Answer :
Sure, let's go through the solution step-by-step.
### Part (a): Expected Frequencies
To calculate the expected frequencies for each type of ice cream flavor based on the general American preferences, we need to apply the given percentages to the total number of patrons, which is 880.
1. Vanilla:
Expected frequency = [tex]\( \frac{22.7}{100} \times 880 \approx 200 \)[/tex]
2. Chocolate:
Expected frequency = [tex]\( \frac{22.9}{100} \times 880 \approx 202 \)[/tex]
3. Butter Pecan:
Expected frequency = [tex]\( \frac{10}{100} \times 880 \approx 88 \)[/tex]
4. Strawberry:
Expected frequency = [tex]\( \frac{8.8}{100} \times 880 \approx 77 \)[/tex]
5. Other Flavors:
Since we know the total number of patrons (880) and the expected counts for the other flavors, we can find this by subtracting the sum of the above expected frequencies from 880.
Expected frequency for Other = [tex]\( 880 - (200 + 202 + 88 + 77) = 313 \)[/tex]
### Part (b): Statistical Test
The correct statistical test to use in this scenario is the Goodness-of-Fit test. This test is used to determine if a sample matches a population with a specific distribution.
### Part (c): Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): The distribution of favorite ice cream for customers at her shop is the same as it is for Americans in general.
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The distribution of favorite ice cream for customers at her shop is not the same as it is for Americans in general.
### Part (d): Degrees of Freedom
The degrees of freedom for a Goodness-of-Fit test is calculated by the formula: [tex]\( \text{df} = k - 1 \)[/tex], where [tex]\( k \)[/tex] is the number of categories. Here, since there are 5 categories (Vanilla, Chocolate, Butter Pecan, Strawberry, Other), the degrees of freedom is:
[tex]\[ \text{df} = 5 - 1 = 4 \][/tex]
### Part (e): Test Statistic
The test statistic for the Goodness-of-Fit test is the Chi-square statistic, which can be calculated as follows:
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
where [tex]\( O_i \)[/tex] is the observed frequency and [tex]\( E_i \)[/tex] is the expected frequency for each category.
Performing the calculations, we find:
[tex]\[ \chi^2 = 17.404 \][/tex]
### Part (f): p-value
The p-value corresponding to the Chi-square statistic of 17.404 and 4 degrees of freedom is computed to be 0.0016.
Since the p-value (0.0016) is less than the significance level ([tex]\(\alpha = 0.01\)[/tex]), we reject the null hypothesis. This indicates that the distribution of favorite ice cream flavors at the ice cream shop is significantly different from the distribution in the general American population.
### Part (a): Expected Frequencies
To calculate the expected frequencies for each type of ice cream flavor based on the general American preferences, we need to apply the given percentages to the total number of patrons, which is 880.
1. Vanilla:
Expected frequency = [tex]\( \frac{22.7}{100} \times 880 \approx 200 \)[/tex]
2. Chocolate:
Expected frequency = [tex]\( \frac{22.9}{100} \times 880 \approx 202 \)[/tex]
3. Butter Pecan:
Expected frequency = [tex]\( \frac{10}{100} \times 880 \approx 88 \)[/tex]
4. Strawberry:
Expected frequency = [tex]\( \frac{8.8}{100} \times 880 \approx 77 \)[/tex]
5. Other Flavors:
Since we know the total number of patrons (880) and the expected counts for the other flavors, we can find this by subtracting the sum of the above expected frequencies from 880.
Expected frequency for Other = [tex]\( 880 - (200 + 202 + 88 + 77) = 313 \)[/tex]
### Part (b): Statistical Test
The correct statistical test to use in this scenario is the Goodness-of-Fit test. This test is used to determine if a sample matches a population with a specific distribution.
### Part (c): Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): The distribution of favorite ice cream for customers at her shop is the same as it is for Americans in general.
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The distribution of favorite ice cream for customers at her shop is not the same as it is for Americans in general.
### Part (d): Degrees of Freedom
The degrees of freedom for a Goodness-of-Fit test is calculated by the formula: [tex]\( \text{df} = k - 1 \)[/tex], where [tex]\( k \)[/tex] is the number of categories. Here, since there are 5 categories (Vanilla, Chocolate, Butter Pecan, Strawberry, Other), the degrees of freedom is:
[tex]\[ \text{df} = 5 - 1 = 4 \][/tex]
### Part (e): Test Statistic
The test statistic for the Goodness-of-Fit test is the Chi-square statistic, which can be calculated as follows:
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
where [tex]\( O_i \)[/tex] is the observed frequency and [tex]\( E_i \)[/tex] is the expected frequency for each category.
Performing the calculations, we find:
[tex]\[ \chi^2 = 17.404 \][/tex]
### Part (f): p-value
The p-value corresponding to the Chi-square statistic of 17.404 and 4 degrees of freedom is computed to be 0.0016.
Since the p-value (0.0016) is less than the significance level ([tex]\(\alpha = 0.01\)[/tex]), we reject the null hypothesis. This indicates that the distribution of favorite ice cream flavors at the ice cream shop is significantly different from the distribution in the general American population.
