categories a “success” and “failure”, but they are just a name • What is defined as a “success” and “failure” is up to the experimenter

5. Binomial Experiments Examples (with successes and failures) • Flips of a coin – heads and tails • Rolls of a dice – “6” and “not six” • True-False exams – true and false • Multiple choice exams – correct and incorrect • Carnival games (fish bowls, etc.) – wins and losses

6. The sampling distribution of p • In order to test hypotheses about p, we need to know something about the sampling distribution: Approximately normal

7. Hypothesis Test of π • Professors act the local university claim that their research uses samples that are representative of the undergraduate population, at large • We suspect, however, that women are represented disproportionately in their studies

8. Hypothesis Test of π • The proportion of women at the university is: π = 0.57 • In the study of interest: n = 80 Number of women = 56 • Is the π in this study different than that of the university (0.57)?

9. 1. State and Check Assumptions • Sampling • n observations obtained through independent random sampling • The sample is large (n = 80) • Data • Mutually exclusive and exhaustive (gender)

10. 2. Null and Alternative Hypotheses H0 : π = 0.57 HA : π ≠ 0.57

11. 3. Sampling Distribution • We will use the normal distribution as an approximation to the binomial and a z-score transformation:

12. 4. Set Significance Level α = .05 Non-directional HA: Reject H0 if z ≥ 1.96 or z ≤ -1.96, or Reject H0if p < .05

13. 5. Compute π = 0.57 n= 80 Number of women = 56 The p of women in the sample = 56/80 p = .70

14. 5. Compute (in Excel)

15. 5. Computation results

16. Note on computations • All computations were performed in Excel • The p-value was determined using the function =NORM.S.DIST • This function returns the proportion of zs LESS than or equal to our z value • However, we need the proportion of zs greater than our z • Thus, we subtracted the result of NORM.S.DIST from 1

17. 6. Conclusions • Since our p < .05, we Reject the H0 and accept the HA and conclude • That the sample of students used in this report over-represent women in comparison to the general university population