Question```latex
The accompanying table lists distances (mm) between pupils of randomly selected U.S. Army personnel collected as part of a large reputable study. Results from two-way analysis of variance. Use the displayed results and use a 0.05 significance level. What do you conclude?
Click the icon to view the data and two-way analysis of variance results.
D. Fail to reject . There is insufficient evidence to support the alternative hypothesis. There does not appear to be an interaction between gender and handedness.
If appropriate, test for an effect from the row factor. Choose the correct answer below.
A. : Left-handed people and right-handed people have the same population mean distance between pupils. : Left-handed people and right-handed people have different population mean distances between pupils.
B. : Men and women have different population mean distances between pupils. : Men and women have the same population mean distance between pupils.
C. : Left-handed people and right-handed people have different population mean distances between pupils. : Left-handed people and right-handed people have the same population mean distance between pupils.
D. : Men and women have the same population mean distance between pupils. : Men and women have different population mean distances between pupils.
E. This test is not appropriate due to the results of the test for interaction between the two factors.
The user took a picture with their phone and the text was extracted above. The user then had a dialogue with an AI Assistant to help clarify the instructions.
Dialogue Transcript:
Hello! It looks like you have a two-way analysis of variance (ANOVA) problem here involving gender and handedness as factors. However, I need a bit more information to assist you properly. Could you please provide the specific results from the two-way ANOVA, such as the p-values for the main effects and interactions? Additionally, let me know if there is anything else specific you'd like help with regarding this problem.
Feel free to type out any important details that might help me understand the context better!
Extracted text from attached image:
Data and Two-Way ANOVA Results
\begin{tabular}{l|c|c}
\hline & \text{Right-Handed} & \text{Left-Handed} \\
\hline \text{Female} & 6663596056 & 7162616962 \\
\hline \text{Male} & 6764666971 & 6767656864 \\
\hline
\end{tabular}
\begin{tabular}{|lllllll|}
\hline \text{Source:} & \text{DF:} & \text{SS:} & \text{MS:} & \text{Test Stat, F:} & \text{Critical F:} & \text{P-Value:} \\
\text{Interaction:} & 1 & 36.45 & 36.45 & 3.15584 & 4.49401 & 0.09467 \\
\text{Row Variable:} & 1 & 76.05 & 76.05 & 6.58442 & 4.49401 & 0.02072 \\
\text{Column Variable:} & 1 & 11.25 & 11.25 & 0.97403 & 4.49401 & 0.33837 \\
\hline
\end{tabular}
```
Studdy Solution
STEP 1
What is this asking?
We need to figure out if there's a difference in pupil distances between men and women, considering we already know there's no interaction between gender and handedness.
Watch out!
Don't mix up rows and columns in the ANOVA table!
Also, remember that a small p-value means there's strong evidence *against* the null hypothesis.
STEP 2
1. Check Interaction
2. State the Hypotheses
3. Analyze Row Factor
STEP 3
The problem states that there is no interaction between gender and handedness.
The provided data confirms this.
The p-value for the interaction effect is , which is greater than our significance level of .
This means we **fail to reject the null hypothesis** that there's no interaction.
Great! This allows us to proceed with analyzing the main effects.
STEP 4
Since the row variable represents gender (Male/Female), our hypotheses focus on potential differences between men and women.
The **null hypothesis** () states there's *no difference* in the mean pupil distance between men and women.
The **alternative hypothesis** () states there *is* a difference.
STEP 5
So, formally: : Men and women have the same population mean distance between pupils. : Men and women have different population mean distances between pupils.
STEP 6
Now, let's look at the "Row Variable" in the ANOVA table.
This corresponds to the effect of gender.
The p-value for the row variable is ****.
STEP 7
Since (our significance level), we **reject the null hypothesis**.
This means we have statistically significant evidence to suggest that there *is* a difference in the mean pupil distance between men and women!
STEP 8
The correct hypotheses are given by option D.
Since the p-value for the row factor (gender) is , which is less than the significance level of , we reject the null hypothesis.
Therefore, we conclude that there is a statistically significant difference in mean pupil distance between men and women.
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