Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problema 8.4.11

Capitolo 8, Problema 8.4.11

Seat Belt Use In a survey of 1000 drivers from the West, 934 wear a seat belt. In a survey of 1000 drivers from the Northeast, 909 wear a seat belt. At α=0.05, can you support the claim that the proportion of drivers who wear seat belts is greater in the West than in the Northeast? (Adapted from National Highway Traffic Safety Administration)

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Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). Here, H₀: p_West ≤ p_Northeast (the proportion of seat belt users in the West is less than or equal to that in the Northeast), and H₁: p_West > p_Northeast (the proportion in the West is greater).

Calculate the sample proportions for each region:

\(\hat{p}\)_{West} = \(\frac{934}{1000}\)

and

\(\hat{p}\)_{Northeast} = \(\frac{909}{1000}\)

Compute the pooled proportion

\(\hat{p}\)

since the null hypothesis assumes the proportions are equal:

\(\hat{p}\) = \(\frac{934 + 909}{1000 + 1000}\)

Calculate the standard error (SE) of the difference between the two sample proportions using the formula:

SE = \(\sqrt{\hat{p}\)(1 - \(\hat{p}\)) \(\left\)( \(\frac{1}{n_{West}\)} + \(\frac{1}{n_{Northeast}\)} \(\right\))}

, where

n_{West} = 1000

and

n_{Northeast} = 1000

Compute the test statistic (z) using:

z = \(\frac{\hat{p}\)_{West} - \(\hat{p}\)_{Northeast}}{SE}

. Then, compare this z-value to the critical z-value for α = 0.05 in a one-tailed test to decide whether to reject H₀.

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Hypothesis Testing for Two Proportions

This involves comparing two population proportions to determine if there is a statistically significant difference between them. We set up a null hypothesis (no difference) and an alternative hypothesis (one proportion is greater), then use sample data to test these claims at a chosen significance level.

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Percorso guidato

09:27

Difference in Proportions: Hypothesis Tests

Significance Level (α) and p-value

The significance level, α, is the threshold for rejecting the null hypothesis, commonly set at 0.05. The p-value measures the probability of observing the sample data if the null hypothesis is true. If the p-value is less than α, we reject the null hypothesis, supporting the alternative claim.

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Percorso guidato

06:50

Step 3: Get P-Value

Calculation of Test Statistic for Two Proportions

The test statistic compares the difference between sample proportions relative to the variability expected under the null hypothesis. It is calculated using the pooled proportion and standard error, then converted to a z-score to assess significance against the standard normal distribution.

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Percorso guidato

09:27

Difference in Proportions: Hypothesis Tests

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