Question

An article in the Journal of Sound and Vibration (Vol. 151, 1991, pp. 383-394) described a study investigating the relationship between noise exposure and hypertension. The following data are representative of those reported in the article. a. Create a scatter diagram of y (blood pressure rise in millimeters of mercury) versus x (sound pressure level in decibels). Does a simple linear regression model seem reasonable in this situation? b. Test the hypothesis H0​ : β1​=0 using alpha =0.05. What do you determine about the slope of the regression line? c. What is the estimate of σ2 ? d. Find the predicted mean rise in blood pressure level associated with a sound pressure level of 85 decibels. e. Would it be reasonable to estimate the rise in blood pressure level associated with a sound pressure level of 120 decibels? Explain. f. What proportion of total variability in blood pressure rise is accounted for by sound pressure level?

Answer 1

a) A scatter diagram of blood pressure rise (y) versus sound pressure level (x) should be created to visually assess the relationship. Whether a simple linear regression model is reasonable depends on the pattern observed in the scatter plot.

b)The hypothesis test for β =0 is conducted using alpha = 0.05 to determine if the slope of the regression line significantly differs from zero. The result of the test informs whether there is a statistically significant linear relationship between sound pressure level and blood pressure rise.

c) The estimate of the variance of the errors in the regression model, is obtained to understand the variability in the blood pressure rise not explained by the linear relationship with sound pressure level.

d)The predicted mean rise in blood pressure level associated with a sound pressure level of 85 decibels is calculated using the regression equation.

e) The reasonableness of estimating the rise in blood pressure level associated with a sound pressure level of 120 decibels is discussed, considering the range of observed data and the potential for extrapolation beyond the observed values.

f. The proportion of total variability in blood pressure rise accounted for by sound pressure level is determined using the coefficient of determination , providing insight into the strength of the linear relationship between the variables.

Step-by-step explanation:

a. The scatter diagram visually assesses the pattern between blood pressure rise and sound pressure level. If the points exhibit a linear trend, a simple linear regression model is reasonable. If no clear pattern is discernible, other models or factors may need consideration.

b. The hypothesis test β =0 evaluates whether the slope of the regression line is significantly different from zero. This test helps determine the significance of the relationship between sound pressure level and blood pressure rise.

c. The estimate of provides an understanding of the variability in blood pressure rise not explained by the linear relationship. It is a crucial measure for assessing he goodness of fit of the regression model.

d. Calculating the predicted mean rise in blood pressure associated with a sound pressure level of 85 decibels offers a practical interpretation of the regression model within the observed range of sound pressure levels.

e. Extrapolating the regression model to a sound pressure level of 120 decibels requires caution. The reasonableness of this estimation is discussed, considering potential limitations and uncertainties associated with extrapolation.

f. The proportion of total variability accounted for by sound pressure level, expressed by , provides insight into how well the linear model explains the variability in blood pressure rise. A higher indicates a stronger linear relationship.