Answer :
To thoroughly analyze the data from this survey, let's fill in the missing entries in the frequency table and then answer each part of the question.
1. Complete the Frequency Table:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & Likes hamburgers & Does not like hamburgers & Total \\ \hline Likes burritos & 29 & 41 & 70 \\ \hline Does not like burritos & 81 & 54 & 135 \\ \hline Total & 110 & 95 & 205 \\ \hline \end{tabular} \][/tex]
We know the following:
- Total survey respondents: 205.
- Likes hamburgers and burritos: 29.
- Likes hamburgers: 110.
- Does not like hamburgers, does not like burritos: [tex]\(135 - 29 - 41 = 65\)[/tex].
- Likes burritos: [tex]\(29 + 41 = 70\)[/tex].
Using these values, we can determine the other entries:
- Does not like burritos, likes hamburgers: [tex]\(110 - 29 = 81\)[/tex].
- Total that like neither hamburgers nor burritos is already calculated as 65.
So, the completed table should look like:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & Likes hamburgers & Does not like hamburgers & Total \\ \hline Likes burritos & 29 & 41 & 70 \\ \hline Does not like burritos & 81 & 54 & 135 \\ \hline Total & 110 & 95 & 205 \\ \hline \end{tabular} \][/tex]
### Part A: Percentage of Survey Respondents Who Liked Neither Hamburgers nor Burritos
To find the percentage of respondents who liked neither hamburgers nor burritos, use the following calculation:
[tex]\[ \text{Neither percentage} = \frac{\text{Number who liked neither}}{\text{Total number of respondents}} \times 100 \][/tex]
[tex]\[ \text{Neither percentage} = \frac{65}{205} \times 100 \approx 31.71\% \][/tex]
### Part B: Marginal Relative Frequency of Customers Who Like Hamburgers
The marginal relative frequency of all customers who like hamburgers is calculated by dividing the number of people who like hamburgers by the total number of respondents:
[tex]\[ \text{Marginal relative frequency of hamburgers} = \frac{\text{Number who like hamburgers}}{\text{Total number of respondents}} \][/tex]
[tex]\[ \text{Marginal relative frequency of hamburgers} = \frac{110}{205} \approx 0.537 \][/tex]
### Part C: Association Between Liking Burritos and Liking Hamburgers
To determine if there is an association between liking burritos and liking hamburgers, we can compare the joint and marginal frequencies.
1. Calculate the joint relative frequency of people who like both:
[tex]\[ \text{Joint relative frequency (likes both)} = \frac{\text{Number who like both}}{\text{Total number of respondents}} \][/tex]
[tex]\[ \text{Joint relative frequency} = \frac{29}{205} \approx 0.141 \][/tex]
2. Calculate the marginal relative frequency of people who like burritos:
[tex]\[ \text{Marginal relative frequency (likes burritos)} = \frac{\text{Number who like burritos}}{\text{Total number of respondents}} \][/tex]
[tex]\[ \text{Marginal relative frequency} = \frac{70}{205} \approx 0.341 \][/tex]
3. Compare this with the expected joint frequency, assuming independence:
[tex]\[ \text{Expected joint frequency} = (\text{Marginal relative frequency of hamburgers}) \times (\text{Marginal relative frequency of burritos}) \][/tex]
[tex]\[ \text{Expected joint frequency} = (0.537) \times (0.341) \approx 0.183 \][/tex]
4. Measure the ratio of observed to expected joint frequency:
[tex]\[ \text{Ratio} = \frac{\text{Joint relative frequency}}{\text{Expected joint frequency}} \][/tex]
[tex]\[ \text{Ratio} = \frac{0.141}{0.183} \approx 0.772 \][/tex]
Since the ratio is less than 1 (0.772), it indicates that the actual joint frequency is less than the expected joint frequency if the events were independent. Therefore, there may be a negative association between liking burritos and liking hamburgers, meaning people who like one are somewhat less likely to like the other.
To summarize the answers:
- Part A: 31.71% of respondents liked neither hamburgers nor burritos.
- Part B: The marginal relative frequency of customers who like hamburgers is approximately 0.537.
