Questions: MATH 1342 Name Assignment 4 For 1 and 2, find the mean and standard deviation of the probability distribution. 1. Data for the number of dogs per household in a neighborhood has been provided. Dogs: 0, 1, 2, 3, 4, 5 Probability: 0.686, 0.195, 0.077, 0.022, 0.013, 0.007 2. Data for the number of school-related extracurricular activities per high school student. Activities: 0, 1, 2, 3, 4, 5, 6, 7 Probability: 0.059, 0.122, 0.163, 0.178, 0.213, 0.128, 0.084, 0.053 3. A high school basketball team is selling 10 raffle tickets as part of a fund-raising program. The first prize is a trip to the Bahamas worth 5,460, and the second prize is a weekend ski package worth 496. The remaining 18 prizes are 100 gift cards to a local restaurant. The number of tickets to be sold is 3,500. Find the expected value E(x) for someone buying one raffle ticket. Hint: It wouldn't be a fund-raiser if the E(x) was a positive value for a raffle ticket purchase. 4. Fill in the missing value to complete the probability distribution. X: 1, 2, 3, 4, 5 P(x): 0.35, , 0.17, 0.21, 0.03

MATH 1342 Name Assignment 4 For 1 and 2, find the mean and standard deviation of the probability distribution. 1. Data for the number of dogs per household in a neighborhood has been provided. Dogs: 0, 1, 2, 3, 4, 5 Probability: 0.686, 0.195, 0.077, 0.022, 0.013, 0.007 2. Data for the number of school-related extracurricular activities per high school student. Activities: 0, 1, 2, 3, 4, 5, 6, 7 Probability: 0.059, 0.122, 0.163, 0.178, 0.213, 0.128, 0.084, 0.053 3. A high school basketball team is selling 10 raffle tickets as part of a fund-raising program. The first prize is a trip to the Bahamas worth 5,460, and the second prize is a weekend ski package worth 496. The remaining 18 prizes are 100 gift cards to a local restaurant. The number of tickets to be sold is 3,500. Find the expected value E(x) for someone buying one raffle ticket. Hint: It wouldn't be a fund-raiser if the E(x) was a positive value for a raffle ticket purchase. 4. Fill in the missing value to complete the probability distribution. X: 1, 2, 3, 4, 5 P(x): 0.35, , 0.17, 0.21, 0.03
Transcript text: MATH 1342 Name Assignment 4 For $1$ and $2$, find the mean and standard deviation of the probability distribution. 1. Data for the number of dogs per household in a neighborhood has been provided. \begin{tabular}{|l|c|c|c|c|c|c|} \hline Dogs & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline Probability & 0.686 & 0.195 & 0.077 & 0.022 & 0.013 & 0.007 \\ \hline \end{tabular} 2. Data for the number of school-related extracurricular activities per high school student. \begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline Activities & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline Probability & 0.059 & 0.122 & 0.163 & 0.178 & 0.213 & 0.128 & 0.084 & 0.053 \\ \hline \end{tabular} 3. A high school basketball team is selling $10 raffle tickets as part of a fund-raising program. The first prize is a trip to the Bahamas worth $5,460, and the second prize is a weekend ski package worth $496. The remaining 18 prizes are $100 gift cards to a local restaurant. The number of tickets to be sold is 3,500 . Find the expected value $E(x)$ for someone buying one raffle ticket. Hint: It wouldn't be a fund-raiser if the $E(x)$ was a positive value for a raffle ticket purchase. 4. Fill in the missing value to complete the probability distribution. \begin{tabular}{|c|c|} \hline $\mathbf{X}$ & $\mathbf{P ( x )}$ \\ \hline 1 & 0.35 \\ \hline 2 & \\ \hline 3 & 0.17 \\ \hline 4 & 0.21 \\ \hline 5 & 0.03 \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Mean and Standard Deviation for Dogs

To find the mean and standard deviation of the probability distribution for the number of dogs per household, we calculated the following:

  • Mean \( E(X) \) is given by: \[ E(X) = \sum (x_i \cdot P(x_i)) = 0 \cdot 0.686 + 1 \cdot 0.195 + 2 \cdot 0.077 + 3 \cdot 0.022 + 4 \cdot 0.013 + 5 \cdot 0.007 = 0.502 \]

  • The standard deviation \( \sigma \) is calculated as: \[ \sigma = \sqrt{E(X^2) - (E(X))^2} \] Where \( E(X^2) = \sum (x_i^2 \cdot P(x_i)) \). The result is: \[ \sigma \approx 0.912 \]

Step 2: Mean and Standard Deviation for Activities

Next, we calculated the mean and standard deviation for the number of school-related extracurricular activities per high school student:

  • Mean \( E(X) \) is given by: \[ E(X) = \sum (x_i \cdot P(x_i)) = 0 \cdot 0.059 + 1 \cdot 0.122 + 2 \cdot 0.163 + 3 \cdot 0.178 + 4 \cdot 0.213 + 5 \cdot 0.128 + 6 \cdot 0.084 + 7 \cdot 0.053 \approx 3.349 \]

  • The standard deviation \( \sigma \) is calculated similarly: \[ \sigma \approx 1.841 \]

Step 3: Expected Value for Raffle Ticket

For the raffle ticket, we calculated the expected value \( E(X) \) for someone buying one ticket:

  • The total value of prizes is: \[ \text{Total Prizes} = 5460 + 496 + 18 \cdot 100 = 5460 + 496 + 1800 = 6756 \]

  • The expected value \( E(X) \) for one ticket is: \[ E(X) = \frac{6756}{3500} \approx 1.93 \]

  • The net expected value after subtracting the ticket price is: \[ E(X) - \text{Ticket Price} = 1.93 - 10 = -7.78 \]

Final Answer

  • For the number of dogs per household:

    • Mean: \( \boxed{0.502} \)
    • Standard Deviation: \( \boxed{0.912} \)

  • For the number of activities per student:

    • Mean: \( \boxed{3.349} \)
    • Standard Deviation: \( \boxed{1.841} \)

  • For the expected value of the raffle ticket:

    • Expected Value: \( \boxed{-7.78} \)
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