Answer :
Let's begin by completing the given frequency table.
First, we are given the following values:
- Number of people who like both burritos and hamburgers: 29
- Number of people who like burritos but do not like hamburgers: 41
- Number of people who do not like either burritos or hamburgers: 54
- Total number of people surveyed: 205
- Total number of people who like hamburgers: 110
- Total number of people who do not like burritos (including both those who like and those who don’t like hamburgers): 135
We need to find the missing values in the table. Let's start with "Does not like burritos but likes hamburgers":
[tex]\[ \text{Number of people who do not like burritos but like hamburgers} = \text{Total who like hamburgers} - \text{Number of people who like both burritos and hamburgers} \][/tex]
[tex]\[ = 110 - 29 \][/tex]
[tex]\[ = 81 \][/tex]
Now we can fill in the table:
\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}
### Part A:
Calculate the percentage of the survey respondents who liked neither hamburgers nor burritos.
The number of people who liked neither hamburgers nor burritos is already given as 54. The total number of people surveyed is 205.
[tex]\[ \text{Percentage liking neither} = \left( \frac{\text{Number liking neither}}{\text{Total number surveyed}} \right) \times 100 \][/tex]
[tex]\[ = \left( \frac{54}{205} \right) \times 100 \][/tex]
[tex]\[ = 26.34146341463415\% \][/tex]
Thus, [tex]\(26.34%\)[/tex] of the respondents liked neither hamburgers nor burritos.
### Part B:
Calculate the marginal relative frequency of all customers who like hamburgers.
The total number of people who like hamburgers is 110, and the total number of people surveyed is 205.
[tex]\[ \text{Marginal relative frequency (hamburgers)} = \left( \frac{\text{Number who like hamburgers}}{\text{Total number surveyed}} \right) \times 100 \][/tex]
[tex]\[ = \left( \frac{110}{205} \right) \times 100 \][/tex]
[tex]\[ = 53.65853658536586\% \][/tex]
So, the marginal relative frequency of customers who like hamburgers is [tex]\(53.66\%\)[/tex].
### Part C:
Determine if there is an association between liking burritos and liking hamburgers.
We need to compare the joint and marginal frequencies:
1. Joint frequency of liking both burritos and hamburgers:
[tex]\[ \text{Joint frequency (likes both)} = \frac{\text{Number who like both hamburgers and burritos}}{\text{Total number surveyed}} \][/tex]
[tex]\[ = \frac{29}{205} \][/tex]
2. Marginal frequency of liking burritos:
[tex]\[ \text{Marginal frequency (likes burritos)} = \frac{\text{Number who like burritos}}{\text{Total number surveyed}} \][/tex]
[tex]\[ = \frac{70}{205} \][/tex]
3. Marginal frequency of liking hamburgers:
[tex]\[ \text{Marginal frequency (likes hamburgers)} = \frac{110}{205} \][/tex]
Now, we compute the ratio of the joint frequency to the marginal frequency of liking burritos:
[tex]\[ \text{Ratio} = \frac{\text{Joint frequency (likes both)}}{\text{Marginal frequency (likes burritos)}} \][/tex]
[tex]\[ = \frac{\left(\frac{29}{205}\right)}{\left(\frac{70}{205}\right)} \][/tex]
[tex]\[ = \frac{29}{70} \][/tex]
[tex]\[ = 0.4142857142857142 \][/tex]
Since the ratio is less than 1 (0.414 < 1), it suggests that there is no positive association between liking burritos and liking hamburgers.
Thus, there is no positive association between liking burritos and liking hamburgers. The ratio of joint to marginal frequencies being less than 1 indicates that those who like burritos are less likely to also like hamburgers.
First, we are given the following values:
- Number of people who like both burritos and hamburgers: 29
- Number of people who like burritos but do not like hamburgers: 41
- Number of people who do not like either burritos or hamburgers: 54
- Total number of people surveyed: 205
- Total number of people who like hamburgers: 110
- Total number of people who do not like burritos (including both those who like and those who don’t like hamburgers): 135
We need to find the missing values in the table. Let's start with "Does not like burritos but likes hamburgers":
[tex]\[ \text{Number of people who do not like burritos but like hamburgers} = \text{Total who like hamburgers} - \text{Number of people who like both burritos and hamburgers} \][/tex]
[tex]\[ = 110 - 29 \][/tex]
[tex]\[ = 81 \][/tex]
Now we can fill in the table:
\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}
### Part A:
Calculate the percentage of the survey respondents who liked neither hamburgers nor burritos.
The number of people who liked neither hamburgers nor burritos is already given as 54. The total number of people surveyed is 205.
[tex]\[ \text{Percentage liking neither} = \left( \frac{\text{Number liking neither}}{\text{Total number surveyed}} \right) \times 100 \][/tex]
[tex]\[ = \left( \frac{54}{205} \right) \times 100 \][/tex]
[tex]\[ = 26.34146341463415\% \][/tex]
Thus, [tex]\(26.34%\)[/tex] of the respondents liked neither hamburgers nor burritos.
### Part B:
Calculate the marginal relative frequency of all customers who like hamburgers.
The total number of people who like hamburgers is 110, and the total number of people surveyed is 205.
[tex]\[ \text{Marginal relative frequency (hamburgers)} = \left( \frac{\text{Number who like hamburgers}}{\text{Total number surveyed}} \right) \times 100 \][/tex]
[tex]\[ = \left( \frac{110}{205} \right) \times 100 \][/tex]
[tex]\[ = 53.65853658536586\% \][/tex]
So, the marginal relative frequency of customers who like hamburgers is [tex]\(53.66\%\)[/tex].
### Part C:
Determine if there is an association between liking burritos and liking hamburgers.
We need to compare the joint and marginal frequencies:
1. Joint frequency of liking both burritos and hamburgers:
[tex]\[ \text{Joint frequency (likes both)} = \frac{\text{Number who like both hamburgers and burritos}}{\text{Total number surveyed}} \][/tex]
[tex]\[ = \frac{29}{205} \][/tex]
2. Marginal frequency of liking burritos:
[tex]\[ \text{Marginal frequency (likes burritos)} = \frac{\text{Number who like burritos}}{\text{Total number surveyed}} \][/tex]
[tex]\[ = \frac{70}{205} \][/tex]
3. Marginal frequency of liking hamburgers:
[tex]\[ \text{Marginal frequency (likes hamburgers)} = \frac{110}{205} \][/tex]
Now, we compute the ratio of the joint frequency to the marginal frequency of liking burritos:
[tex]\[ \text{Ratio} = \frac{\text{Joint frequency (likes both)}}{\text{Marginal frequency (likes burritos)}} \][/tex]
[tex]\[ = \frac{\left(\frac{29}{205}\right)}{\left(\frac{70}{205}\right)} \][/tex]
[tex]\[ = \frac{29}{70} \][/tex]
[tex]\[ = 0.4142857142857142 \][/tex]
Since the ratio is less than 1 (0.414 < 1), it suggests that there is no positive association between liking burritos and liking hamburgers.
Thus, there is no positive association between liking burritos and liking hamburgers. The ratio of joint to marginal frequencies being less than 1 indicates that those who like burritos are less likely to also like hamburgers.
