To determine which sections have a significant difference between the empirical model (observed frequencies) and the theoretical model (expected frequencies), follow these steps:
1. Find the Theoretical Probability: Since the spinner is divided evenly into four sections, the theoretical probability for each section is [tex]\( \frac{1}{4} \)[/tex].
2. Calculate the Expected Frequency:
- The total number of spins is 24.
- The expected frequency for each section = theoretical probability [tex]\(\times\)[/tex] total number of spins = [tex]\( \frac{1}{4} \times 24 \)[/tex] = 6.
3. Determine the Differences between Observed and Expected Frequencies:
- For "Forward One Space": Observed frequency = 8, Expected frequency = 6, Difference = [tex]\( 8 - 6 = 2 \)[/tex].
- For "Forward Two Spaces": Observed frequency = 6, Expected frequency = 6, Difference = [tex]\( 6 - 6 = 0 \)[/tex].
- For "Backward One Space": Observed frequency = 4, Expected frequency = 6, Difference = [tex]\( 4 - 6 = -2 \)[/tex].
- For "Backward Two Spaces": Observed frequency = 6, Expected frequency = 6, Difference = [tex]\( 6 - 6 = 0 \)[/tex].
4. Identify Sections with Significant Differences:
- Any section with a non-zero difference indicates a discrepancy between the empirical and theoretical models.
Here are the results:
- "Forward One Space" has a difference of 2, which is a significant deviation.
- "Backward One Space" has a difference of -2, which is also a significant deviation.
- "Forward Two Spaces" and "Backward Two Spaces" have zero differences, so their empirical models are not significantly different from the theoretical model.
Based on this analysis, the correct locations in the table where the empirical model significantly differs from the theoretical model are:
- Forward One Space
- Backward One Space
