Behavioral economists were interested whether high-priced fake medicine is more effective in lowering blood pressure than low-priced fake medicine. 141 individuals with high blood pressure were selected to participate in the experiment. The subjects were given placebos with varying prices and blood pressure was measured in each subject before and after the study period. 1. In the above scenario, which of the following variables is the predictor and which one is the response? a. Price of the placebo b. Change in blood pressure. 2. Write out the simple linear regression model equation. 3. The correlation between the price and the blood pressure change was estimated as -0.4228474. Interpret the coefficient. 4. The standard deviation of placebo price is $457.2175, and the standard deviation of the blood pressure change is 11.65404 mmHg. What is the value of the estimated slope parameter in the simple linear regression model? Interpret the slope parameter. 5. The standard error of the slope parameter is 0.001959. Test, whether the slope parameter is significantly different from 0. Write out all steps of hypothesis testing. 6. Is the true correlation coefficient significantly different from zero? Why or why not? 7. What is the value of the coefficient of determination (R^2) in this model? Interpret the coefficient. 8. What is your conclusion? Is fake medicine price associated with the medicine\'s effectiveness? If yes, how? 9. [UPDATE]: The estimated intercept of the simple linear regression model is -3.328. What change in blood pressure do you predict for an individual taking a placebo that costs: a. $100 b. $1,000 c. $10,000 Hint: See the chart on the next page. 10. Construct confidence intervals for the predictions above. Note that the residual standard error sigma = 10.6, the mean placebo price in the sample is $835.8865, and sigma(xi - x)^2 = 29,266,700. Interpret the constructed prediction intervals. Consider the level of significance alpha = 0.05. You may assume that all assumptions of the simple linear model were satisfied.
1. Predictor (independent variable) - Price of Placebo
Response (dependent variable) - Change in Blood pressure
2. Linear regression:
Y = a. X + b where
Y: Change in blood pressure, X: Price of placebo & a: Slope coefficient, b: Y-coefficient
3. This coefficient means that if price increases by 1 unit, blood pressure decreases by 0.4228474 units.
4. Estimated Slope Parameter, a = (Correlation Coefficient x S.D. of dependent variable) / S.D. of independent variable
= (0.4228474 x 11.65404) / 457.2175 = 0.0108
5. Hypotheses:
Null Hypothesis H0: a = 0
Alternate Hypothesis H1: a not equal to 0
Level of significance 5%
We need to perform a t-test for this hypothesis testing.
Calculated t-statistic = (a - 0) / Standard Error = 0.0108 / 0.001959 = 5.513
degree of freedom (n - 2), or df = 139 (since n = 141)
This is a 2-tailed test. The critical t-value corresponding to df 139 & level of significance 5% is 1.9772. Since the calculated t-statistic 5.513 > critical t-value 1.9772, we accept the null hypothesis, that is,Slope parameter is not significantly different from 0.
6. Skipped - not sufficient data
7. R2 = Square of correlation coefficient = (0.4228474)2 = 0.1788
This means that, the simple regression model explains 17.88% of the variance in Y.
8. Looking at the R2 value, it is unlikely that there is significant correlation between placebo price and medicinal effectiveness.
9. Estimated intercept b = - 3.328, so our regression equation becomes
Y = a.X + b = 0.4228474 X - 3.328
(a) Y = (0.4228474) x 100 - 3.328 = $38.96
(b) Y = (0.4228474) x 1000 - 3.328 = $419.52
(c) Y = (0.4228474) x 10000 - 3.328 = $4225.15
10. Skipped - too many questions clubbed inside this question
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