- Part C: The ratio 0.772 suggests a possible negative association between liking burritos and liking hamburgers.
1. Complete the Frequency Table:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & Likes hamburgers & Does not like hamburgers & Total \\ \hline Likes burritos & 29 & 41 & 70 \\ \hline Does not like burritos & 81 & 54 & 135 \\ \hline Total & 110 & 95 & 205 \\ \hline \end{tabular} \][/tex]
We know the following:
- Total survey respondents: 205.
- Likes hamburgers and burritos: 29.
- Likes hamburgers: 110.
- Does not like hamburgers, does not like burritos: [tex]\(135 - 29 - 41 = 65\)[/tex].
- Likes burritos: [tex]\(29 + 41 = 70\)[/tex].
Using these values, we can determine the other entries:
- Does not like burritos, likes hamburgers: [tex]\(110 - 29 = 81\)[/tex].
- Total that like neither hamburgers nor burritos is already calculated as 65.
So, the completed table should look like:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & Likes hamburgers & Does not like hamburgers & Total \\ \hline Likes burritos & 29 & 41 & 70 \\ \hline Does not like burritos & 81 & 54 & 135 \\ \hline Total & 110 & 95 & 205 \\ \hline \end{tabular} \][/tex]
### Part A: Percentage of Survey Respondents Who Liked Neither Hamburgers nor Burritos
To find the percentage of respondents who liked neither hamburgers nor burritos, use the following calculation:
[tex]\[ \text{Neither percentage} = \frac{\text{Number who liked neither}}{\text{Total number of respondents}} \times 100 \][/tex]
[tex]\[ \text{Neither percentage} = \frac{65}{205} \times 100 \approx 31.71\% \][/tex]
### Part B: Marginal Relative Frequency of Customers Who Like Hamburgers
The marginal relative frequency of all customers who like hamburgers is calculated by dividing the number of people who like hamburgers by the total number of respondents:
[tex]\[ \text{Marginal relative frequency of hamburgers} = \frac{\text{Number who like hamburgers}}{\text{Total number of respondents}} \][/tex]
[tex]\[ \text{Marginal relative frequency of hamburgers} = \frac{110}{205} \approx 0.537 \][/tex]
### Part C: Association Between Liking Burritos and Liking Hamburgers
To determine if there is an association between liking burritos and liking hamburgers, we can compare the joint and marginal frequencies.
1. Calculate the joint relative frequency of people who like both:
[tex]\[ \text{Joint relative frequency (likes both)} = \frac{\text{Number who like both}}{\text{Total number of respondents}} \][/tex]
[tex]\[ \text{Joint relative frequency} = \frac{29}{205} \approx 0.141 \][/tex]
2. Calculate the marginal relative frequency of people who like burritos:
[tex]\[ \text{Marginal relative frequency (likes burritos)} = \frac{\text{Number who like burritos}}{\text{Total number of respondents}} \][/tex]
[tex]\[ \text{Marginal relative frequency} = \frac{70}{205} \approx 0.341 \][/tex]
3. Compare this with the expected joint frequency, assuming independence:
[tex]\[ \text{Expected joint frequency} = (\text{Marginal relative frequency of hamburgers}) \times (\text{Marginal relative frequency of burritos}) \][/tex]
[tex]\[ \text{Expected joint frequency} = (0.537) \times (0.341) \approx 0.183 \][/tex]
4. Measure the ratio of observed to expected joint frequency:
[tex]\[ \text{Ratio} = \frac{\text{Joint relative frequency}}{\text{Expected joint frequency}} \][/tex]
[tex]\[ \text{Ratio} = \frac{0.141}{0.183} \approx 0.772 \][/tex]
Since the ratio is less than 1 (0.772), it indicates that the actual joint frequency is less than the expected joint frequency if the events were independent. Therefore, there may be a negative association between liking burritos and liking hamburgers, meaning people who like one are somewhat less likely to like the other.
To summarize the answers:
- Part A: 31.71% of respondents liked neither hamburgers nor burritos.
- Part B: The marginal relative frequency of customers who like hamburgers is approximately 0.537.
- Part C: The ratio 0.772 suggests a possible negative association between liking burritos and liking hamburgers.
