91Ó°ÊÓ

Vegetarians more liberal? When a sample of social science graduate students at the University of Florida gave their responses on political ideology (ranging from \(1=\) very liberal to \(7=\) very conservative \(),\) the mean was \(3.18(s=1.72)\) for the 51 nonvegetarian students and \(2.22(s=0.67)\) for the 9 vegetarian students. Software for comparing the means provides the printout, which shows results first for inferences that assume equal population standard deviations and then for inferences that allow them to be unequal. \(\begin{array}{ccclc}\text { Sample } & \mathrm{N} & \text { Mean } & \text { StDev } & \text { SE Mean } \\ 1 & 51 & 3.18 & 1.72 & 0.24 \\ 2 & 9 & 2.220 & 0.670 & 0.22 \\ \text { Difference } & =\mu(1)-\mu(2) & & \end{array}\) Estimate for difference: 0.960 95?a CI for difference: (-0.210,2.130) T-Test of difference \(=0\) (vs \(\neq):\) T-Value \(=1.64\) \(\mathrm{P}\) -Value \(=0.106 \mathrm{DF}=58\) Both use Pooled StDev \(=1.6162\) 95? CI for difference: (0.289,1.631) T-Test of difference \(=0(\mathrm{vs} \neq):\) T-Value \(=2.92\) P-Value \(=0.007 \quad \mathrm{DF}=30\) a. Explain why the results of the two approaches differ so much. Which do you think is more reliable? b. State your conclusion about whether the true means are plausibly equal.

Step by step solution

01

Understanding the Problem

We need to compare two different approaches to evaluating the differences in political ideology scores between vegetarian and non-vegetarian students at the University of Florida's social science graduate program. We're provided with outputs for both equal and unequal variance assumptions.

02

Examining the Results with Equal Variance Assumption

When assuming equal population standard deviations, the software found a difference in the means of 0.960, with a 95% confidence interval of (-0.210, 2.130). The T-Test gave a T-Value of 1.64 and a P-Value of 0.106, and degrees of freedom (DF) = 58. This P-Value suggests that there's no statistically significant difference at the usual 0.05 significance level because it is greater than 0.05.

03

Examining the Results with Unequal Variance Assumption

When allowing for unequal population standard deviations, the 95% confidence interval for the difference is (0.289, 1.631). The T-Test here provided a T-Value of 2.92, a P-Value of 0.007, with DF = 30. This P-Value indicates a statistically significant difference because it is less than 0.05.

04

Comparing the Two Approaches

The differing outcome between the two methods arises because the assumption about population standard deviations greatly affects the degrees of freedom and, consequently, the test statistics. The method assuming unequal variances is typically more flexible and appropriate for small sample sizes or when standard deviations differ significantly between groups.

05

Conclusion about True Means

Based on the unequal variance approach, the P-Value is 0.007, suggesting strong evidence against the null hypothesis of equal means. Thus, we conclude there is a statistically significant difference in political ideology scores between vegetarians and non-vegetarians.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

T-Test

A T-Test is a statistical method used to compare the means of two groups. In this context, it helps to determine if there's a significant difference in political ideology scores between vegetarian and non-vegetarian students. The test examines the means of both groups, considering their standard deviations and sample sizes.
A T-Test can either be one-sample, paired-sample, or independent sample. Here, we use an independent sample T-Test, which is ideal for comparing two independent groups.
The T-Value derived from the test indicates how many standard deviations the observed difference is from the hypothesized difference, typically zero. A higher absolute T-Value suggests a greater difference and possibly significant results. However, to make a decision, we look at the P-Value. A small P-Value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, suggesting a significant difference.

Confidence Interval

A Confidence Interval (CI) provides a range of values that likely contain the true difference between the group means. It offers a way to understand both the magnitude and uncertainty of the estimated difference.
For instance, in the exercise, different results were found depending on variance assumptions:

• Under the equal variance assumption, with a wider CI of (-0.210, 2.130), the interval includes zero, indicating no definite conclusion about a difference.
• Under the unequal variance assumption, the CI of (0.289, 1.631) shows a positive range, suggesting a difference that is likely not by chance.

The reliability of a CI is anchored on its width and location: narrower CIs suggest more precise estimates, while intervals not crossing zero indicate significant differences.

Equal Variance Assumption

The Equal Variance Assumption means both groups are believed to have the same population standard deviation. It's sometimes referred to as "homogeneity of variance."
This assumption simplifies calculations and can lead to more stable estimates in large sample sizes. However, when this assumption isn't met, such as with different sample sizes or when the group standard deviations are visibly distinct, results can be misleading.
If equal variances are wrongly assumed, it could lead to an inflated Type I error rate, meaning we might incorrectly find a difference when none exists. The equal variance method in the exercise showed no significant difference, suggesting it might not fully account for variations in the smaller vegetarian group.

Unequal Variance Assumption

The Unequal Variance Assumption, also known as Welch's T-Test, is a more flexible approach that doesn't require equal standard deviations between groups. It's particularly useful when dealing with real-world data where equal variances are unlikely.
This test adjusts the degrees of freedom, often leading to different critical values as seen in the example where the P-Value was significantly lower with this assumption (0.007 compared to 0.106 under equal variances).
Using the unequal variance assumption generally provides a more robust analysis, especially in cases of differing group sizes and variances, as it better reflects the underlying data variability.