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 8.2.21

Chapter 8, Problem 8.2.21

[APPLET] Teaching Methods
A new method of teaching reading is being tested on third grade students. A group of third grade students is taught using the new curriculum. A control group of third grade students is taught using the old curriculum. The reading test scores for the two groups are shown in the back-to-back stem-and-leaf plot.
At , α=0.10 is there enough evidence to support the claim that the new method of teaching reading produces higher reading test scores than the old method does? Assume the population variances are equal.

Verified step by step guidance

Step 1: Formulate the null and alternative hypotheses. The null hypothesis (H₀) states that the mean reading test scores for the old and new curricula are equal, i.e., μ₁ = μ₂. The alternative hypothesis (H₁) states that the mean reading test scores for the new curriculum are higher than those for the old curriculum, i.e., μ₁ < μ₂.

Step 2: Identify the test statistic to use. Since the population variances are assumed to be equal, a two-sample t-test for independent samples is appropriate. The test statistic formula is:

\frac{(X_{1} - X_{2})}{\sqrt{(\frac{S_{p} ² (\frac{1}{n_{1}}}{)}}}

, where Sₚ² is the pooled variance.

Step 3: Calculate the pooled variance (Sₚ²). Use the formula:

S_{p} ² = \frac{((n_{1} - 1) S_{1} ² + (n_{2} - 1) S_{2} ²)}{n_{1} + n_{2} - 2}

, where S₁² and S₂² are the sample variances for the old and new curricula, respectively.

Step 4: Compute the test statistic using the formula provided in Step 2. Plug in the sample means, sample sizes, and pooled variance calculated in Step 3.

Step 5: Compare the calculated test statistic to the critical value from the t-distribution table at α = 0.10 with degrees of freedom df = n₁ + n₂ - 2. If the test statistic exceeds the critical value, reject the null hypothesis and conclude that the new curriculum produces higher reading test scores.

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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 a population based on sample data. In this context, the null hypothesis (H0) posits that there is no difference in reading test scores between the two teaching methods, while the alternative hypothesis (H1) suggests that the new method leads to higher scores. The significance level (α) indicates the probability of rejecting the null hypothesis when it is true, guiding the decision-making process.

Stem-and-Leaf Plot

A stem-and-leaf plot is a graphical representation of quantitative data that helps visualize the distribution of scores. Each number is split into a 'stem' (the leading digit) and a 'leaf' (the trailing digit), allowing for easy identification of the shape and spread of the data. In this case, the plot compares the reading test scores of students taught with the old and new curricula, facilitating a direct visual comparison.

Equal Variance Assumption

The equal variance assumption, also known as homoscedasticity, is a key condition in many statistical tests, including t-tests. It assumes that the variances of the two groups being compared are equal. In this scenario, the assumption allows for the use of pooled variance in the analysis, which can lead to more accurate results when determining if the new teaching method significantly improves reading scores compared to the old method.

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Related Practice

Textbook Question

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

Textbook Question

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[Image]

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Textbook Question

What conditions are necessary to use the dependent samples t-test for the mean of the differences for a population of paired data?

Textbook Question

What conditions are necessary to use the t-test for testing the difference between two population means?

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.

Textbook Question

Explain how to perform a two-sample z-test for the difference between two population means using independent samples with and known.

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