91Ó°ÊÓ

Ideal number of children Let \(X\) denote the response of a randomly selected person to the question, "What is the ideal number of children for a family to have?" The probability distribution of \(X\) in the United States is approximately as shown in the table, according to the gender of the person asked the question. $$ \begin{array}{lcc} \hline \begin{array}{l} \text { Probability Distribution of } \boldsymbol{X}=\text { Ideal Number } \\\ \text { of Children } \end{array} \\ \hline \boldsymbol{x} & \mathbf{P}(\boldsymbol{x}) \text { Females } & \mathbf{P}(\boldsymbol{x}) \text { Males } \\ \hline 0 & 0.01 & 0.02 \\ 1 & 0.03 & 0.03 \\ 2 & 0.55 & 0.60 \\ 3 & 0.31 & 0.28 \\ 4 & 0.11 & 0.08 \\ \hline \end{array} $$ a. Show that the means are similar, 2.50 for females and 2.39 for males. b. The standard deviation for the females is 0.770 and 0.758 for the males. Explain why a practical implication of the values for the standard deviations is that males hold slightly more consistent views than females about the ideal family size.

Step by step solution

01

Understand the Probability Distribution Table

The table provides the probability distribution of the ideal number of children as perceived by females and males. Each value of \(x\) (number of children) has corresponding probabilities for both females and males. These probabilities must sum to 1 for each category (females and males).

02

Calculate the Mean for Females

To calculate the expected value (mean) \(E(X)\) for females, use the formula \(E(X) = \sum x_i P(x_i)\). For females, plug in the values: \[ 0(0.01) + 1(0.03) + 2(0.55) + 3(0.31) + 4(0.11) \]. Simplifying this gives: \(E(X) = 2.50\).

03

Calculate the Mean for Males

Use the same formula \(E(X) = \sum x_i P(x_i)\) for males: \[ 0(0.02) + 1(0.03) + 2(0.60) + 3(0.28) + 4(0.08) \]. Simplifying these terms gives: \(E(X) = 2.39\).

04

Compare the Means

The calculated means for females and males are 2.50 and 2.39, respectively. This shows that both genders, on average, prefer a similar number of ideal children.

05

Understand Standard Deviation

Standard deviation measures how much the responses are spread out from the mean. Lower standard deviation indicates more consistency in responses.

06

Interpret the Standard Deviations

The standard deviation for females is 0.770 and for males is 0.758. Since the male standard deviation is slightly lower, it suggests that males' responses about the ideal number of children are slightly more consistent and closely clustered around their mean value than females' responses.

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Key Concepts

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

Expected Value

The expected value is a fundamental concept in probability theory, representing the average or mean value that you would anticipate if you repeated an experiment a large number of times. In our exercise, the expected value helps us find the average ideal number of children as perceived by each gender.To calculate the expected value, you multiply each possible outcome of a random variable by its corresponding probability and sum all those products. For example, the formula to find the expected value for a given probability distribution like ours is:\[ E(X) = \sum x_i P(x_i) \]**Calculating Expected Value for Females and Males**- **For Females:** We multiply each number of children by their probability and sum: \[ E(X) = 0 \times 0.01 + 1 \times 0.03 + 2 \times 0.55 + 3 \times 0.31 + 4 \times 0.11 = 2.50 \]- **For Males:** Using the same method: \[ E(X) = 0 \times 0.02 + 1 \times 0.03 + 2 \times 0.60 + 3 \times 0.28 + 4 \times 0.08 = 2.39 \]Both genders show similar average preferences, with males slightly leaning towards a slightly lower average ideal.

Standard Deviation

Standard deviation is a key statistic that quantifies the amount of variation or dispersion of a set of data points. A low standard deviation means the values are closely clustered around the mean, while a high standard deviation indicates a large variance from the mean. **Practical Implication of Standard Deviation** - The standard deviation for females, at 0.770, indicates more spread in their responses about the ideal number of children. - For males, the standard deviation is 0.758, slightly lower than for females. It shows their responses are a bit more consistent and less spread out from the mean of 2.39. This measure provides insights into how tightly grouped each gender's preferences are. Males show marginally more uniform responses, suggesting a more consistent view on family size among them compared to females.

Gender Comparison

When comparing gender preferences in a statistical study, such as ideal family size, it's essential to look at both the mean (expected value) and the standard deviation. Each tells a story about central tendency and variability. **Mean Comparison** - Females' expected mean value is 2.50, whereas males have a slightly lower mean at 2.39. - This suggests that while both genders prefer an ideal family size of around 2 to 3 children, females might favor slightly larger families. **Standard Deviation Comparison** - The smaller standard deviation for males (0.758 vs. 0.770 for females) indicates that male responses are more clustered, suggesting a consensus. From a gender-comparison perspective, this analysis highlights subtle differences in family size ideals and consistency in viewpoints. It provides a constructive view on how societal and possibly cultural influences can shape gender perceptions on family planning.