Chapter 11 Inferences about population proportions using the z statistic

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

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

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

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

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

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

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)?

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)

2. Null and Alternative Hypotheses H 0 : π = 0.57 H A : π ≠ 0.57

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

4. Set Significance Level α =.05 Non-directional H A : Reject H 0 if z ≥ 1.96 or z ≤ -1.96, or Reject H 0 if p <.05

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

5. Compute (in Excel)

5. Computation results

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

6. Conclusions Since our p <.05, we Reject the H 0 and accept the H A and conclude That the sample of students used in this report over-represent women in comparison to the general university population