Question Type 7: Finding expected frequencies for a chi-squared goodness of fit test Exercises

A coin is flipped 30 times, yielding 18 heads and 12 tails. Assuming the coin is fair, calculate the $\chi^2$ statistic.

Using the chi-squared result $\chi^2=1.2$ from a test with $1$ degree of freedom, find the approximate $p\text{-value}$ and state whether you reject the null hypothesis at $\alpha=0.05$.

A study classifies families by the number of children, $X$. The probabilities are given as $P(X=0)=0.1$, $P(X=1)=0.3$, $P(X=2)=0.4$, and $P(X=3)=0.2$.

If 80 families are sampled, find the expected frequencies for each number of children.

Given observed frequencies for four categories as $[12, 15, 13, 10]$ and expected frequencies $[12.5, 14.5, 13.0, 10.0]$, calculate the chi-squared ($\chi^2$) statistic.

A fair die is rolled 60 times with observed counts $(8, 10, 12, 6, 14, 10)$. Calculate the $\chi^2$ statistic to test the fairness of the die.

A goodness-of-fit test yields $\chi^2 = 9.348$ with $df = 4$. Using technology, find the $p$-value and make a decision at the $5\%$ significance level.

A fair six-sided die is rolled 60 times. Calculate the expected frequency for each face.

A market share model predicts that car brand preferences in a specific region follow these probabilities:

• VW: 0.30
• Ford: 0.25
• Toyota: 0.20
• Honda: 0.15
• Nissan: 0.10

In a random sample of 200 recent car buyers, the following counts were observed:

• VW: 70
• Ford: 50
• Toyota: 40
• Honda: 25
• Nissan: 15

Perform a chi-squared goodness-of-fit test at the 1% significance level to determine whether the observed preferences are consistent with the predicted market share model.

A test yields $\chi^2 = 3.84$ with $3$ degrees of freedom. Estimate the $p$-value range from standard chi-squared tables and conclude at the $5\%$ significance level.

Compute the expected frequency for each category.

In a survey of 100 respondents, categories A, B, and C have observed counts 45, 35, and 20, and expected counts 50, 30, and 20 respectively. Calculate the $\chi^2$ statistic.

A survey of 100 people records choices among three options A, B, and C. A theoretical model predicts probabilities $P(A)=0.5$, $P(B)=0.3$, and $P(C)=0.2$. Find the expected frequency for each option. [3 marks]