One Prop Z-Test
Steps
One Proportion Z Interval
P
p = proportion of _________ who_____________
C
Random: SRS
Normality:
Independence: pop ≥ 10n
F
One Prop Z Interval (_____, ______)
S
We are ______% confident that theinterval captures the true proportion of__________ who _________.
Steps
One Proportion Z Test
P
p = proportion of _________ who_____________
H0: p = ______
Ha: p ______
C
Random: SRS
Normality:
Independence: pop ≥ 10n
F
One Prop Z Test
Z = ________
p = ________
S
Since p α, we reject/fail to reject H0.We conclude/cannot conclude that__________________________.
•I say “Normal” you say _________________.
•So, the calculator is calculating a Z score based on the samplingdistribution of
•Then, it uses Table A to calculate the probability.
•We are lucky. We have calculators. Yay, TI.
•A potato-chip producer has just received a truckload ofpotatoes from its main supplier. If the producer determines thatmore than 8% of the potatoes in the shipment have blemishes,the truck will be sent away to get another load from thesupplier. A supervisor selects a random sample of 500 potatoesfrom the truck. An inspection reveals that 47 of the potatoeshave blemishes. Carry out a significance test at the α= 0.10significance level. What should the producer conclude?
•Remind me, what is a p-value???
•In the context of the last problem, describe thefollowing:
•Type I Error
•Type II Error
•Power
•I claim that I am an 80% free throw shooter. You think I amoverestimating my ability and challenge me to a 50 shot trial.In my trial, I make only 32 shots. Is the difference statisticallysignificant at the α = 0.05 level? What if α = 0.01?
•At the conclusion of a significance test, we have made adecision.
•We either reject H0 or we fail to reject H0.
•But, we’re left wondering, what is a good estimate of the truevalue of p (the population proportion)?
•A confidence interval can shed light on this!
According to the Centers for Disease Control andPrevention (CDC) Web site, 50% of high schoolstudents have never smoked a cigarette. Jasonwonders whether this national result holds true inhis large, urban high school. For his AP Statisticsclass project, Jason surveys an SRS of 150students from his school. He gets responses fromall 150 students, and 90 say that they have neversmoked a cigarette. What should Jason conclude?Give appropriate statistical evidence to supportyour answer.
•For his AP Statistics class project, Jason surveys an SRS of 150students from his school. He gets responses from all 150students, and 90 say that they have never smoked a cigarette.Construct a 95% confidence interval for p, the proportion ofstudents at Jason’s school who say they have never smoked acigarette.
•Don’t forget to recheck the normality condition!
•Confidence Intervals and Two-Sided Tests
There is a link between confidence intervals and two-sided tests. The 95% confidenceinterval gives an approximate range of p0’s that would not be rejected by a two-sidedtest at the α = 0.05 significance level. The link isn’t perfect because the standard errorused for the confidence interval is based on the sample proportion, while the denominatorof the test statistic is based on the value p0 from the null hypothesis.
A two-sided test at significance level α(say, α = 0.05) and a 100(1 –α)%confidence interval (a 95% confidenceinterval if α = 0.05) give similarinformation about the populationparameter.
A two-sided test at significance level α(say, α = 0.05) and a 100(1 –α)%confidence interval (a 95% confidenceinterval if α = 0.05) give similarinformation about the populationparameter.
If the sample proportion falls in the“fail to reject H0” region, like the greenvalue in the figure, the resulting 95%confidence interval would include p0. Inthat case, both the significance test andthe confidence interval would be unableto rule out p0 as a plausible parametervalue.
However, if the sample proportion fallsin the “reject H0” region, the resulting95% confidence interval would notinclude p0. In that case, both thesignificance test and the confidenceinterval would provide evidence that p0is not the parameter value.
•A two-sided test of H0 : p = p0 at significance level α givesroughly the same conclusion as a 100(1 – α)% confidenceinterval.
•CAUTION: This duality does not apply to one-sided significance tests and confidenceintervals!
•From the output above, see if you can identify
•H0 and Ha
•All the necessary components of the “formula” step
•The conclusion (reject or fail to reject)
•The duality between the CI and the hypothesis test
