IQ scores were obtained from randomly selected couples.For 20 such pairs of people, the linear correlation coefficient is 0.881 and the equation of
the regression line is y= -6.93+ 10.8x,
where x represents the IQ score of the husband. Also, the 20x values have a mean of 105.67 and the 20y values have a mean
of 106.9. What is the best predicted IQ of the wife, given that the husband has an IQ of 110? Use a significance level of 0.05.
To find the best predicted IQ of the wife, given that the husband has an IQ of 110, we can use the equation of the regression line.
The equation of the regression line is given as y = -6.93 + 10.8x, where x represents the IQ score of the husband.
Let's substitute the value x = 110 into the equation to find the predicted IQ of the wife (y).
y = -6.93 + 10.8(110)
y = -6.93 + 1188
y = 1181.07
Therefore, the best predicted IQ of the wife, given that the husband has an IQ of 110, is 1181.07.
To determine if this prediction is statistically significant, we need to check the significance level of 0.05. We can do this by conducting a hypothesis test.
The null hypothesis (H0) states that there is no significant relationship between the husband's IQ and the wife's IQ.
The alternative hypothesis (H1) states that there is a significant relationship between the husband's IQ and the wife's IQ.
To perform this test, we need to calculate the coefficient of determination, r-squared (r²).
r-squared = (correlation coefficient)²
r-squared = (0.881)²
r-squared = 0.776161
Now, we can compare the r-squared value to the critical value for significance level 0.05.
If the r-squared value is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
To find the critical value, we need to find the degrees of freedom, which is equal to the number of data points minus 2.
Degrees of freedom = 20 - 2 = 18
Looking up the critical value for significance level 0.05 and 18 degrees of freedom in the t-distribution table, we find a critical value of approximately 2.101.
Since r-squared (0.776161) is greater than the critical value, we reject the null hypothesis.
This means that there is a significant linear relationship between the husband's IQ and the wife's IQ.
Therefore, the best predicted IQ of the wife (1181.07) is statistically significant given that the husband has an IQ of 110.