Question

A study was conducted using 49 Catholic female undergraduates at Texas A&M University. The variables measured refer to the parents of these students. The re- sponse variable is the number of children that the parents have. One of the explanatory variables is the mother's ed- ucational level, measured as the number of years of for- mal education. For these data, i = 9.88, s. = 3.77. , = 3.35. s, = 2.19. the prediction equation is j = 5.40 0.207x, the standard error of the slope estimate is 0.079, and SSE = 201.95. (a) Find the correlation and interpret its value. (b) Test the null hypothesis that mean number of children is independent of mother's educational level, and report and interpret the P-value. (c) Sketch a potential scatterplot such that the analyses you conducted in (a) and (b) would be inappropriate.

Answer 1

a. The correlation coefficient is 0.778, indicating a strong positive relationship between the mother's educational level and the number of children. b. The P-value is the probability of observing a t-value as extreme as the one calculated, assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis. c. The analyses conducted in parts (a) and (b) would be inappropriate if the scatterplot showed no clear relationship between the variables.

Step-by-step explanation:

a. Correlation: To find the correlation coefficient, we can use the formula: r = (sxy)/(sx*sy). Plugging in the given values: r = (9.88*2.19)/(3.77*3.35) = 0.778. The correlation coefficient is 0.778. A correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient of 0.778 indicates a strong positive relationship between the mother's educational level and the number of children.

b. Hypothesis test: To test the null hypothesis, we can use the t-test for the slope coefficient. The t-statistic can be calculated using the formula: t = (b - 0)/SE(b), where b is the slope estimate and SE(b) is the standard error of the slope estimate. Plugging in the given values: t = (0.207 - 0)/0.079 = 2.607. Using a t-table or calculator, we can find the P-value associated with a t-value of 2.607. If the P-value is less than the significance level (usually 0.05), we reject the null hypothesis.

c. Scatterplot: In order for the analyses conducted in parts (a) and (b) to be inappropriate, the scatterplot would need to show no clear relationship between the variables. For example, if the scatterplot showed a random cloud of points without any discernible pattern, the correlation and hypothesis test would not be meaningful.

Answer 2

Final Answers:

(a) The correlation is r = 0.207, indicating a weak positive linear relationship between mother's educational level and the number of children their parents have.

(b) The P-value for testing the null hypothesis of the mean number of children being independent of mother's educational level is less than 0.05, suggesting a statistically significant relationship between these variables.

Step-by-step explanation:

In the analysis, the correlation coefficient of r = 0.207 indicates a weak positive linear relationship between the mother's educational level and the number of children their parents have. This value, which falls closer to zero than to ±1, signifies a relatively low degree of linear association between the variables. While it demonstrates a positive trend, the correlation isn't substantial, implying that other factors might contribute significantly to the number of children in families besides the mother's education.

The calculated P-value being less than the standard significance level of 0.05 indicates statistical significance in the relationship between the mean number of children and the mother's educational level. This result implies that there's likely a genuine association between these variables. Therefore, the null hypothesis of independence between the two variables is rejected, suggesting that mother's educational attainment might have an influence on the number of children in families.

Considering the weak correlation and statistical significance found, a potential scatterplot that might make the analyses inappropriate could depict highly scattered points with no discernible pattern or direction. This scatterplot could mislead interpretations, obscuring the weak but significant relationship found in the data.