Q: In a contingency table, how do you calculate the expected frequency for a cell if the variables are independent?

$$E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$$

Question 1

A researcher rolls a fair 6-sided die 60 times and records the observed frequencies for each face. What is the expected frequency for each face if the die is fair?

Accepted Answer

$$E = \frac{60}{6} = 10$$

Question 2

Which of the following is the correct formula for the chi-square test statistic in a goodness-of-fit test?

Accepted Answer

$$\$$chi^2$$ = \sum \frac{(O - E)^2$}{E}$$

Question 3

A gym owner surveys the age distribution of members and wants to test if the number of members is equal across 5 age groups. The observed counts are 54, 46, 53, 49, and 48. What is the null hypothesis for this chi-square goodness-of-fit test?

Accepted Answer

$$H_0$$: The number of members is the same for all age groups.

Question 4

A company claims that coupon distribution matches the following probabilities: 50% ($$5 off$$), 20% ($$10 off$$), 15% ($$15 off$$), 10% ($$20 off$$), 5% ($$50 off). In a $$sample of 1000 coupons, what is the expected frequency for the $$20 off$$ coupon?

Accepted Answer

$$E = 1000 	imes 0.10 = 100$$

Question 5

Which of the following is a requirement for using the chi-square goodness-of-fit test?

Accepted Answer

The expected frequency for each category must be at least 5.

Question 6

In a contingency table, how do you calculate the expected frequency for a cell if the variables are independent?

Accepted Answer

$$E = \frac{(	ext{row total}) 	imes (	ext{column total})}{	ext{grand total}}$$

Question 7

A pharmaceutical company tests a new ADHD medication. The observed frequencies for improvement and no improvement are given for placebo and non-placebo groups. Which test should be used to determine if improvement is independent of group?

Accepted Answer

Chi-square test of independence

Question 8

What are the degrees of freedom for a chi-square test of independence in a $$2 	imes 3$$ contingency table?

Accepted Answer

$$(2-1) 	imes (3-1) = 2$$

Question 9

A one-way ANOVA is used to compare the mean test scores of students taught by three different methods. What is the null hypothesis?

Accepted Answer

$$H_0$$: \$$mu_1$$ = \$$mu_2$$ = \$$mu_3$$

Question 10

Given $$MST = 226.6$$ and $$MSE = 7.944 in a $$one-way ANOVA, what is the value of the $$F-$$statistic?

Accepted Answer

$$F = \frac{226.6}{7.944} = 28.53$$

Question 11

After rejecting the null hypothesis in a one-way ANOVA, which test can be used to determine which specific means are different?

Accepted Answer

Tukey, Tukey-Kramer, or Bonferroni post-hoc tests

Question 12

A nutritionist compares three diet plans and finds a significant ANOVA result. She uses the Tukey-Kramer test to compare means. What is the formula for the Tukey-Kramer test statistic for comparing means $$\bar{x}_1$$ and $$\bar{x}_2$$ with group sizes $$n_1$$ and $$n_2$$ and MSE $$s^2$$?

Accepted Answer

$$q_0$$ = \frac{\bar{x}_2 - \bar{x}_1}{\sqrt{\frac{$$s^2$$}{2} \left( \frac{1}{$$n_1$$} + \frac{1}{$$n_2$$} \right) }}$$

Question 13

A researcher is comparing mean cholesterol levels across 4 diet plans (A, B, C, D) in a One-Way ANOVA test. If $$H_0$$ was rejected and the researcher uses a Bonferroni Test, how many pairwise comparisons will be made?

Accepted Answer

$$\binom{4}{2} = 6$$

Question 14

Which scenario is most appropriate for a Two-Way ANOVA?

Accepted Answer

A nutritionist studies how meal type (breakfast, lunch, dinner) and diet plan (low-carb, low-fat, Mediterranean) affect blood sugar levels.

Question 15

In a two-way ANOVA, what does a significant interaction effect indicate?

Accepted Answer

The effect of one factor depends on the level of the other factor.

Question 16

Which interaction plot best indicates no interaction between two factors?

Accepted Answer

The plot where the lines are parallel.

Question 17

A quality control team finds the following two-way ANOVA output: Machine Type $$p=0.052$$, Operator Experience $$p=0.556$$, Interaction $$p=0.027. At$$ $$\alpha=0.05$$, what can be concluded?

Accepted Answer

There is a significant interaction effect, but no significant main effects.

Chi-Square Tests and Analysis of Variance (ANOVA): Study Notes

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