Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

문제 8.2.21

8장, 문제 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.

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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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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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