 # t-stats

I am sure I am missing/forgetting something very basic and fundamental here. Forgive me if this sounds like a stupid query. My question is - Why is the calculated test statistic coefficient/standard error? Henry Hilton, CFA, is undertaking an analysis of the bicycle industry. He hypothesizes that bicycle sales (SALES) are a function of three factors: the population under 20 (POP), the level of disposable income (INCOME), and the number of dollars spent on advertising (ADV). All data are measured in millions of units. Hilton gathers data for the last 20 years and estimates the following equation (standard errors in parentheses): SALES = α + 0.004 POP + 1.031 INCOME + 2.002 ADV (0.005) (0.337) (2.312) The critical t-statistic for a 95 percent confidence level is 2.120. Which of the independent variables is statistically different from zero at the 95 percent confidence level? A) INCOME and ADV. B) INCOME only. C) POP and ADV. D) ADV only. Your answer: A was incorrect. The correct answer was B) INCOME only. The calculated test statistic is coefficient/standard error. Hence, the t-stats are 0.8 for POP, 3.059 for INCOME, and 0.866 for ADV. Since the t-stat for INCOME is the only one greater than the critical t-value of 2.120, only INCOME is significantly different from zero.

Check on CFAi text, volume 1, page 250-251 about hypothesis testing.

Thanks map1… great to have you back on the forums.

the t-stat is defined as coefficient/standard error. It has, I believe (n-k+1) where n is the number of observations and k is the number of independent variables. In this case, there are 20-4 = 16 degrees of freedom. You need to know the degrees of freedom in cases they ask for it. But for significance, memorize the critical values for 1,5, and 10% significance from a z-statistics (i.e. a standard normal). This saves time on the exam since in most cases (particularly for n>20 or 30), the values for the critical t become pretty close to those for a normal. So, you don;t have to look them up.

SALES = α + 0.004 POP + 1.031 INCOME + 2.002 ADV (0.005) (0.337) (2.312) t-stat POP = .004 / .005 = 0.8 t-stat INCOME = 1.031 / .337 = 3.06 t-stat ADV = 2.002 / 2.312 < 1 t-crit = 2.12 so only Income is > 2.12 and is significant (By significant - they mean significantly different from ZERO – so actually .004 / .005 should be read as = (0.004 - 0) / 0.005

It’s kind of an odd question, because the t-stat is only for testing whether an individual variable is statistically different from 0 but then they show answers giving two variables. Doing the t-test twice is just not a valid test to determine whether two variables are different from 0.