Chapter 11

Inferences about population proportions using the z statistic

The Binomial ExperimentSituations 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 ExhaustiveWhen 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 ExhaustiveFor 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 tailsRolls of a dice – “6” and “not six”True-False exams – true and falseMultiple choice exams – correct and

incorrectCarnival 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 πProfessors 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 πThe 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 AssumptionsSampling

– n observations obtained through independent random sampling

– The sample is large (n = 80)Data

– Mutually exclusive and exhaustive (gender)

2. Null and Alternative Hypotheses

H0 : π = 0.57

HA : π ≠ 0.57

3. Sampling DistributionWe will use the normal distribution as

an approximation to the binomial and a z-score transformation:

4. Set Significance Level

α = .05

Non-directional HA:

Reject H0 if z ≥ 1.96 or z ≤ -1.96, or

Reject H0 if p < .05

5. Compute

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

5. Compute (in Excel)

5. Computation results

Note on computationsAll computations were performed in

ExcelThe 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 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