Textbook Question
What conditions are necessary to use the z-test for testing the difference between two population proportions?
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10. Hypothesis Testing for Two Samples
Two Proportions
6:35 minutesProblem 8.4.3
Textbook Question
In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions and at the level of significance . Assume the samples are random and independent.
Claim: p1≠p2, α=0.01
Sample statistics: x1=35, n1=70, and x2=36, n2=60
Verified step by step guidance1
Step 1: Verify the conditions for using a normal sampling distribution. Check if the sample sizes are large enough by ensuring that both np and n(1-p) are greater than or equal to 5 for each sample. For each sample, calculate p̂ (sample proportion) as p̂ = x/n, where x is the number of successes and n is the sample size.
Step 2: Calculate the pooled sample proportion (p̂_pooled) since the null hypothesis assumes p1 = p2. Use the formula: p̂_pooled = (x1 + x2) / (n1 + n2), where x1 and x2 are the number of successes, and n1 and n2 are the sample sizes.
Step 3: Compute the standard error (SE) for the difference in proportions using the formula: SE = sqrt(p̂_pooled * (1 - p̂_pooled) * (1/n1 + 1/n2)).
Step 4: Calculate the test statistic (z) for the difference in proportions using the formula: z = (p̂1 - p̂2) / SE, where p̂1 and p̂2 are the sample proportions for the two groups.
Step 5: Compare the calculated z-value to the critical z-value for a two-tailed test at the significance level α = 0.01. Alternatively, calculate the p-value and compare it to α. If the z-value falls outside the critical range or the p-value is less than α, reject the null hypothesis; otherwise, fail to reject it.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Sampling Distribution
A normal sampling distribution is a probability distribution of sample means or proportions that approximates a normal distribution as the sample size increases, according to the Central Limit Theorem. For proportions, this approximation is valid when both np and n(1-p) are greater than 5, ensuring that the sample size is sufficiently large to yield reliable results.
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Difference Between Two Population Proportions
The difference between two population proportions involves comparing the proportions of a certain characteristic in two different populations. This is typically analyzed using a hypothesis test, where the null hypothesis states that the two proportions are equal, and the alternative hypothesis states they are not, allowing for statistical inference about the populations based on sample data.
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Level of Significance (α)
The level of significance, denoted as α, is the threshold for determining whether to reject the null hypothesis in a statistical test. It represents the probability of making a Type I error, which occurs when the null hypothesis is true but is incorrectly rejected. In this case, α is set at 0.01, indicating a 1% risk of concluding that a difference exists when there is none.
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