Question

Fifteen individuals were randomly selected from the respondents to a public opinion survey, and their scores are reproduced here. a. Calculate Pearson's \(r\) for age \((X)\) and four dependent variables \(\left(Y^{\prime} s\right):\) prestige, number of children, support for abortion, and hours of TV watching. b. Calculate Pearson's \(r\) for gender \((X)\) and four dependent variables \(\left(Y^{\prime} s\right):\) support for abortion, prestige, hours of TV watching, and number of children. c. Test the four relationships in either \(a\) or \(b\) for significance. $$ \begin{array}{cccccc} \begin{array}{c} \text { Occupational } \\ \text { Prestige } \end{array} & \begin{array}{c} \text { Number } \\ \text { of } \\ \text { Children } \end{array} & \text { Age } & \begin{array}{c} \text { Support for } \\ \text { Legal Abortion } \end{array} & \begin{array}{c} \text { Hours of } \\ \text { TV per } \\ \text { Day } \end{array} & \text { Gender } \\ \hline 32 & 3 & 34 & 3 & 1 & 1 \\ 50 & 0 & 41 & 1 & 3 & 1 \\ 17 & 0 & 52 & 5 & 2 & 1 \\ 69 & 3 & 67 & 1 & 5 & 0 \\ 17 & 0 & 40 & 1 & 5 & 0 \\ 52 & 0 & 22 & 2 & 3 & 1 \\ 32 & 3 & 31 & 3 & 4 & 0 \\ 50 & 0 & 23 & 4 & 4 & 0 \\ 19 & 9 & 64 & 1 & 6 & 1 \\ 37 & 4 & 55 & 4 & 2 & 0 \\ 14 & 3 & 66 & 5 & 5 & 1 \\ 51 & 0 & 22 & 2 & 0 & 0 \\ 45 & 0 & 19 & 3 & 7 & 0 \\ 44 & 0 & 21 & 4 & 1 & 1 \\ 46 & 4 & 58 & 2 & 0 & 1 \end{array} $$

Short answer

Compute means, deviations, covariances, standard deviations, and Pearson's r. Test significance using hypotheses, t-statistics, and compare with critical value.

Step-by-step solution

- Organize the Data

Extract each pair of variables (X,Y) from the table: age-prestige, age-number of children, age-support for abortion, age-hours of TV watching, gender-support for abortion, gender-prestige, gender-hours of TV watching, and gender-number of children.

- Calculate Means

Calculate the mean of each X and Y variable pair. For example, for age and prestige, compute the mean of the age values and the mean of the prestige values.

- Compute Deviations

For each variable pair (X, Y), calculate the deviation of each X and Y from its mean: (xi - mean_x) and (yi - mean_y), where xi and yi are individual observations.

- Calculate Covariances

Multiply the deviations for each pair and then calculate the sum of these products. Finally, divide by the number of pairs minus 1 to get the covariance: COV(X, Y) = Σ (xi - mean_x) * (yi - mean_y) / (n-1)

- Compute Standard Deviations

Calculate the standard deviations of X and Y for each variable pair. The formula for standard deviation is: SD(X) = sqrt [ Σ (xi - mean_x)^2 / (n-1) ] SD(Y) = sqrt [ Σ (yi - mean_y)^2 / (n-1) ]

- Calculate Pearson's r

For each variable pair (X, Y), calculate Pearson's r using the formula: r = COV(X, Y) / (SD(X) * SD(Y))

- Start with Testing Hypotheses

Formulate the null hypothesis (H0: ρ = 0) and the alternative hypothesis (H1: ρ ≠ 0) for significance tests.

- Calculate t-Statistic

Use the formula for the t-statistic: t = r * sqrt((n - 2) / (1 - r^2))

- Determine Degrees of Freedom

Calculate degrees of freedom: df = n - 2

- Compare with Critical Value

Use a t-distribution table to find the critical t-value at a chosen significance level (usually 0.05) and compare the calculated t-value to this critical value.

- Conclusion

If the calculated t-value exceeds the critical value, reject the null hypothesis, meaning the relationship is significant. Otherwise, do not reject the null hypothesis.

Key concepts

Hypothesis Testing

To determine if there is a significant relationship between two variables, you need to perform hypothesis testing. This process involves formulating two hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1).
H0 often states that there is no relationship between the variables (ρ = 0). H1 states that there is a relationship (ρ ≠ 0).
The next step is to calculate a test statistic to evaluate these hypotheses. For Pearson's correlation coefficient, you use the t-statistic: \( t = r \sqrt{(n - 2) / (1 - r^2)} \).
Lastly, you compare your calculated t-value with a critical t-value from the t-distribution table, based on your chosen significance level (e.g., 0.05).
If your calculated t-value exceeds the critical value, you reject H0, suggesting a significant relationship. Otherwise, you do not reject H0.

Covariance

Covariance measures how two variables change together. When variables have a positive covariance, they tend to increase or decrease together. Negative covariance indicates that one variable increases while the other decreases.
To calculate covariance, follow these steps:

• Calculate the deviations of each variable from their respective means.
• Multiply these deviations for each pair of data points.
• Sum these products.
• Divide the result by the number of data pairs minus one.

The mathematical formula for covariance is: \(Cov(X, Y) = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n-1}\).
Understanding covariance helps in understanding correlation, as it directly impacts the calculation of Pearson's r.

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation means the values tend to be close to the mean, while a high standard deviation indicates the values are spread out.
The steps to calculate standard deviation are:

• Find the mean (average) of your data set.
• Subtract the mean from each data point to find the deviation.
• Square each deviation.
• Calculate the average of these squared deviations.
• Take the square root of this average.

Mathematically, it is expressed as: \(SD(X) = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}\).
Standard deviation is critical in calculating Pearson's correlation coefficient, as it appears in the denominator of the formula.

Public Opinion Survey Analysis

Analyzing public opinion surveys helps understand societal trends and opinions. The analysis often involves statistical methods to draw meaningful conclusions from the data.
For instance, Pearson's correlation coefficient can assess relationships between attitudes and demographic factors.
Let's break it down:

• Identify your variables of interest from survey responses.
• Calculate correlation coefficients to see how different responses relate.
• Perform hypothesis testing to determine the significance of your findings.

By meticulously following these steps, valuable insights can be drawn, which are useful in areas like marketing, policy-making, and social research.