[DATA] Height versus Head Circumference [See Problem 13 in Section 12.3] A pediatrician wants to determine the relation that may exist between a child’s height and head circumference. She randomly selects 11 children from her practice, measures their heights and head circumferences, and obtains the following data:

d. State your conclusion to the hypotheses from part (b).

Verified step by step guidance

1. Review the hypotheses stated in part (b). Typically, for testing the relationship between two variables (height and head circumference), the null hypothesis \(H_0\) states that there is no linear relationship (correlation) between the variables, i.e., the population correlation coefficient \(\rho = 0\). The alternative hypothesis \(H_a\) states that there is a linear relationship, i.e., \(\rho \neq 0\).
2. Recall the test statistic used for testing the correlation coefficient, which is based on the sample correlation coefficient \(r\) calculated from the data. The test statistic is given by: \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \] where \(n\) is the sample size (number of paired observations).
3. Determine the degrees of freedom for the test, which is \(df = n - 2\). In this case, since there are 11 children, \(df = 11 - 2 = 9\).
4. Compare the calculated test statistic \(t\) to the critical value from the \(t\)-distribution with 9 degrees of freedom at the chosen significance level (commonly \(\alpha = 0.05\)). Alternatively, find the p-value corresponding to the test statistic.
5. State the conclusion: If the test statistic falls in the rejection region or if the p-value is less than \(\alpha\), reject the null hypothesis and conclude that there is evidence of a linear relationship between height and head circumference. Otherwise, fail to reject the null hypothesis and conclude that there is not enough evidence to support a linear relationship.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. It involves setting up hypotheses, calculating a test statistic, and comparing it to a critical value or p-value to draw conclusions about the population.

Correlation and Linear Relationship

Correlation measures the strength and direction of a linear relationship between two variables. Understanding correlation helps determine if changes in one variable, like height, are associated with changes in another, like head circumference, which is essential for interpreting the pediatrician's data.

Paired Data and Scatterplots

Paired data consists of two related measurements taken from the same subjects, such as height and head circumference for each child. Visualizing paired data with scatterplots helps identify patterns or relationships, which is crucial before performing statistical tests or drawing conclusions.