To solve this problem, we need to determine the proportion of households interested in each food item based on the sample and then predict the total number of households in the entire neighborhood that might have the same preference.
### 4. Spaghetti Preference
#### Step-by-Step Solution:
1. Given Data:
- Total number of households in the neighborhood: [tex]\(350\)[/tex]
- Sample size: [tex]\(20\)[/tex] households
- Number of households in the sample that prefer spaghetti: [tex]\(5\)[/tex]
2. Calculate the proportion of households in the sample that prefer spaghetti:
[tex]\[
\text{Proportion of households that prefer spaghetti} = \frac{\text{Number of spaghetti households in the sample}}{\text{Total sample size}} = \frac{5}{20} = 0.25
\][/tex]
3. Predict the number of households in the entire neighborhood that would prefer spaghetti:
[tex]\[
\text{Predicted number of households that prefer spaghetti} = \text{Proportion of households that prefer spaghetti} \times \text{Total number of households} = 0.25 \times 350 = 87.5
\][/tex]
Therefore, the predicted proportion of households that prefer spaghetti is [tex]\(0.25\)[/tex], and the predicted number of households in the whole neighborhood that would prefer spaghetti is [tex]\(87.5\)[/tex].
### 5. Pizza Preference
#### Step-by-Step Solution:
1. Given Data:
- Total number of households in the neighborhood: [tex]\(350\)[/tex]
- Sample size: [tex]\(20\)[/tex] households
- Number of households in the sample that prefer pizza: [tex]\(6\)[/tex]
2. Calculate the proportion of households in the sample that prefer pizza:
[tex]\[
\text{Proportion of households that prefer pizza} = \frac{\text{Number of pizza households in the sample}}{\text{Total sample size}} = \frac{6}{20} = 0.3
\][/tex]
3. Predict the number of households in the entire neighborhood that would prefer pizza:
[tex]\[
\text{Predicted number of households that prefer pizza} = \text{Proportion of households that prefer pizza} \times \text{Total number of households} = 0.3 \times 350 = 105.0
\][/tex]
Therefore, the predicted proportion of households that prefer pizza is [tex]\(0.3\)[/tex], and the predicted number of households in the whole neighborhood that would prefer pizza is [tex]\(105.0\)[/tex].
In conclusion, based on the sample data:
- Approximately [tex]\(0.25\)[/tex] of the households prefer spaghetti, leading to an expected [tex]\(87.5\)[/tex] households in the entire neighborhood preferring spaghetti.
- Approximately [tex]\(0.3\)[/tex] of the households prefer pizza, leading to an expected [tex]\(105.0\)[/tex] households in the entire neighborhood preferring pizza.
