Lesson 108 — Chi-squared test: goodness of fit and independence

A six-sided die is rolled 60 times. What is the number of degrees of freedom in the goodness-of-fit test for the uniform distribution?

For the die from the previous exercise rolled 60 times, what is the expected frequency per face?

A die is rolled 60 times: observing 12, 8, 11, 9, 13, 7 for faces 1 to 6. Calculate $\chi^2$ and conclude at 5%.

Calculate the degrees of freedom for the independence test in a $3 \times 4$ contingency table.

In a contingency table with $R_1 = 80$, $C_1 = 120$, $n = 300$, calculate $E_{11}$.

In an independence test, we obtained $\chi^2 = 8$ with $df = 2$. The critical value at 5% is 5.99. What is the conclusion?

Calculate Cramér's V: $\chi^2 = 20$, $n = 200$, table $3 \times 4$ (so $\min(r-1,c-1) = 2$).

In which situations should the Yates correction be applied to the chi-squared test?

A researcher has $E_i = 3$ in two of five cells in a table. Is the chi-squared test appropriate? Justify.

Observe $O = (45, 35, 20)$ in $n = 100$ observations with expected proportions $(0,5;\; 0,3;\; 0,2)$. Calculate $\chi^2$.

For the previous exercise (3 categories, fully specified distribution), what is the number of degrees of freedom?

A study measures blood pressure (high/normal) in the same group of patients before and after an exercise program. Why is the standard independence chi-squared test inadequate?

What is the critical value $\chi^2_{0,05;\,1}$ (chi-squared with 1 degree of freedom at the 5% level)?

Calculate the expected frequencies for the $2 \times 2$ table with cells $a = 80$, $b = 20$, $c = 60$, $d = 40$.

With the expected values from the previous exercise, calculate $\chi^2$ and conclude at 5%.

Why is $df = 1$ in every $2 \times 2$ contingency table? Explain geometrically or algebraically.

What are the mean and variance of $\chi^2_k$? For $k = 20$, is the distribution approximately symmetric?

In a goodness-of-fit test with $k$ categories, how do the degrees of freedom change when we estimate $r$ parameters of the distribution from the data itself?

Show that the chi-squared statistic $\chi^2 = \sum (O_i - E_i)^2 / E_i$ is always non-negative.

Is the goodness-of-fit chi-squared test one-tailed (right tail) or two-tailed? Why?

$O = (10, 20, 30, 40)$ in $n = 100$ with expected uniform distribution. Calculate $\chi^2$ and conclude at 1%.

What is the conceptual difference between homogeneity test and independence test? Does the formula for $\chi^2$ change?

$\chi^2 = 14,5$ with $df = 5$. What is the conclusion at 5% and at 1%? (Critical values: 11.07 and 15.09 respectively.)

What would it mean to obtain $\chi^2 = 0$ in a goodness-of-fit test? Is this possible in real data?

Why do very large samples make $\chi^2$ a problematic measure? What alternative should be used?

Describe the shape of the chi-squared curve for small $df$ (e.g. $df = 2$) vs. large $df$ (e.g. $df = 20$). How does this relate to its origin as a sum of squares?

Which formula below is Pearson's chi-squared statistic?

Explain why Cochran's rule ($E_i \geq 5$) is necessary for the validity of the chi-squared test.

Dihybrid cross in peas predicts phenotypes in ratio 9:3:3:1. In 160 offspring observe 95, 30, 27, 8. Test goodness of fit at 5%.

Survey of 400 university students (200 men, 200 women) tabulates opinion on quotas (Favorable/Neutral/Against): men 70/60/70, women 110/50/40. Test independence at 5%.

A sample of 200 M&M's from a package shows: 30 red, 35 orange, 22 yellow, 40 green, 55 blue, 18 brown. According to the manufacturer, proportions are 13%, 20%, 14%, 16%, 24%, 13%. Test goodness of fit at 5%.

A/B/C test on landing page: 200 visitors per variation. Conversions: A = 24, B = 30, C = 40. Test homogeneity of conversion rates at 5%.

Four machines produce defects: 30, 40, 25, 35 defects respectively (total 130). Test whether the defect rate is uniform among machines at the 5% level.

Clinical trial with 50 patients (25 per group): vaccine resulted in 18 cures, placebo in 12 cures. Build the $2 \times 2$ table and apply the chi-squared test with Yates correction at 5%.

Do DNIT highway accident data follow a Poisson distribution? Describe the complete flowchart for the goodness-of-fit test, including how to handle the unknown parameter.

Which condition below is necessary for the validity of the independence chi-squared test?

In a before-after study, the same 80 patients are classified as hypertensive or normal before and after intervention. Why use McNemar instead of standard chi-squared?

A survey of 500 Brazilians records region (North, Southeast, South) and payment preference (cash vs. installment). Which test is most appropriate to verify whether preference and region are independent?

An emergency room recorded 210 visits in one week (30 per day expected). Observed: Sun=18, Mon=40, Tue=28, Wed=25, Thu=29, Fri=42, Sat=28. Is the flow uniform across days? Test at 5%.

Electoral survey in 3 Brazilian states (SP, RJ, MG) with 600 voters (200 per state) records candidate preference (A, B, C). Data: SP=(80,70,50), RJ=(60,90,50), MG=(70,60,70). Test independence between state and candidate at 5% and calculate Cramér's V.

Show that for $k = 2$ categories, $\chi^2 = Z^2$ where $Z$ is the bilateral $z$ statistic for proportion test. This explains why $\chi^2_{0,05;\,1} = (1,96)^2 \approx 3,84$.

Prove the formula $df = (r-1)(c-1)$ for the independence test in an $r \times c$ table, explaining how many independent constraints the margins impose on the count vector.