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A study was performed in South Wales to investigate the degree of heredity in blood pressure. A group of 623 individuals (called propositii) over 5 years of age were randomly selected in two geographically defined populations in South Wales (Miall and Oldham [20]). The individuals and their first-degree relatives participated in the survey. The data from this study are provided in the dataset WALES. DAT, at www.cengagebrain.com, with documentation in WALES.DOC. They had their blood pressure measured by one observer in their homes at baseline and at 3 follow-up exams. A correlation coefficient was computed based on 248 families between the blood pressure of the propositus and the blood pressure of their fathers. The correlation coefficient for systolic blood pressure (SBP) was 0.313. Test whether this correlation coefficient is significantly different from \(0,\) and report a \(p\) -value (two-tailed). Hint: Assume that a \(t\) distribution with \(>200\) df is approximately normal.

Step by step solution

01

Formulate Hypotheses

We want to test whether the correlation coefficient between the systolic blood pressure of the propositus and their fathers is significantly different from 0. The null hypothesis \(H_0\) is that the true correlation \(\rho = 0\), and the alternative hypothesis \(H_a\) is that \(\rho eq 0\).

02

Determine the Test Statistic

To test the significance of the correlation coefficient \(r = 0.313\), we use the test statistic formula for correlation: \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \] where \(n\) is the number of data pairs. Here \(n = 248\).

03

Calculate the Test Statistic

Substitute \(r = 0.313\) and \(n = 248\) into the formula: \[ t = \frac{0.313 \sqrt{248-2}}{\sqrt{1-0.313^2}} \] Simplifying: \[ t = \frac{0.313 \times 15.716}{\sqrt{0.902}} \approx 5.064 \]

04

Determine the Critical Value and Compare

With \(n-2 = 246\) degrees of freedom, we can use the normal distribution approximation. For a two-tailed test at \(\alpha = 0.05\), the critical \(t\) value is \(\pm 1.96\). Our calculated \(t = 5.064\) is greater than 1.96, indicating a significant result.

05

Calculate the p-value

Using the normal distribution approximation, calculate the \(p\)-value for \(t = 5.064\). The two-tailed \(p\)-value is less than 0.05. Typically, a \(p\)-value this size with a large \(t\)-statistic indicates significance far less than 0.05.

06

Conclusion

Since the \(p\)-value is less than 0.05, we reject the null hypothesis. Therefore, the correlation coefficient is significantly different from 0, suggesting a significant relationship between the blood pressure of the propositus and their fathers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), is a statistical measure that describes the degree and direction of a linear relationship between two variables. In our example, it measures the strength of the relationship between the systolic blood pressure of propositus individuals and their fathers. If \( r \) is positive, it indicates a direct relationship, meaning as one variable increases, the other tends to increase as well.
A correlation coefficient of 0.313 suggests a moderate positive correlation, suggesting that as the father’s blood pressure increases, the propositus blood pressure tends to increase, although the relationship is not strong. The value ranges from -1 to 1, where -1 denotes a perfect negative linear relationship, 0 denotes no linear relationship, and 1 denotes a perfect positive linear relationship.

• A correlation close to 0 suggests weak or no association between the variables.
• The closer the value is to 1 or -1, the stronger the relationship.

Hypothesis Testing Basics

Hypothesis testing is a critical process in statistics used to decide whether there is enough evidence to reject a null hypothesis. The null hypothesis (\( H_0 \)) often represents the status quo or a statement of no effect. In contrast, the alternative hypothesis (\( H_a \)) reflects the opposite statement.For the exercise in question, we wanted to determine whether the correlation between systolic blood pressure of propositus individuals and their fathers was significantly different from 0. Thus, our hypotheses were:

• \( H_0 \): \( \rho = 0 \) (The true correlation is zero, suggesting no relationship.)
• \( H_a \): \( \rho eq 0 \) (There exists a difference from zero, suggesting a relationship.)

A significant test result supports the alternative hypothesis, suggesting the correlation is indeed different from zero and noteworthy.

The Role of the t-Distribution

The t-distribution is essential in hypothesis testing, especially with smaller sample sizes. It accounts for additional variability expected with smaller datasets and is similar in shape to the normal distribution but has heavier tails. In our case, the t-distribution is used to evaluate the significance of our correlation coefficient.When the sample size exceeds 30, the t-distribution approximates the normal distribution. For our study with a sample size of 248, the t-distribution approaches normality, making it a suitable choice. The formula \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \]calculates the test statistic used to determine if our correlation significantly deviates from zero. This takes into account the sample size, ensuring the decision regarding the null hypothesis is robust.

Insight Into Systolic Blood Pressure

Systolic blood pressure (SBP) measures how much pressure your blood is exerting against your artery walls when the heart beats. It's presented as the higher number in a blood pressure reading. Regular monitoring of SBP is crucial, as elevated levels can indicate health issues like hypertension. In the given study, researchers were interested in SBP to explore genetic influences. Seeing notable correlations between the SBP of propositus individuals and their fathers can indicate hereditary patterns in blood pressure, helping in understanding the potential genetic underpinning of blood pressure levels. Understanding the factors influencing SBP can aid in managing heart health and preventing cardiovascular diseases.

Conducting Significance Testing

Significance testing is a systematic way to test hypotheses by evaluating the probability of observing the data if the null hypothesis were true. The primary aim is to determine if the results observed (like our correlation of 0.313) could be due to chance.A computed \( t \)-statistic is compared to a critical value derived from the t-distribution; if it exceeds the critical value, the null hypothesis could be rejected. In our study, \( t = 5.064 \) exceeded the critical value \( 1.96 \), suggesting the correlation was significantly different from zero. The two-tailed \( p \)-value was also less than 0.05. These facts led us to reject the null hypothesis, supporting a significant relationship between the blood pressures of father-propositus pairs. Statistical significance doesn't imply practical significance but indicates strong evidence against the null hypothesis, reinforcing our understanding of blood pressure's hereditary influence.