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11 questions · 1 formulas

t-statistic

t = (b − β_H0) / SE(b)

Trap: Usually β_H0 = 0. Critical 1.645 / 1.96 / 2.576 for 10/5/1%.

Hypothesis TestingE-level★ High priority — official Canvas example

sqft coefficient = 144.862 and SE = 6.346. What is the t-statistic?

→ B) 22.827

Canvas Example MCQ — Q59

  • A) 0.044
  • B) 22.827
  • C) 2.283
  • D) 0.000
  • E) None of A-D is close; write the correct answer.

t = 144.862 / 6.346 ≈ 22.827.

Rule: For H0: β=0, t = coefficient / standard error.

Hypothesis TestingE-level★ High priority — official Canvas example

What is the general form of a t-statistic?

→ C) (estimator - hypothesized value) / standard error of estimator

Canvas Example MCQ — Q60

  • A) (estimate - hypothesized value) / standard error of the estimate
  • B) estimator / standard error of estimator
  • C) (estimator - hypothesized value) / standard error of estimator
  • D) (estimator - hypothesized value) / (standard error of estimator / sqrt(n))

A t-statistic measures how many standard errors the estimate lies from the null value.

Rule: General t = (estimate - null value)/SE.

Hypothesis TestingE-level★ High priority — official Canvas example

STR coefficient = -1.10 and |t| = 2.56. Approximate SE?

→ D) 0.43

Canvas Example MCQ — Q62

  • A) 0.25
  • B) 1.96
  • C) 0.650
  • D) 0.43

SE = 1.10/2.56 ≈ 0.4297.

Rule: Same t-stat rearrangement applies in multiple regression.

Hypothesis TestingE-level★ High priority — official Canvas example

Correct decision rule for a two-sided test at 5%?

→ B) Reject H0 if p < 0.05.

Canvas Example MCQ — Q69

  • A) Reject H0 if p > 0.05.
  • B) Reject H0 if p < 0.05.
  • C) Reject H0 only if the coefficient is positive.
  • D) Reject H0 only if the t-statistic is positive.

The p-value is the smallest significance level at which H0 would be rejected.

Rule: Reject when p is smaller than α.

Normal DistributionD–A levelVery high priority

The t-statistic for the ln(SQFT) (see ln_sqft) is closest to_________.

→ C) 56.99

Old Exam — Oct 2021, Q19

Given: ln(SQFT) coefficient=.892479, SE=.015659.

  • A) 5.69
  • B) 0.00
  • C) 56.99
  • D) Not possible to compute given the information in the summary output.

t=b/SE=0.892479/0.015659≈56.99 -> C.

Rule: Identify model units and form before calculating. Never confuse coefficient size, significance, R2 and prediction accuracy.

Omitted Variable BiasD–A levelHigh priority

The STR coefficient falls in magnitude from -2.280 (Model 1, no controls) to -1.014 (Model 5, with controls), while its sign stays negative. What is the best interpretation?

→ A) This pattern is consistent with omitted variable bias in Model 1, but the coefficient change alone does not prove STR's effect is causal

Computer Lab 4 — Exercise 2, Task 6 (comparison)

  • A) This pattern is consistent with omitted variable bias in Model 1, but the coefficient change alone does not prove STR's effect is causal
  • B) This proves Model 1 was causal and Model 5 is biased
  • C) This proves STR has no real relationship with test scores
  • D) This change in magnitude automatically makes the Model 5 coefficient statistically insignificant

A shrinking coefficient magnitude after adding plausible confounders is consistent with OVB in the simpler model, but establishing causality requires more than observing a coefficient change; significance must be checked separately using SEs/t-stats.

Rule: Coefficient movement across specifications is suggestive of OVB, not conclusive proof of causality or (in)significance.

Hypothesis TestingD–A levelHigh priority

With df = n − k − 1 = 420 − 4 − 1 = 415, the two-sided 5% critical value is |t| = 1.9657. The reported t-statistic for el_pct is -3.581. Should H0: β_el_pct = 0 be rejected at the 5% level?

