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Understanding the correlation between two variables, such as infant weight and chest size, is critical in many research areas, particularly in health sciences where it can indicate if and how strongly the pairs of variables are related. The Pearson correlation coefficient, denoted as r, is a measure of the linear correlation between two variables. To calculate r, you need to first know the individual means of both variables. Next, you determine the deviation of each variable's individual data points from their respective means, square these deviations, and sum them up.

More formally, you calculate the product of the deviations from the means for the two variables and sum this over all data points. Afterwards, r is found by dividing the sum of these products by the square root of the product of the sum of squared deviations for each variable. This result can range from -1 to +1, with -1 meaning a perfect inverse relationship, +1 meaning a perfect direct relationship, and 0 indicating no linear relationship.

The formula for r could be expressed as:

Hypothesis testing is vital for making inferences about populations from sample data. It allows us to determine if there is enough evidence to support a specific claim or hypothesis about a population parameter.

For the given exercise, the specific hypothesis we are testing is whether the correlation coefficient in the population is zero (the null hypothesis H₀: 蟻 = 0) against the alternative hypothesis that it is greater than zero (H_a: 蟻 > 0). The process involves calculating a test statistic from the sample data which, under the null hypothesis, should follow a known probability distribution. In the case of correlation coefficients, the test statistic follows a t-distribution.

To test this hypothesis, we use the calculated r to obtain the test statistic using the formula:
\( t = r \times \frac{\text{\textsqrt}{n - 2}}{1 - r^2} \)
where n is the number of data pairs. This t-statistic is then compared to a critical value from the t-distribution table at a specified level of significance. If the t-statistic is greater than the critical value, the null hypothesis is rejected, suggesting a significant relationship exists.

In simpler terms, we're trying to find out if the observed relationship between weights and chest sizes in babies could reasonably occur if there was actually no relationship in the wider population of babies.

Once we've established that a relationship exists between two variables, it's often helpful to quantify how much of the variation in one variable can be 'explained' by variations in the other. This concept is captured by the coefficient of determination, which is denoted as R² and is the square of the Pearson correlation coefficient, r.

For the scenario in question, by squaring the computed value of r, we obtain R², which tells us what proportion of the total variability in the chest size of infants can be explained by their weights. Multiplying R² by 100 gives us the percentage of variation in chest size that can be accounted for by the variation in weight.

For example, if r was calculated to be 0.9, R² would be 0.81, indicating that 81% of the variability in chest size is related to weight. This means chest size variability can be largely predicted by how much the infant weighs, which might be crucial for pediatric health assessments.