Section 9 - Breazeal



Chapter 11

The student will be able to:

1. Use the P-value approach for hypothesis testing of claims about two dependent means, two independent means, and two independent proportions.

2. Recommended: Report confidence intervals to assess the size and importance of a significant difference.

3. State real world conclusions to hypothesis tests using appropriate terminology.

4. Use a graphing calculator when testing hypotheses about two populations.

Section 11.1 – Inference About Two Population Proportions

Objectives

1. Distinguish between independent and dependent sampling

2. Test hypotheses regarding two proportions from independent samples

3. Construct and interpret confidence intervals for the difference between two population proportions

4. Test hypotheses regarding two proportions from dependent samples

5. Determine the sample size necessary for estimating the difference between two population proportions

Objective 1 – Distinguish between independent and dependent sampling

[pic]

Examples

For each of the following, determine whether the sampling method is independent or dependent.

a) A researcher wants to know whether the price of a one-night stay at a Holiday Inn Express is less than the price of a one-night stay at a Red Roof Inn. She randomly selects 8 towns where the location of the hotels is close to each other and determines the price of a one-night stay.

b) A researcher wants to know whether the “state” quarters (introduced in 1999) have a mean weight that is different from “traditional” quarters. He randomly selects 18 “state” quarters and 16 “traditional” quarters and compares their weights.

Objective 2 – Test Hypothesis Regarding Two Proportions from Independent Samples

Notation:

n1 = size of sample 1 n2 = size of sample 2

x1 = number of successes in sample 1 x2 = number of successes in sample 2

p1 = proportion for population 1 p2 = proportion for population 2

[pic] (sample 1 proportion) [pic] (sample 2 proportion)

The inferential methods of this section apply to analysis of data from a completely randomized design with two treatment levels for which the response is a Bernoulli (success or failure) variable.

The below requirements ensure randomness, normality, and independence.

[pic]

[pic]

[pic]

Example

State the null and alternative hypotheses only.

The proportion of female students (1) is different than the proportion of male students (2).

A. H0: p1 < p2 H1: p1 > p2

B. H0: p1 = p2 H1: p1 ≠ p2

C. H0: p1 = p2 H1: p1 < p2

D: H0: p1 ≠ p2 H1: p1 = p2

Example

An economist believes that the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access. He obtains a random sample of 800 urban households and finds that 338 of them have Internet access. He obtains a random sample of 750 rural households and finds that 292 of them have Internet access. Test the economist’s claim at the α = 0.05 level of significance.

Example

In October of 1947, the Gallup organization surveyed 1100 adult Americans and asked, “Are you a total abstainer from alcoholic beverages?” Of the 1100 adults surveyed, 407 answered “yes.” In July 2010, the same question was asked of 1100 adult Americans and 333 answered “yes.” Has the proportion of adult Americans who totally abstain from alcohol changed? Use a 0.05 significance level.

Objective 3 – Construct and Interpret Confidence Intervals for the Difference between two Population Proportions

The objective is to find a confidence interval for p1 – p2 where p1 and p2 are population proportions.

Remember a confidence interval is given by

Point estimate ( Margin of error

The point estimate for p1 is [pic]and the point estimate for p2 is [pic], so the point estimate for p1 – p2 is [pic].

The margin of error is given by

[pic]

So the confidence interval for p1 – p2 is given by the following formula

([pic]) ( [pic]

However, the calculator does all of this work for us. The following requirements still must be met.

The below requirements ensure randomness, normality, and independence.

[pic]

[pic]

If the confidence interval contains 0, we say there is no significant difference between the values.

If the confidence interval does not contain 0 we say there is a significant difference between the values.

Example

Suppose the Cartoon Network conducts a nation-wide survey to assess viewer attitudes toward Superman. Using a simple random sample, they select 400 boys and 300 girls to participate in the study. 160 of the boys say that Superman is their favorite character, compared to 90 of the girls. What is the 90% confidence interval for the true difference in attitudes toward Superman?

Example

Among 5000 items of randomly selected baggage handled by American Airlines, 22 were lost. Among 4000 items of randomly selected baggage handled by Delta Airlines, 15 were lost. Use the sample data to construct a 95% confidence interval estimate of the difference between the two rates of lost baggage. Based on the result, does there appear to be a significant difference in the lost luggage rates?

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A sampling method is independent when the individuals selected for one sample do not dictate which individuals are to be in the second sample.

A sampling method is dependent when the individuals selected to be in one sample are used to determine the individuals in the second sample.

Dependent samples are often referred to as “matched-pairs” samples (see section 11.2). It is possible for an individual to be matched against him- or herself.

Calculator Instructions

1. Press STAT, highlight TESTS, and select 6:2-PropZTest

2. Enter the values for x1, n1, x2, n2

3. Highlight the appropriate relationship between p1 and p2 in the alternative hypothesis

4. Highlight Calculate or Draw and press ENTER

• Calculate gives the test statistic and P-value

• Draw shows the Z-distribution with the P-value shaded

Requirements

To test hypotheses regarding two population proportions, p1 and p2, we can use the steps that follow, provided that:

• the samples are independently obtained using simple random sampling,

• [pic]

• [pic]

• n1 d" 0.05N1 and n2 d" 0.05N2 (the sample size is no more than 5% of the population size); this requirement ensures the independence necessary for a binomial experime≤ 0.05N1 and n2 ≤ 0.05N2 (the sample size is no more than 5% of the population size); this requirement ensures the independence necessary for a binomial experiment.

Steps in Hypothesis testing two proportions from independent samples

1. Determine the null and alternative hypothesis. The hypothesis are structured in one of three ways

[pic]

2. Determine the level of significance, (, depending on the seriousness of making a Type I error

3. Compute the test statistic and P-value

4. Compare P-value to (

If P–value < α, Reject H0

If P–value > α, Do Not Reject H0

5. State the conclusion

Requirements

To test hypotheses regarding two population proportions, p1 and p2, we can use the steps that follow, provided that:

• the samples are independently obtained using simple random sampling,

• [pic]

• [pic]

• n1 ≤ 0.05N1 and n2 ≤ 0.05N2 (the sample size is no more than 5% of the population size); this requirement ensures the independence necessary for a binomial experiment.

Calculator Instructions

1. Press STAT, highlight TESTS, and select B:2-PropZInt

2. Enter the values for x1, n1, x2, n2

3. Enter a confidence level, C-Level:

4. Highlight Calculate and press ENTER

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