→ A) Yes, because |-3.581| = 3.581 > 1.9657

Computer Lab 4 — Exercise 3, Task 2c

  • A) Yes, because |-3.581| = 3.581 > 1.9657
  • B) No, because 3.581 < 1.9657 is false reasoning so fail to reject
  • C) Cannot be determined without the p-value
  • D) No, because t-statistics cannot be used for two-sided tests

Reject H0 when |t| exceeds the critical value: |-3.581| = 3.581, which exceeds 1.9657, so H0 is rejected at the 5% level.

Rule: Two-sided rule: reject H0 if |t| > t_critical.

Hypothesis TestingE-levelHigh priority

The reported t-statistic for calw_pct is -0.816, and the two-sided critical value at 5% (df=415) is 1.9657. What is the correct conclusion?

→ A) Fail to reject H0, since |-0.816| = 0.816 < 1.9657

Computer Lab 4 — Exercise 3, Task 2c

  • A) Fail to reject H0, since |-0.816| = 0.816 < 1.9657
  • B) Reject H0, since the t-statistic is negative
  • C) Reject H0 at 5% but not at 1%
  • D) The test is inconclusive without more data

Since 0.816 is less than every critical value in the table (including the loosest, 1.9657 at 5%), H0 is not rejected at any conventional significance level.

Rule: A negative t-statistic is judged by its absolute value against the critical value, not its sign alone.

Hypothesis TestingE-levelHigh priority

For the same α and the same two-sided test specification, do the p-value method and the critical-value (t-statistic) method always produce the same reject/fail-to-reject decision?

→ A) True — they are equivalent decision rules for the same test

Computer Lab 4 — Exercise 3, Task 2d

  • A) True — they are equivalent decision rules for the same test
  • B) False — the t-statistic method is more conservative
  • C) False — the p-value method is only valid for one-sided tests
  • D) True, but only when the sample size exceeds 1000

Both approaches test the same null against the same alternative using the same α, so 'p-value < α' and '|t| > t_critical' always agree for a correctly specified two-sided test.

Rule: p-value method and critical-value method are mathematically equivalent given the same α and test.

Regression BasicsE-levelVery high priority

Switching from Excel's standard OLS output to Python's HC1 robust standard errors for the same ln(PRICE) regression, which outputs should remain identical, and which can change?

→ A) Coefficients and fitted values stay the same; standard errors, t-statistics and p-values can differ

Computer Lab 5 — Exercise 4, Robust SE

  • A) Coefficients and fitted values stay the same; standard errors, t-statistics and p-values can differ
  • B) Everything, including coefficients, changes
  • C) Only the R² changes
  • D) Coefficients change but standard errors stay identical

OLS point estimates and fitted values depend only on the OLS fitting procedure, which is unaffected by the choice of variance estimator; only the estimated standard errors (and resulting t-stats/p-values) can change with HC1.

Rule: HC1 robust SEs change only the measure of estimation uncertainty, not the OLS coefficients themselves.

ProbitE-levelVery high priority

In a Probit model P(ADJUST=1) = Φ[-7.0166 + 0.5301·FIXRATE - 0.2675·MARGIN + 0.0864·NETWORTH], with t-statistics of 2.09, -2.05, and 2.44 respectively and a two-sided 5% critical t-value of about 1.993 (df=74), which variables are statistically significant at the 5% level?

→ A) All three: FIXRATE, MARGIN, and NETWORTH

Computer Lab 6 — Exercise 3, Model 1 signs

  • A) All three: FIXRATE, MARGIN, and NETWORTH
  • B) Only NETWORTH
  • C) None are significant
  • D) Only FIXRATE and NETWORTH

All three absolute t-statistics (2.09, 2.05, 2.44) exceed the critical value of 1.993, so all three coefficients are statistically significant at the 5% level.

Rule: Compare |t| to the critical value for each coefficient individually to assess significance.