Warm-up8.2 Testing a proportion

Data Analysis 9 If ten executives have salaries of $80,000, six salaries of $75,000, and three have salaries of $70,000, what is the median salary?

A. $75,000 B. $76,842 C. $77,500

D. $80,000 E. None of the above

Experimental Design 14In general, for a survey to yield unusable results:

I. A sample size of n = 30 is usually sufficient. II. Researchers must be careful in the way questions are worded. III. Researchers must carefully choose people who they think are representative of the population. A. I only B. II only C. III only D. II and III E. I, II, and III

Inference for Distributions

p̂ m x mx m*

ˆˆ pp z SE *xx z SE *

xx t SE

* ˆ ˆ(1 )ˆ

p pp z

n

*x zn

* s

x tn

Confidence Intervals

8.2 Testing a Proportion 9.1 Confidence Interval of the Mean 9.2 Significance Testing for the Mean

Answers to 8.1 E #7 – 10, 16 and 17

E#7. The randomization condition is met since the teens were randomly selected. A sample of 549 is less than 10% of the population of teens in the U.S.. 52% of 549 is 285.48 > 10 and 48% of 549 is 263.52 > 10 so it meets the success/failure condition.

E#8 The sample was not random. 885 is less than 10% of thepopulation. Both 40% and 60% of 885 are greater than 10.

E #9 The diagram of 8.2can not be used becausethe graph on the rightis for a sample size of 40and not for 100. A samplesize of 100 would have a smaller confidence interval.

E #10 For a sample size of 40 and a proportion of 5%, the 8.2diagram could be used. The only issue is that a sample size of40 is not big enough to meet the success failure condition.40 (0.05) = 8 < 10

E #16. Since the margin of error is to make it

one quarter of its original size, the sample size has to be 16 times greater.E#17. A, D, F and H are correct interpretations

n

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Hypothesis Testing

We will now watch a video called

P-valueExtravaganza

FYI: The guys are drinking soda in the video.

8.2 Significance TestJenny and Maya wonder if heads and tails are equally likelywhen a penny is spun. They spin pennies 40 times and get17 heads. Later… Miguel and Kevin spin pennies and get 10 heads outof 40 spins for a sample proportion, p-hat of 0.25. A sample proportion is said to be statistically significant ifit isn’t a reasonably likely outcome when the proposedstandard is true.

P-valuesInstead of simply reporting whether a result is statisticallysignificant, it is common practice also to report a P-value. The P-value for a test is the probability of seeing a result from arandom sample that is as extreme as or more extreme than theresult you got from your random sample if the null hypothesis istrue. Find the P-value for Jenny and Maya. They spin pennies 40times and get 17 heads.

A large P-value tells you that the sample proportion youobserved is near p0, so your result isn’t statistically significant.

Finding Critical Values

• Suppose you want to reject the null hypothesiswhen the test statistic, z, is in the outer 10% of thestandard normal distribution—that is, with a level ofsignificance equal to 0.10. What should you use ascritical values? α = 0.1

To reject or not to rejectIn a courtroom, a defendant is either guilty or notguilty. Notice they don’t say innocent.

Same with the null hypothesis.

Either reject the null hypothesis or fail to reject thenull hypothesis.

The rejection or lack of rejection is based on the p-valueand whether or not it is statistically significant.

Pg 498 Components of Significant Test1. Give the name of the test and check the conditions for

its use. For a significance test for a proportion, three conditions must be met: The sample is a simple random sample from a binomial population. Both np0 and n(1 − p0) are at least 10. The population size is at least 10 times the sample size.

2. State the hypotheses, defining any symbols. When testing a proportion, the null hypothesis, H0, isH0: The percentage of successes, p, in the population from which the sample came is equal to p0.The alternative hypothesis, Ha, can be of three forms:

porporp0

Components continued…3. Compute the test statistic, z, and find the criticalvalues, z*, and the P-value. Include a sketch that illustratesthe situation. If the alt. hypothesis is ≠, the α is applied to two sides.4. Write a conclusion. There are two parts to stating aconclusion: Compare the value of z to the predeterminedcritical values, or compare the P-value to α . Then saywhether you reject the null hypothesis or don’t reject thenull hypothesis, linking your reason to the P-value or to thecritical value.

Problem 1: A large city’s Dept. of Motor Vehicles claimed80% of candidates pass the driving test, but newspaperreports of 90 randomly selected local teens who had takenthe test only 61 passed.

STEP 1: STEP 1: Check the 3 conditions then give the name of the test you are about to perform.

STEP 2: STEP 2: State the hypothesis. (Null and possible Alternative Hypothesis)

STEP 3: STEP 3: Calculate the critical value, test statistic and p-value. *Demonstrate these with a sketch of the distribution curve as

well!

STEP 4: STEP 4: Write a conclusion. Include an explanation of the pvalue in context and in the format of a ratio.Because the P-value of 0.002 is very low, I reject the nullhypothesis in favor of the alternate hypothesis. These survey dataprovide strong evidence that the passing rate for teenagers takingthe driving test is lower than 80%. If the passing rate for teenagedriving candidates were actually 80%, we’d expect to see successrates this low in only about 1 in 500 (0.2%). This seems quiteunlikely, casting doubt that the DMV’s stated success rate appliesto teens.

Problem 2Advances in medical care such as prenatal ultrasoundexamination now makes it possible to determine a child’s sexearly in the pregnancy. A study from Punjab, India reportsthat, in 1993, one hospital reported 56.9% boys in 550 livebirths. It’s a medical fact that male babies are slightly morecommon than female babies. The baseline is 51.7%boys in all live births. Use α = 0.1. Is this a one-tailed or two-tailed significance test?

STEP 1: STEP 1: Check the 3 conditions then give the name of the test you are about to perform.

H.W. P#20, 23 and 24

STEP 2: STEP 2: State the hypothesis. (Null and possible Alternative Hypothesis)

STEP 3: STEP 3: Calculate the critical value, test statistic and p-value.

*Demonstrate these with a sketch of the distribution curve as well!

STEP 4: STEP 4: Write a conclusion.

Things to watch for with one proportion significance test

• Don’t base your null hypothesis on what you see in the data (p-hat).

• Don’t base your alternative hypothesis on the data (p-hat).• Don’t forget to check the conditions.• Don’t accept the alternate hypothesis.• If you fail to reject the null hypothesis, don’t think a bigger

sample would be more likely to lead to a rejection.

Ch. 7 TestSince 8.1,8.2, 9.1 and 9.2 takes the concepts in Ch. 7further, I will replace your Ch. 7 test score with yourtest score from the next test.

The only common mistake I noticed on the test wasthat sometimes you forget for a sampling distributionyou have to adjust the S.E.. And you adjust the S.E.using the information from the population.