Question

Data on three variables have been collected for 15 nations. The variables are fertility rate (average number of children born to each woman), education for females (expressed as a percentage of all students at the secondary level who are female), and percent of married women ages \(15-49\) using all forms of contraception. $$ \begin{array}{cccc} & & & \text { Percent of } \\ \multicolumn{1}{c} {\text { Nation }} & \begin{array}{c} \text { Fertility } \\ \text { Rate* } \end{array} & \begin{array}{c} \text { Education of } \\ \text { Females* } \end{array} & \begin{array}{c} \text { Women Using } \\ \text { Contraceptiont } \end{array} \\ \hline \text { Niger } & 7.3 & 38.4 & 11 \\ \text { Cambodia } & 3.0 & 35.6 & 40 \\ \text { Guatemala } & 3.6 & 46.9 & 43 \\ \text { Ghana } & 3.8 & 44.7 & 24 \text { Bolivia } & 2.7 & 48.2 & 61 \\ \text { Kenya } & 4.7 & 47.6 & 39 \\ \text { Dominican } & & & \\ \text { Republic } & 2.8 & 54.8 & 73 \\ \text { Mexico } & 2.4 & 50.5 & 71 \\ \text { Vietnam } & 1.9 & 47.1 & 79 \\ \text { Turkey } & 1.9 & 37.5 & 71 \\ \text { United States } & 2.1 & 49.0 & 73 \\ \text { China } & 1.8 & 45.3 & 90 \\ \text { United } & & & \\ \text { Kingdom } & 1.7 & 52.7 & 84 \\ \text { Japan } & 1.2 & 49.0 & 52 \\ \text { Italy } & 1.3 & 48.1 & 60 \end{array} $$ a. Compute \(r\) and \(r^{2}\) for each combination of variables. b. Summarize these relationships in terms of strength and direction.

Short answer

1. Calculate Pearson's r for each variable pair. 2. Square each r to get r². 3. Interpret r and r² for strength and direction.

Step-by-step solution

- Organize the Data

Gather the values for each of the three variables: fertility rate, education of females, and percent of women using contraception for each nation.

- Calculate Correlation Coefficient (r)

Use the Pearson correlation formula to compute the correlation coefficient (r) between each pair of variables.

- Compute r²

Square the correlation coefficient (r) calculated in Step 2 to find r² for each combination of variables.

- Analyze Strength and Direction

Interpret the values of r and r² to determine the strength and direction of the relationships between the variables. Recall that r indicates direction (positive or negative relationship) and magnitude (strength of the relationship), while r² indicates the proportion of variance in one variable explained by the other.

Key concepts

fertility rate

The fertility rate is a key demographic measure that indicates the average number of children a woman is expected to have during her lifetime. It is crucial for understanding population growth and societal changes.

Higher fertility rates generally reflect higher rates of population growth. This can impact a country's economy, healthcare, and education systems.

Conversely, lower fertility rates can indicate better access to education and healthcare, as well as higher female participation in the workforce.

female education

Female education refers to the percentage of female students at the secondary education level. This metric is essential for understanding gender equality in education.

Higher female education levels often correlate with numerous positive outcomes, such as reduced fertility rates, higher contraceptive use, and improved economic growth.

Educated women are more likely to participate in the workforce and make informed choices about their health and family planning.

contraceptive use

Contraceptive use measures the percentage of married women aged 15-49 who use any form of contraception. This statistic is pivotal for understanding reproductive health and family planning in a given context.

High contraceptive use can lead to lower fertility rates, which can reduce the strain on a country's resources and improve overall quality of life.

Access to contraception is often linked with better educational and economic opportunities for women, allowing them to have greater control over their reproductive health.

Pearson correlation coefficient

The Pearson correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables.

It ranges from -1 to 1. A value of 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.

Positive correlations signify that as one variable increases, the other also increases. Negative correlations indicate that as one variable increases, the other decreases.

In this exercise, r helps us understand the relationships between fertility rate, female education, and contraceptive use.

r-squared (r²)

r-squared (r²) is derived from the Pearson correlation coefficient and represents the proportion of variance in one variable explained by the other variable.

It ranges from 0 to 1. Higher values indicate that a larger proportion of the variability in one variable is accounted for by the other.

For example, if r² is 0.75, it means that 75% of the variation in one variable can be explained by the variation in the other variable.

In summary, r² provides insight into the predictability and explanatory power of the relationship between two variables.