Answer:
Explanation:
Using the given data, we can calculate the regression equation as follows:
First, calculate the means of x and y:
mean(x) = (7.3 + 23.6 + 4.3 + 22.1 + 7.1 + 3.9 + 21.2) / 7 = 13.1143
mean(y) = (22.9 + 74.1 + 13.5 + 69.4 + 22.3 + 12.3 + 66.6) / 7 = 42.3286
Then, calculate the sample standard deviations of x and y:
s_x = sqrt(((7.3 - 13.1143)^2 + (23.6 - 13.1143)^2 + ... + (21.2 - 13.1143)^2) / 6) = 7.8836
s_y = sqrt(((22.9 - 42.3286)^2 + (74.1 - 42.3286)^2 + ... + (66.6 - 42.3286)^2) / 6) = 25.3612
Calculate the sample covariance:
cov(x,y) = ((7.3 - 13.1143)(22.9 - 42.3286) + (23.6 - 13.1143)(74.1 - 42.3286) + ... + (21.2 - 13.1143)*(66.6 - 42.3286)) / 6 = 194.4714
Calculate the slope of the regression line:
b = cov(x,y) / s_x^2 = 194.4714 / (7.8836^2) = 3.1343
Calculate the intercept of the regression line:
a = mean(y) - b * mean(x) = 42.3286 - 3.1343 * 13.1143 = 1.1639
Therefore, the regression equation is:
y = 1.1639 + 3.1343x
To find the best predicted circumference for a diameter of 42.5 cm, we substitute x = 42.5 into the regression equation:
y = 1.1639 + 3.1343(42.5) = 133.262
The predicted circumference is 133.262 cm.
To determine whether the predicted value is significantly different from the actual circumference of 133.5 cm, we can calculate the standard error of the estimate:
s_e = sqrt(((22.9 - 133.262)^2 + (74.1 - 133.262)^2 + ... + (66.6 - 133.262)^2) / 5) = 17.2538
Using a significance level of 0.05 and a t-distribution with 5 degrees of freedom (n - 2), the critical t-value is 2.571. The margin of error is then:
ME = t_critic * s_e = 2.571 * 17.2538 = 44.3726
Since the actual circumference of 133.5 cm falls within the margin of error (133.262 - 44.3726 to 133.262 + 44.3726), we can conclude that the predicted value is not significantly different from the actual circumference. Therefore, the answer is:
b) Even though 42.5 cm is beyond the scope of the sample diameters, the predicted value yields the actual circumference.