Hypothesis testing for proportions. Essential statistics

Сентябрь 13, 2022

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Hypothesis testing for proportions. Essential statistics

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2. In this section, you will learn how to test a population proportion, p. If np ≥
3. Assumptions Write hypotheses & define parameter Calculate the test statistic & p-value Write a statement in
4. The P-Value is the probability of obtaining a test statistic that is at least as extreme
5. Under Stat – Tests Select 1 Prop Z-test Input p, x, and n P is claim
6. Provides you with the z score P-Value Sample proportion Interpret the p-value based off of your
7. Draw & shade a curve & calculate the p-value: 1) right-tail test z = 1.6 2)
8. α Represents the remaining percentage of our confidence interval. 95% confidence interval has a 5% alpha.
9. A medical researcher claims that less than 20% of American adults are allergic to a medication.
10. The products np = 100(0.20)= 20 and nq = 100(0.80) = 80 are both greater than
11. Because the test is a left-tailed test and the level of significance is α = 0.01,
12. SOLUTION Continued . . . The graph shows the location of the rejection region and the
13. Solutions Continued……
14. Interpretation Since the .1056 > .01, I fail to reject the H0 There is not sufficient
15. Harper’s Index claims that 23% of Americans are in favor of outlawing cigarettes. You decide to
16. The products np = 200(0.23) = 45 and nq = 200(0.77) = 154 are both greater
17. Because the test is a two-tailed test, and the level of significance is α = 0.05.
18. SOLUTION Continued . . . The graph shows the location of the rejection regions and the
19. The Pew Research Center claims that more than 55% of American adults regularly watch a network
20. The products np = 425(0.55) = 235 and nq = 425(0.45) = 191 are both greater
21. Because the test is a right-tailed test, and the level of significance is α = 0.05.
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In this section, you will learn how to test a population

proportion, p. If np ≥ 10 and n(1-p) ≥ 10 for a binomial distribution, then the sampling distribution for is normal with and

Hypothesis Test for Proportions

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Assumptions
Write hypotheses & define parameter
Calculate the test statistic & p-value
Write a

statement in the context of the problem.

Steps for doing a hypothesis test

“Since the p-value < (>) α, I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha (in context).”

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The P-Value is the probability of obtaining a test statistic that

is at least as extreme as the one that was actually observed, assuming the null is true.
p-value < (>) α, I reject (fail to reject) the H0.

What is the p-value

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Under Stat – Tests
Select 1 Prop Z-test
Input p, x, and n
P

is claim proportion
X is number of sampling matching claim
N is number sampled
Select correct Alternate Hypothesis
Calculate

How to calculate the P-value

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Provides you with the z score
P-Value
Sample proportion
Interpret the p-value based off

of your Confidence interval

Reading the Information

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Draw & shade a curve & calculate the p-value:
1) right-tail test

z = 1.6
2) two-tail test z = 2.3

P-value = .0548

P-value = (.0107)2 = .0214

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α Represents the remaining percentage of our confidence interval. 95% confidence

interval has a 5% alpha.

What is α

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A medical researcher claims that less than 20% of American adults

are allergic to a medication. In a random sample of 100 adults, 15% say they have such an allergy. Test the researcher’s claim at α = 0.01.

Ex. 1: Hypothesis Test for a Proportion

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The products np = 100(0.20)= 20 and nq = 100(0.80) =

80 are both greater than 10. So, you can use the z-test. The claim is “less than 20% are allergic to a medication.” So the null and alternative hypothesis are:
Ho: p = 0.2 and Ha: p < 0.2 (Claim)

SOLUTION

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Because the test is a left-tailed test and the level of

significance is α = 0.01, the critical value is zo = -2.33 and the rejection region is z < -2.33. Using the z-test, the standardized test statistic is:

Solution By HAND continued . . .

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SOLUTION Continued . . .
The graph shows the location of

the rejection region and the standardized test statistic, z. Because z is not in the rejection region, you should decide not to reject the null hypothesis. In other words, there is not enough evidence to support the claim that less than 20% of Americans are allergic to the medication.

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Solutions Continued……

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Interpretation
Since the .1056 > .01, I fail to reject the H0

There is not sufficient evidence to suggest that 20% of adults are allergic to medication.

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Harper’s Index claims that 23% of Americans are in favor of

outlawing cigarettes. You decide to test this claim and ask a random sample of 200 Americas whether they are in favor outlawing cigarettes. Of the 200 Americans, 27% are in favor. At α = 0.05, is there enough evidence to reject the claim?

Ex. 2 Hypothesis Test for a Proportion

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The products np = 200(0.23) = 45 and nq = 200(0.77)

= 154 are both greater than 5. So you can use a z-test. The claim is “23% of Americans are in favor of outlawing cigarettes.” So, the null and alternative hypotheses are:
Ho: p = 0.23 (Claim) and Ha: p ≠ 0.23

SOLUTION:

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Because the test is a two-tailed test, and the level of

significance is α = 0.05.
Z = 1.344
P = .179
Since the .179 > .05, I fail to reject the H0 There is not sufficient evidence to suggest that more or less than 23% of Americans are in favor of outlawing cigarette’s.

SOLUTION continued . . .

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SOLUTION Continued . . .
The graph shows the location of

the rejection regions and the standardized test statistic, z.
Because z is not in the rejection region, you should fail to reject the null hypothesis. At the 5% level of significance, there is not enough evidence to reject the claim that 23% of Americans are in favor of outlawing cigarettes.

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The Pew Research Center claims that more than 55% of American

adults regularly watch a network news broadcast. You decide to test this claim and ask a random sample of 425 Americans whether they regularly watch a network news broadcast. Of the 425 Americans, 255 responded yes. At α = 0.05, is there enough evidence to support the claim?

Ex. 3 Hypothesis Test a Proportion

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The products np = 425(0.55) = 235 and nq = 425(0.45)

= 191 are both greater than 5. So you can use a z-test. The claim is “more than 55% of Americans watch a network news broadcast.” So, the null and alternative hypotheses are:
Ho: p = 0.55 and Ha: p > 0.55 (Claim)

SOLUTION:

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Because the test is a right-tailed test, and the level of

significance is α = 0.05.
Z = 2.072
P-value = .019
Since the 0.019 < .05, I reject the H0. There is sufficient evidence to suggest that 20% of adults are allergic to medication.

SOLUTION continued . . .