m09-parametric-tests

M09 – Parametric Tests: CFAI Practice Problems

Source: CFAI CFA1 Quant Practice 2026, pp.261–262 Back to module: M09 Glossary: M09 Terms

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Question 1

Batten is testing whether the correlation between Stellar Energy stock returns and the CPIENG energy index returns is significantly different from zero. The sample correlation is $r=-0.1452$ based on $n=248$ monthly observations. The critical value at the 0.05 significance level is $\pm 1.96$.

Batten should conclude that the relationship between Stellar Energy and CPIENG is:

• A. significant, because the test statistic falls outside the critical value bounds
• B. significant, because the test statistic has a lower absolute value than the critical value
• C. insignificant, because the test statistic falls outside the critical value bounds

Answer

A. significant, because the test statistic falls outside the critical value bounds

The test statistic for a hypothesis test of zero correlation, $H_0:\rho =0$, is:

$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$

Step 1 – Compute the test statistic:

$t=\frac{-0.1452\times \sqrt{248-2}}{\sqrt{1-(-0.1452{)}^2}}$

$t=\frac{-0.1452\times \sqrt{246}}{\sqrt{1-0.02108}}$

$t=\frac{-0.1452\times 15.684}{\sqrt{0.97892}}$

$t=\frac{-2.2773}{0.98941}\approx -2.302$

Step 2 – Compare to critical value:

$|t|=2.302>1.96=\text{critical value}$

The test statistic falls outside the bounds $[-1.96,+1.96]$, which is the rejection region.

Conclusion: Reject $H_0:\rho =0$. The correlation between Stellar Energy and CPIENG is statistically significant at the 5% level.

Why B is wrong: The absolute value of the test statistic (2.302) is greater than the critical value (1.96), not lower. If the test statistic had a lower absolute value than the critical value, we would fail to reject $H_0$.

Why C is wrong: Falling outside the critical value bounds means the result is significant (reject $H_0$), not insignificant. C correctly identifies the location but draws the wrong conclusion.

Economic interpretation: The negative correlation ($r=-0.1452$, $t=-2.30$) indicates a statistically significant (though weak) inverse relationship between Stellar Energy stock returns and CPIENG energy index returns. This may seem counterintuitive — one possible explanation is that Stellar uses significant energy as an input, so rising energy prices compress its margins and depress its stock price.

📖 Giải thích chi tiết

Ôn lại khái niệm: Để kiểm định correlation $\rho$ có khác 0 không, dùng test statistic: $t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$, phân phối t với $df=n-2$. Với $n=248$ lớn, critical value $\approx \pm 1.96$ (xấp xỉ z).

Tại sao A đúng: Tính $t=\frac{-0.1452\times \sqrt{246}}{\sqrt{1-0.0211}}=\frac{-0.1452\times 15.684}{0.9895}\approx -2.302$. Vì $|-2.302|=2.302>1.96$ (critical value), test statistic nằm ngoài bounds → reject $H_0:\rho =0$ → correlation là significant. Tại sao B sai: “Lower absolute value than the critical value” là điều kiện để fail to reject $H_0$ — ở đây $|t|=2.302>1.96$, tức là lớn hơn critical value, không phải nhỏ hơn. Tại sao C sai: C đúng khi nói test statistic “falls outside the critical value bounds”, nhưng lại kết luận “insignificant” — ngược lại! Nằm ngoài bounds = vùng rejection = significant.

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Question 2

Which of the following is correct about the chi-square test of independence?

• A. It has a one-sided rejection region
• B. The null hypothesis is that the two groups are dependent
• C. When there are two categories, each with three levels, there are six degrees of freedom

Answer

A. It has a one-sided rejection region

The chi-square test of independence uses the statistic:

${\chi }^2={\sum }_{i,j}\frac{(O_{ij}-E_{ij}{)}^2}{E_{ij}}$

Since this statistic involves squared deviations, it is always non-negative (${\chi }^2\ge 0$). Large values of ${\chi }^2$ indicate deviation from independence. Therefore, the rejection region is always in the right tail only — this is a one-sided (right-tailed) test.

Why B is wrong: The null hypothesis in a chi-square test of independence is that the two categorical variables are independent (not dependent). The alternative is that they are dependent:

$H_0:\text{The two variables are independent}$ $H_a:\text{The two variables are dependent}$

Why C is wrong: The degrees of freedom for a chi-square test of independence in a contingency table are:

$df=(r-1)(c-1)$

where $r$ = number of rows (levels of first variable) and $c$ = number of columns (levels of second variable).

For two categories each with three levels (a $3\times 3$ table):

$df=(3-1)(3-1)=2\times 2=4$

Not 6. The value 6 would be incorrect — it may arise from multiplying $3\times 2=6$ (confusing number of levels with degrees of freedom).

Degrees of freedom summary:

Table dimensions	$df$
$2\times 2$	$(2-1)(2-1)=1$
$2\times 3$	$(2-1)(3-1)=2$
$3\times 3$	$(3-1)(3-1)=4$
$3\times 4$	$(3-1)(4-1)=6$

📖 Giải thích chi tiết

Ôn lại khái niệm: Chi-square test of independence kiểm định xem hai biến categorical có độc lập nhau không. Test statistic ${\chi }^2=\sum \frac{(O-E{)}^2}{E}$ luôn $\ge 0$ (bình phương), nên rejection region chỉ ở đuôi phải. Degrees of freedom: $df=(r-1)(c-1)$.

Tại sao A đúng: Vì ${\chi }^2$ luôn dương (bình phương) và giá trị lớn mới cho thấy sự phụ thuộc, rejection region luôn ở đuôi phải — đây là one-sided test (right-tail). Đây là đặc điểm cốt lõi phân biệt chi-square với t-test hay z-test (có thể two-tailed). Tại sao B sai: $H_0$ trong chi-square test of independence là hai biến độc lập (independent), không phải dependent. $H_a$ là dependent. Bác bỏ $H_0$ nghĩa là có bằng chứng về sự phụ thuộc. Tại sao C sai: $df=(r-1)(c-1)=(3-1)(3-1)=4$, không phải 6. Con số 6 có thể bị nhầm từ $3\times 2=6$ (nhân số levels với nhau thay vì áp dụng công thức đúng). Chỉ table $3\times 4$ mới cho $df=6$.