Under the usual regression testing assumptions, a coefficient t test with null hypothesis beta_j = 0 assesses whether:
Correct Answer
A) There is statistical evidence against a zero conditional coefficient
For H0: beta_j = 0, the usual coefficient t statistic is estimated coefficient divided by its standard error. Under the stated model and test assumptions, it assesses evidence against that zero-coefficient null. It does not test residual normality, sample adequacy or whether a model explains most variation. Statistical significance alone establishes neither causation nor a practically material appraisal adjustment.
Why This Is the Correct Answer
Option A is correct because the coefficient t test compares the estimated coefficient to its standard error to judge whether the data provide statistical evidence against the null that the conditional coefficient is zero. It is a test about one coefficient, given the other variables in the model. A significant result does not by itself prove causation or a practically material adjustment.
Why the Other Options Are Wrong
Option B: The model as a whole explains most of the variation in the response variable
Option B describes whether the model as a whole explains most of the variation in the response, which is the territory of R-squared and the overall F test. A single coefficient t test examines one coefficient, not the overall explanatory power of the model.
Option C: The sample size is adequate for the intended use
Option C claims the test shows the sample size is adequate for the intended use. Sample size affects the standard error, but the coefficient t test does not judge whether the sample is adequate for an assignment.
Option D: The residuals follow a normal distribution
Option D says the test checks whether the residuals follow a normal distribution. Residual normality is one of the assumptions behind the t test, and it is checked with residual plots or separate normality tests, not with the coefficient t test itself.
t equals signal over noise
Think of the t statistic as the coefficient (signal) divided by its standard error (noise). A big ratio means the coefficient stands out from zero.
How to use: Use this when a question asks what a coefficient t test measures, to keep it tied to one coefficient against zero.
Exam Tip
Match each regression test to its target: the coefficient t test addresses one coefficient, while R-squared and the F test describe the model as a whole.
Common Mistakes to Avoid
- -Treating a significant t value as proof that the variable causes the price difference
- -Confusing the coefficient t test with the overall model fit measured by R-squared
- -Assuming the t test checks the normality of the residuals
Concept Deep Dive
Analysis
This question tests what a single-coefficient t test in regression actually examines. The null hypothesis beta_j = 0 says that, holding the other predictors constant, variable j has no linear association with the response. The t statistic divides the estimated coefficient by its standard error, so a large absolute value is evidence against that zero-coefficient null under the model assumptions.
Background Knowledge
In multiple regression each coefficient estimates the change in the response associated with a one-unit change in that predictor, holding the others constant. Its reliability can be gauged from its standard error, t value, significance level and confidence interval. Statistical significance is distinct from practical importance and from causation.
Real-World Application
An appraiser builds a regression of sale prices on living area, lot size and garage count. The garage coefficient has a small t value, so there is little statistical evidence that it differs from zero once size and lot are controlled. The appraiser still considers whether paired sales or other market evidence support a garage adjustment.
More Statistics Questions
Sale prices are $270,000, $290,000, $300,000, $310,000, $330,000. Treat the listed observations as the complete population and divide by n when computing the standard deviation. Report to the nearest dollar.
A sales sampling frame systematically excludes a relevant part of the target market. What is the principal risk?
An appraiser estimates a market-wide price trend using only sales data from one brokerage that specializes in luxury homes, without checking how other segments are represented. What is the principal concern?
Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
The base-period median sale price is $400,000 and the current-period median is $460,000. Compute the price relative defined as current ÷ base, as a decimal to three places.
The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. Calculate COD as [average absolute deviation from the median ÷ median]×100. For even n, average the two middle ratios. Report the COD as a percentage to two decimals.
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