CHAPTER 8 Confidence Intervals for p - One Sample Sample Size

MATH 115 PROBABILITY & STATISTICS

COLLEGE OF MARIN N. PSOMAS

CHAPTER 8 Tests for the Population Proportion p - Sections: 8.1 & 8.2

1. -- Confidence interval for p

2. -- Sample size for estimating p

3. -- Hypothesis testing for p

4. -- Comparing two proportions - confidence intervals

5. -- Comparing two proportions - hypothesis testing

Confidence Intervals for p - One Sample

Note: p here stands for the true proportion (fraction, also expressed as %) of all members in a

population of a particular attribute. p is most commonly known as the population proportion.

Sample Statistic Standard Error

of the Sample Statistic

Level C

Confidence Interval

note: z* = zα/2 (where α = 1-C)

Sample Size For a Margin of Error m For Estimating p

The level C confidence interval for the population proportion p will have margin of error at most

m, if n is chosen to be:



Hypotheses Testing for p - One Sample

NULL HYPOTHESIS

Ho : p = po

ALTERNATIVE

HYPOTHESIS

REJECT Ho AT

LEVEL αααα IF:

P-VALUE

ONE SAMPLE Z-STATISTIC

Ha : p > po zobs > z*α P[Z ≥ zobs]

Ha : p < po zobs < − z*α P[Z ≤ - zobs]

Ha : p ≠ po zobs < − z*α/2 OR

zobs > z*α/2

2P[Z ≥ |zobs|]

Assumptions:

1. The sample must come from a large population

2. np >5 and n(1-p) > 5

Notation and Conditions

1. The Z-statistic has the Standard Normal distribution

2. zobs is the observed value of the statistic, which is computed from the sample data.

3. z*α or z*α/2 is the critical z-value based on the given α (one-sided alternative) or α/2

(two-sided alternative).

TEST ABOUT THE POPULATION PROPORTION p OF A SINGLE POPULATION

EXAMPLE

How common is behavior that puts people at risk of AIDS? The National AIDS Behavioral

Surveys interviewed a random sample of 2,673 adult heterosexuals. Of these, 170 admitted to

having more than one sexual partner in the past year. That's 6.36% of the sample.

Based on these data, what can we say about the percent of all adult heterosexuals who admit to

having multiple partners?

a) What is a 99% confidence interval for p, the true population proportion of all

heterosexuals who admit to having multiple partners?

b) Based on this sample, can we conclude that the true population proportion of all adult

heterosexuals who admit to having multiple partners is less than 7.00%. Use α = 0.01.



SOLUTION:

Population parameter under investigation

p = True population proportion of all heterosexuals who admit to having multiple partners.

a) A 99% confidence interval for p is given by:

====================================================================

Where �̂ = 170/2673 = 0.0636, zα/2 = 2.576, n = 2673

Therefore, the 99% CI = 0.636 ± 0.122 = (5.14%, 7.58%)

b) Testing the hypothesis p = 0.07

===================================================================

Null hypothesis: Ho: p = 0.07

Alternative hypothesis: Ha: p < 0.07

Test Statistic

Rejection region based on α = 0.01α = 0.01α = 0.01α = 0.01.



Reject Ho at the 1% level of significance if zobs < - 2.326.

Calculating Z, the P-value, and drawing a conclusion based on the available data.

Since -1.30 does not fall inside the rejection region, our test does not reject the null hypothesis. We cannot, with α = 1%, conclude that the true population proportion of all adult heterosexuals

who admit to having multiple partners is less than 7%.

P-Value = P[Z ≤ - 1.30] = 0.0968 or 9.7%

Using the TI-83/84 to Compute CI & Test Hypothesis for p

(single population)

(a) Confidence Intervals

STAT > TESTS > A: 1-PropZInt...

(b) Testing Hypotheses

STAT > TESTS > 5: 1-PropZTest...



Confidence Interval for Comparing Two Population Proportions

(Two Samples)

A level C confidence interval for p1- p2 is given by:

note: z* = zα/2 (where α = 1-C)

Hypothesis Tests for Comparing Two Population Proportions

(Two Samples)

NULL HYPOTHESIS

Ho : p1 = = = = p2

ALTERNATIVE

HYPOTHESIS

REJECT Ho AT

LEVEL αααα IF:

P-VALUE

TWO SAMPLE Z-STATISTIC

Ha : p1 > p2

zobs ≥ z*α

P[Z ≥ zobs]

Ha : p1 < p2 zobs≤ − z*α P[Z ≤ - zobs]

Ha : p1 ≠ p2 zobs ≤ − z*α/2 OR

zobs ≥ z*α/2

2P[Z ≥ |zobs|]

Assumptions:

1. The samples cmust come from large populations

2. n1p1 >5 and n1(1-p) > 5

3. n2p2 >5 and n2(1-p) > 5

Notation and Conditions

1. The Z-statistic has the Standard Normal distribution

2. zobs is the observed value of the statistic, which is computed from the sample data.

3. z*α or z*α/2 is the critical z-value based on the given α (one-sided alternative) or α/2

(two-sided alternative).



COMPARING TWO POPULATION PROPORTIONS

EXAMPLE. In 1954 an experiment was conducted to test the effectiveness of the Salk vaccine as protection

against polio. Approximately 200,000 children were injected with an ineffective salt solution,

and 200,000 other children were injected with the vaccine. The experiment was "double blind"

because the children being injected didn't know whether they were given the real vaccine or the

placebo, and the doctors giving the injection and evaluating the results didn't know either. Of the

200,000 children that were given the vaccine 33 later developed paralytic polio, whereas 115 of

the 200,000 injected with the salt solution later developed paralytic polio. Could this difference

be attributed to chance alone, or can we conclude that the vaccine was effective in preventing

polio?

SOLUTION:

Population parameters under investigation:

p1 = true proportion of children who will develop polio if given the vaccine

p2 = true proportion of children who will develop polio if not given the vaccine (or if given a

placebo)

Null Hypothesis: Ho: p1 = p2 (vaccine has no effect)

Alternative Hypothesis: Ha: p1 > p2 (vaccine prevents polio, i.e., % of polio cases in

group of children who took the vaccine is reduced)

Test Statistic:

Rejection region using αααα = 0.005 (i.e., we set the level of significance to 0.5%)



Decision Rule: Reject Ho if P-value < 0.5% (i.e., zobs < zα = -2.576)

Calculating Z, the P-value, and drawing a conclusion based on the available data.

Since -6.742 falls well into the rejection region, the tests rejects the null hypothesis. The data

support that the vaccine is effective in preventing polio.

P-Value = P[Z ≤ - 6.742] = 7.85x10-12 or 0.00000000000785 well below 0.5% (0.005)



Using the TI-83/84 to Compute CI & Test Hypotheses for the

difference between two population proportions p1 - p2

(a) Confidence Intervals

STAT > TESTS > B: 2-PropZInt...

(b) Testing Hypotheses

STAT > TESTS > 6: 2-PropZTest...

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CHAPTER 8 Confidence Intervals for p - One Sample Sample Size