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

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

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.

In a theoretical distribution with four categories having probabilities $0.1$, $0.2$, $0.3$ and $0.4$, 120 observations are recorded. Compute the expected frequency for each category.

A biased coin is flipped 30 times, yielding 18 heads and 12 tails. Assuming a fair coin, compute the chi-squared statistic.

A study classifies families by number of children: probabilities are $P(0)=0.1$, $P(1)=0.3$, $P(2)=0.4$, $P(3)=0.2$. If 80 families are sampled, find the expected frequencies in each class.

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

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 statistic.

A fair die is rolled 60 times with observed counts $(8,10,12,6,14,10)$. Compute the chi-squared statistic to test fairness.

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

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% level.

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

A market share model predicts car brand preferences with probabilities: VW 0.30, Ford 0.25, Toyota 0.20, Honda 0.15, Nissan 0.10. In a sample of 200 buyers, observed counts are: VW 70, Ford 50, Toyota 40, Honda 25, Nissan 15. Perform a chi-squared goodness-of-fit test at $\alpha=0.01$.