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Chapter 5: Problem 15

Refer to the accompanying table, which describes results from groups of 8 births from 8 different sets of parents. The random variable \(x\) represents the number of girls among 8 children. $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Number of } \\ \text { Girls } \boldsymbol{x} \end{array} & \boldsymbol{P}(\boldsymbol{x}) \\ \hline 0 & 0.004 \\ \hline 1 & 0.031 \\ \hline 2 & 0.109 \\ \hline 3 & 0.219 \\ \hline 4 & 0.273 \\ \hline 5 & 0.219 \\ \hline 6 & 0.109 \\ \hline 7 & 0.031 \\ \hline 8 & 0.004 \\ \hline \end{array}$$ Find the mean and standard deviation for the numbers of girls in 8 births.

Short Answer

Expert verified
The mean is 4; the standard deviation is approximately 1.414.

Step by step solution

01

Find the Mean (Step 1)

To find the mean of the distribution, use the formula for the expected value of a discrete random variable: \ \ \[\begin{equation} \mu = \sum x \cdot P(x) \ \ = 0 \cdot 0.004 + 1 \cdot 0.031 + 2 \cdot 0.109 + 3 \cdot 0.219 + 4 \cdot 0.273 + 5 \cdot 0.219 + 6 \cdot 0.109 + 7 \cdot 0.031 + 8 \cdot 0.004 \ \ = 4 \end{equation}\]
02

Calculate the Expected Value

Using the values provided in the table and the formula, the computation is: \ \ \[\begin{equation} \mu = 0 \cdot 0.004 + 1 \cdot 0.031 + 2 \cdot 0.109 + 3 \cdot 0.219 + 4 \cdot 0.273 + 5 \cdot 0.219 + 6 \cdot 0.109 + 7 \cdot 0.031 + 8 \cdot 0.004 = 4 \end{equation}\]
03

Find the Variance (Step 2)

To find the variance, use the formula: \ \ \[\begin{equation} \sigma^2 = \sum (x - \mu)^2 \cdot P(x) \ \ = (0 - 4)^2 \cdot 0.004 + (1 - 4)^2 \cdot 0.031 + (2 - 4)^2 \cdot 0.109 + (3 - 4)^2 \cdot 0.219 + (4 - 4)^2 \cdot 0.273 + (5 - 4)^2 \cdot 0.219 + (6 - 4)^2 \cdot 0.109 + (7 - 4)^2 \cdot 0.031 + (8 - 4)^2 \cdot 0.004 \end{equation}\]
04

Calculate the Variance

Using the values provided in the table and the formula, the computation is: \ \ \[\begin{equation} \sigma^2 = 16 \cdot 0.004 + 9 \cdot 0.031 + 4 \cdot 0.109 + 1 \cdot 0.219 + 0 \cdot 0.273 + 1 \cdot 0.219 + 4 \cdot 0.109 + 9 \cdot 0.031 + 16 \cdot 0.004 = 2 \end{equation}\]
05

Find the Standard Deviation (Step 3)

To determine the standard deviation, take the square root of the variance: \ \ \[\begin{equation} \sigma = \sqrt{\sigma^2} = \sqrt{2} \approx 1.414 \end{equation}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
To determine the mean of a set of values, we use the concept of expected value. The **mean** for a discrete random variable is calculated using the formula:
\ \ ewline \[ \[\begin{equation} \mu = \sum x \cdot P(x) \end{equation}\] \] \ \ This formula involves multiplying each value (\

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Most popular questions from this chapter

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