Answer :
Certainly! Let's analyze the probabilities and test the fairness of the spinner based on the observed frequencies using the chi-squared test.
### Step-by-Step Solution:
1. Total Spins and Observed Frequencies
- The total number of spins: [tex]\( 625 \)[/tex]
- Observed frequencies:
- Orange: [tex]\( 118 \)[/tex]
- Purple: [tex]\( 137 \)[/tex]
- Brown: [tex]\( 122 \)[/tex]
- Yellow: [tex]\( 106 \)[/tex]
- Green: [tex]\( 142 \)[/tex]
2. Expected Frequencies for Each Section
Since the spinner is divided into five equal-sized sections, we assume that each section should have an equal probability of landing on. Hence, if the spinner is fair, each color should appear approximately the same number of times.
- There are 5 colors.
- Expected frequency for each color = [tex]\(\frac{\text{Total spins}}{\text{Number of sections}} = \frac{625}{5} = 125 \)[/tex]
3. Chi-Squared Statistic Calculation
The chi-squared statistic is used to test the hypothesis that the observed frequency distribution differs from the expected distribution.
The formula for the chi-squared statistic is:
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
where:
- [tex]\( O_i \)[/tex] = Observed frequency for the [tex]\( i \)[/tex]-th color
- [tex]\( E_i \)[/tex] = Expected frequency for the [tex]\( i \)[/tex]-th color
Plugging in the values, we calculate for each color:
[tex]\[ \chi^2 = \frac{(118 - 125)^2}{125} + \frac{(137 - 125)^2}{125} + \frac{(122 - 125)^2}{125} + \frac{(106 - 125)^2}{125} + \frac{(142 - 125)^2}{125} \][/tex]
[tex]\[ \chi^2 = \frac{(-7)^2}{125} + \frac{(12)^2}{125} + \frac{(-3)^2}{125} + \frac{(-19)^2}{125} + \frac{(17)^2}{125} \][/tex]
[tex]\[ \chi^2 = \frac{49}{125} + \frac{144}{125} + \frac{9}{125} + \frac{361}{125} + \frac{289}{125} \][/tex]
[tex]\[ \chi^2 = 0.392 + 1.152 + 0.072 + 2.888 + 2.312 \][/tex]
[tex]\[ \chi^2 = 6.816 \][/tex]
4. Final Results
- The expected frequency for each color (if the spinner is fair): 125
- The calculated chi-squared value: 6.816
### Interpretation
To interpret the chi-squared value, one typically compares it against a critical value from the chi-squared distribution table with the appropriate degrees of freedom. However, in this solution, our focus was on the calculation steps, and we derived the values needed for further hypothesis testing.
Thus, the expected frequency for each section is [tex]\(125\)[/tex] and the chi-squared statistic is [tex]\(6.816\)[/tex].
### Step-by-Step Solution:
1. Total Spins and Observed Frequencies
- The total number of spins: [tex]\( 625 \)[/tex]
- Observed frequencies:
- Orange: [tex]\( 118 \)[/tex]
- Purple: [tex]\( 137 \)[/tex]
- Brown: [tex]\( 122 \)[/tex]
- Yellow: [tex]\( 106 \)[/tex]
- Green: [tex]\( 142 \)[/tex]
2. Expected Frequencies for Each Section
Since the spinner is divided into five equal-sized sections, we assume that each section should have an equal probability of landing on. Hence, if the spinner is fair, each color should appear approximately the same number of times.
- There are 5 colors.
- Expected frequency for each color = [tex]\(\frac{\text{Total spins}}{\text{Number of sections}} = \frac{625}{5} = 125 \)[/tex]
3. Chi-Squared Statistic Calculation
The chi-squared statistic is used to test the hypothesis that the observed frequency distribution differs from the expected distribution.
The formula for the chi-squared statistic is:
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
where:
- [tex]\( O_i \)[/tex] = Observed frequency for the [tex]\( i \)[/tex]-th color
- [tex]\( E_i \)[/tex] = Expected frequency for the [tex]\( i \)[/tex]-th color
Plugging in the values, we calculate for each color:
[tex]\[ \chi^2 = \frac{(118 - 125)^2}{125} + \frac{(137 - 125)^2}{125} + \frac{(122 - 125)^2}{125} + \frac{(106 - 125)^2}{125} + \frac{(142 - 125)^2}{125} \][/tex]
[tex]\[ \chi^2 = \frac{(-7)^2}{125} + \frac{(12)^2}{125} + \frac{(-3)^2}{125} + \frac{(-19)^2}{125} + \frac{(17)^2}{125} \][/tex]
[tex]\[ \chi^2 = \frac{49}{125} + \frac{144}{125} + \frac{9}{125} + \frac{361}{125} + \frac{289}{125} \][/tex]
[tex]\[ \chi^2 = 0.392 + 1.152 + 0.072 + 2.888 + 2.312 \][/tex]
[tex]\[ \chi^2 = 6.816 \][/tex]
4. Final Results
- The expected frequency for each color (if the spinner is fair): 125
- The calculated chi-squared value: 6.816
### Interpretation
To interpret the chi-squared value, one typically compares it against a critical value from the chi-squared distribution table with the appropriate degrees of freedom. However, in this solution, our focus was on the calculation steps, and we derived the values needed for further hypothesis testing.
Thus, the expected frequency for each section is [tex]\(125\)[/tex] and the chi-squared statistic is [tex]\(6.816\)[/tex].
