Suppose IQ scores were obtained from randomly selected twins. For 20 such pairs of people, the linear correlation coefficient is 0.876 and the regression line is y-hat = 11,76 + 0.89x, where x represents the IQ score of the twin born second. Also,the 20x values have a mean of 101.23 and the 20 y values have a mean of 101.7. What is the best predicted IQ of the twin born first , given that the twin born second has an IQ of 97? Use a significance level of 0.05.
To find the best predicted IQ of the twin born first, given that the twin born second has an IQ of 97, we can use the equation of the regression line.
The regression line is given as:
y-hat = 11.76 + 0.89x
In this equation, y-hat represents the predicted IQ of the twin born first, and x represents the IQ score of the twin born second.
We are given that the twin born second has an IQ of 97, so we can substitute this value into the equation:
y-hat = 11.76 + 0.89(97)
= 11.76 + 86.33
= 98.09
Therefore, the best predicted IQ of the twin born first, given that the twin born second has an IQ of 97, is approximately 98.09.
To understand how we arrived at this answer and what significance level means, here is an explanation of the steps we took:
Step 1: Linear correlation coefficient
The linear correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. In this case, it tells us how closely related the IQ scores of the twins are. The correlation coefficient of 0.876 indicates a fairly strong positive correlation between the IQ scores of the twins.
Step 2: Regression line
The regression line represents the relationship between the IQ scores of the twins. It helps us predict the IQ score of one twin based on the IQ score of the other twin. The equation of the regression line is determined using statistical techniques, such as least squares regression. In this case, the equation is y-hat = 11.76 + 0.89x, where x represents the IQ score of the twin born second.
Step 3: Calculating the predicted IQ
To find the predicted IQ of the twin born first, we substitute the IQ score of the twin born second (97) into the regression equation.
Step 4: Significance level
The significance level refers to the probability threshold used to determine statistical significance. In this case, a significance level of 0.05 means that there is a 5% chance of making a Type I error, which is rejecting the null hypothesis when it is actually true. It represents the level of confidence we have in our results.