Choose the right answer, and provide the reasons and calculations.
A palmist claims that he can accurately predict the age at death (y) of an individual from the length
(centimeters) of age line (x) on the right hand of the individual. A random sample of 20 dead
individuals from the database maintained by the palmist gave SSyy=2160 and SSxx=28.and the point
prediction ŷ = 45 + 3.5x to predict the age at death. The claim of the palmist can be accepted only at
i. α=0.01 ii. α=0.05 iii. α=0.1 iv. None of them
To determine whether the claim of the palmist can be accepted, we need to perform a significance test using the given information.
Given:
SSyy = 2160
SSxx = 28
Point prediction: ŷ = 45 + 3.5x
To perform the significance test, we need to calculate the coefficient of determination (R^2) and use it to determine the significance level.
The coefficient of determination (R^2) is calculated as follows:
R^2 = SSyy / SSyy_total
where SSyy_total is the total sum of squares for the response variable (y). In this case, we don't have the value of SSyy_total, so we need to estimate it using the sample size (n) and the variability of the age line (SSxx).
We can estimate SSyy_total as:
SSyy_total = SSyy + ŷ^2 * n
Calculating the estimated value of SSyy_total:
SSyy_total = 2160 + (45 + 3.5x)^2 * 20
Next, we can calculate R^2 by dividing SSyy by SSyy_total.
R^2 = SSyy / SSyy_total
Finally, the significance level (α) can be determined using the test statistic F, which is calculated using R^2:
F = (R^2 / q) / [(1 - R^2) / (n - p - 1)]
where q = p = 1 (because we have only one predictor variable, x) and n = 20 (sample size).
Once we have calculated the test statistic F, we can compare it to the F distribution table for the given significance levels (α) to determine whether the claim can be accepted.
So, using the calculations above, we can determine the significance level (α) that would lead to accepting the claim of the palmist.