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Height and the big picture. Forty-six college students were randomly divided into two groups of size 23. One group was asked to imagine being on the upper floor of a tall building and the other on the lowest floor. Participants were then asked to choose between a job that required more detail orientation versus a job that required a more big-picture orientation. They rated their job preferences on an 11-point scale, with higher numbers corresponding to a greater preference for the big-picture job. Here are the summary statistics: 17 $$ \begin{array}{lccc} \hline \text { Group } & \text { Group Size } & \text { Mean } & \text { Standard Deviation } \\ \hline \text { Low } & 23 & 4.61 & 3.08 \\ \hline \text { High } & 23 & 6.68 & 3.45 \\ \hline \end{array} $$ (a) What degrees of freedom would you use in the conservative two-sample \(t\) procedures to compare the lower and higher floor groups? (b) What is the two-sample \(t\) test statistic for comparing the mean job preference ratings for the two groups? (c) Test the null hypothesis of no difference between the two population means against the two-sided alternative. Use your statistic from part (b) with degrees of freedom from part (a).

Step by step solution

01

Identify the Test Type

Since we are comparing the means of two independent groups (low floor and high floor), we'll perform a two-sample t-test.

02

Calculate Degrees of Freedom

For a conservative two-sample t-procedure, we use the formula \( df = \min(n_1 - 1, n_2 - 1) \). Here, \( n_1 = 23 \) and \( n_2 = 23 \). Therefore, \( df = \min(23 - 1, 23 - 1) = 22 \).

03

Formulate Hypotheses

The null hypothesis \( H_0 \) states there is no difference between the population means, i.e., \( \mu_1 = \mu_2 \). The alternative hypothesis \( H_a \) is that there is a difference, i.e., \( \mu_1 eq \mu_2 \).

04

Calculate the Test Statistic

Use the two-sample t-test statistic formula:\[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]Substitute the values:\( \bar{x}_1 = 4.61, \bar{x}_2 = 6.68, s_1 = 3.08, s_2 = 3.45, n_1 = 23, n_2 = 23 \).\[t = \frac{4.61 - 6.68}{\sqrt{\frac{3.08^2}{23} + \frac{3.45^2}{23}}} = \frac{-2.07}{\sqrt{0.41284 + 0.51702}} = \frac{-2.07}{\sqrt{0.92986}} \approx \frac{-2.07}{0.9645} \approx -2.15\]

05

Conduct the Two-Sided Test

Using the t-statistic calculated (-2.15) and degrees of freedom (22), look up the p-value for the two-sided test. For \( df = 22 \), a t-value of approximately -2.15 corresponds to a p-value less than 0.05. Thus, we reject the null hypothesis.

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Key Concepts

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

Degrees of Freedom

Degrees of freedom (df) are vital in statistical tests when determining the amount of information your data can provide. They help assess the variability of the sample means. For a two-sample t-test, this metric is crucial because it influences the shape of the t-distribution you reference when interpreting your results.
This exercise involves two independent groups of 23 students each. The formula for finding the conservative number of degrees of freedom in a two-sample t-test is:

• \( df = \min(n_1 - 1, n_2 - 1) \)
• For our example: \( df = \min(23 - 1, 23 - 1) = 22 \)

By using the smaller of the two group sizes minus one, you ensure that you're taking the most conservative approach, minimizing the risk of incorrect conclusions due to sample variability.
Understanding degrees of freedom helps you interpret statistical results more accurately. It is a foundational concept in hypothesis testing.

Null Hypothesis

A null hypothesis is a starting point for many statistical tests. It proposes that there is no significant effect or difference in your study. In our two-sample t-test example, the null hypothesis assumes that the two group means are equal.

• Null Hypothesis \( H_0 : \mu_1 = \mu_2 \)

It's important because it sets a baseline that your data needs to provide evidence against to show a statistically significant effect. When you conduct t-tests, your whole analysis revolves around determining whether your data provide enough evidence to reject this null hypothesis.
Rejecting the null hypothesis suggests there's a significant effect, while failing to reject it implies no detectable difference with the given data. The null hypothesis serves as the anchor point around which hypothesis testing and p-value calculations revolve.

Test Statistic

The test statistic from a statistical test, like a t-test, is a calculated value that you compare against a known distribution to ascertain the significance of your results. For a two-sample t-test, the test statistic determines if the difference in means from the two groups is significant.

Test Statistic Formula for Two-sample t-test

The formula is:

• \[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]

• \( \bar{x}_1 \) and \( \bar{x}_2 \) are the means of the two groups.
• \( s_1 \) and \( s_2 \) are the standard deviations.
• \( n_1 \) and \( n_2 \) are the sample sizes.

In our example:

• Calculated \( t \) value is approximately \(-2.15\).

A significant test statistic indicates that the observed data differs significantly from what was expected under the null hypothesis. It's a numerical summary that guides the decision-making in hypothesis testing.

Independent Groups

Independent groups are fundamental in many statistical tests, including the two-sample t-test, as each member’s data point in one group is independent of any member's data in another group. This independence ensures that the actions or variables in one group do not influence the other group's results.
In the given exercise, we are comparing two separate sets of college students. One group was placed at a high building floor, while the other was on a lower floor. Since their group memberships do not affect each other, we treat these as independent groups.
This independence assumption is essential because it supports the validity of the two-sample t-test. If the groups were not independent, the statistical test results might be flawed, leading to potentially incorrect conclusions about population means.

P-value

The p-value is a crucial concept in hypothesis testing. It helps to determine the significance of your results. It indicates the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, under the assumption that the null hypothesis is true.
For a two-sample t-test, you calculate a p-value to decide whether your group means are statistically different. In this case study, our calculated t-value was approximately \(-2.15\), and with degrees of freedom \( df = 22 \), the corresponding p-value is less than 0.05.

• A small p-value (typically \(<0.05\)) signals strong evidence against the null hypothesis, suggesting it might be rejected.
• A large p-value shows weak evidence against the null hypothesis, indicating you cannot reject it.

In this scenario, since the p-value is less than 0.05, it implies that there is sufficient evidence to reject the null hypothesis, suggesting a significant difference between the two group means.