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

Problem 8.1.14

Chapter 8, Problem 8.1.14

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1<μ2; α=0.03
Population statistics:σ1=136 and σ2=215
Sample Statistics: x̅1=5004, n1=144, x̅2=4895, n2=156

Verified step by step guidance

Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) states that there is no difference or μ₁ ≥ μ₂. The alternative hypothesis (Hₐ) states that μ₁ < μ₂. This is a left-tailed test.

Step 2: Calculate the test statistic using the formula for the z-test for two population means: z = (x̄₁ - x̄₂) / √((σ₁² / n₁) + (σ₂² / n₂)). Substitute the given values: x̄₁ = 5004, x̄₂ = 4895, σ₁ = 136, σ₂ = 215, n₁ = 144, and n₂ = 156.

Step 3: Determine the critical value for the z-test at the given significance level α = 0.03. Use a z-table or statistical software to find the z-value corresponding to a left-tailed test with α = 0.03.

Step 4: Compare the calculated test statistic to the critical value. If the test statistic is less than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Step 5: State the conclusion in the context of the problem. Based on the comparison in Step 4, determine whether there is sufficient evidence to support the claim that μ₁ < μ₂ at the 0.03 significance level.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1. In this case, the claim is that the mean of population one (μ1) is less than the mean of population two (μ2), which sets the stage for testing this hypothesis.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether the observed data is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error). In this scenario, α is set at 0.03, indicating a 3% risk of concluding that μ1 is less than μ2 when it is not, which guides the decision-making process in hypothesis testing.

Standard Error and Z-Test

The standard error measures the variability of the sample mean estimates and is crucial for conducting a Z-test, which compares the means of two populations. It is calculated using the population standard deviations (σ1 and σ2) and the sample sizes (n1 and n2). In this case, the Z-test will help determine if the difference between the sample means (x̅1 and x̅2) is statistically significant, given the specified significance level.

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Related Practice

Textbook Question

Test the claim about the difference between two population means and at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2, α=0.10, Assume (σ1)^2=(σ2)^2

Sample statistics:

x̅1=0.345, s1=0.305 , n1=11 and x̅2=0.515, s2=0.215, n2=9

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Textbook Question

What conditions are necessary in order to use the z-test to test the difference between two population means?

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Textbook Question

Blue Crabs A marine researcher claims that the stomachs of blue crabs from one location contain more fish than the stomachs of blue crabs from another location. The stomach contents of a sample of 25 blue crabs from Location A contain a mean of 320 milligrams of fish and a standard deviation of 60 milligrams. The stomach contents of a sample of 15 blue crabs from Location B contain a mean of 280 milligrams of fish and a standard deviation of 80 milligrams. At , α= 0.01can you support the marine researcher’s claim? Assume the population variances are equal.

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Textbook Question

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Braking Distances To compare the dry braking distances from 60 to 0 miles per hour for two makes of automobiles, a safety engineer conducts braking tests for 16 compact SUVs and 11 midsize SUVs. The mean braking distance for the compact SUVs is 131.8 feet. Assume the population standard deviation is 5.5 feet. The mean braking distance for the midsize SUVs is 132.8 feet. Assume the population standard deviation is 6.7 feet. At α=0.10 , can the engineer support the claim that the mean braking distances are different for the two categories of SUVs? (Adapted from Consumer Reports)

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Textbook Question

ACT English and Reading Scores The mean ACT English score for 120 high school students is 19.9. Assume the population standard deviation is 7.2. The mean ACT reading score for 150 high school students is 21.2. Assume the population standard deviation is 7.1. At α=0.10, can you support the claim that ACT reading scores are higher than ACT English scores? (Source: ACT, Inc.)

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Textbook Question

Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal and (b) not equal.

Ha:μ1>μ2 , α=0.01 , n1=12 , n2=15

128

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