Question
Construct the probability distribution and calculate the expected value for the given scenario. Show all of your work and round your answers to the nearest hundredth, if needed. Then summarize your answer in the context of the problem.
A sample of GOC students were surveyed and asked, "How many pets do you have?" The results are shown below.
# of Pets 0 1 2 3 4
Frequency 10 23 11 4 2
Question 6 options:
A sample of GOC students were surveyed and asked, "How many pets do you have?" The results are shown below.
# of Pets 0 1 2 3 4
Frequency 10 23 11 4 2
Question 6 options:
Answers
Answered by
GPT-4o mini
To construct the probability distribution and calculate the expected value for the given scenario, we will follow these steps:
1. **Determine the total number of students surveyed.**
2. **Calculate the probability for each number of pets.**
3. **Calculate the expected value (mean) using the probabilities.**
### Step 1: Total Frequency
We calculate the total number of responses by summing the frequencies:
\[
\text{Total} = 10 + 23 + 11 + 4 + 2 = 60
\]
### Step 2: Probability Distribution
Next, we calculate the probability for each number of pets by dividing the frequency of each category by the total number of students surveyed.
| # of Pets | Frequency | Probability \( P(X) \) |
|-----------|-----------|-------------------------------|
| 0 | 10 | \( P(0) = \frac{10}{60} = \frac{1}{6} \approx 0.1667 \) |
| 1 | 23 | \( P(1) = \frac{23}{60} \approx 0.3833 \) |
| 2 | 11 | \( P(2) = \frac{11}{60} \approx 0.1833 \) |
| 3 | 4 | \( P(3) = \frac{4}{60} = \frac{1}{15} \approx 0.0667 \) |
| 4 | 2 | \( P(4) = \frac{2}{60} = \frac{1}{30} \approx 0.0333 \) |
Thus, the probability distribution is:
\[
\begin{array}{|c|c|c|}
\hline
\text{# of Pets (X)} & \text{Probability P(X)} \\
\hline
0 & 0.1667 \\
1 & 0.3833 \\
2 & 0.1833 \\
3 & 0.0667 \\
4 & 0.0333 \\
\hline
\end{array}
\]
### Step 3: Expected Value Calculation
To calculate the expected value (mean), we use the formula:
\[
E(X) = \sum (x \cdot P(X))
\]
Where \( x \) is the number of pets and \( P(X) \) is the probability of \( x \).
Calculating \( E(X) \):
\[
E(X) = (0 \cdot 0.1667) + (1 \cdot 0.3833) + (2 \cdot 0.1833) + (3 \cdot 0.0667) + (4 \cdot 0.0333)
\]
Calculating each term:
\[
= 0 + 0.3833 + 0.3666 + 0.2001 + 0.1332
\]
\[
= 0 + 0.3833 + 0.3666 + 0.2001 + 0.1332 = 1.0832
\]
Rounding to the nearest hundredth:
\[
E(X) \approx 1.08
\]
### Summary
The expected number of pets among the surveyed GOC students is approximately **1.08**. This means that, on average, a student in this sample has just over **1 pet**. This result provides insight into the pet ownership habits of the students surveyed.
1. **Determine the total number of students surveyed.**
2. **Calculate the probability for each number of pets.**
3. **Calculate the expected value (mean) using the probabilities.**
### Step 1: Total Frequency
We calculate the total number of responses by summing the frequencies:
\[
\text{Total} = 10 + 23 + 11 + 4 + 2 = 60
\]
### Step 2: Probability Distribution
Next, we calculate the probability for each number of pets by dividing the frequency of each category by the total number of students surveyed.
| # of Pets | Frequency | Probability \( P(X) \) |
|-----------|-----------|-------------------------------|
| 0 | 10 | \( P(0) = \frac{10}{60} = \frac{1}{6} \approx 0.1667 \) |
| 1 | 23 | \( P(1) = \frac{23}{60} \approx 0.3833 \) |
| 2 | 11 | \( P(2) = \frac{11}{60} \approx 0.1833 \) |
| 3 | 4 | \( P(3) = \frac{4}{60} = \frac{1}{15} \approx 0.0667 \) |
| 4 | 2 | \( P(4) = \frac{2}{60} = \frac{1}{30} \approx 0.0333 \) |
Thus, the probability distribution is:
\[
\begin{array}{|c|c|c|}
\hline
\text{# of Pets (X)} & \text{Probability P(X)} \\
\hline
0 & 0.1667 \\
1 & 0.3833 \\
2 & 0.1833 \\
3 & 0.0667 \\
4 & 0.0333 \\
\hline
\end{array}
\]
### Step 3: Expected Value Calculation
To calculate the expected value (mean), we use the formula:
\[
E(X) = \sum (x \cdot P(X))
\]
Where \( x \) is the number of pets and \( P(X) \) is the probability of \( x \).
Calculating \( E(X) \):
\[
E(X) = (0 \cdot 0.1667) + (1 \cdot 0.3833) + (2 \cdot 0.1833) + (3 \cdot 0.0667) + (4 \cdot 0.0333)
\]
Calculating each term:
\[
= 0 + 0.3833 + 0.3666 + 0.2001 + 0.1332
\]
\[
= 0 + 0.3833 + 0.3666 + 0.2001 + 0.1332 = 1.0832
\]
Rounding to the nearest hundredth:
\[
E(X) \approx 1.08
\]
### Summary
The expected number of pets among the surveyed GOC students is approximately **1.08**. This means that, on average, a student in this sample has just over **1 pet**. This result provides insight into the pet ownership habits of the students surveyed.