Question Type 8: Finding degree of freedom and performing chi-squared goodness of fit test Exercises

Given a categorical variable with 5 categories, state the degrees of freedom for a chi-squared goodness of fit test.

An unfair coin is flipped 200 times yielding 130 heads and 70 tails. Test at the 5% level if the coin is fair.

Observed counts of successes $k=0,1,2,3,4$ in $n=4$ trials from 80 experiments are [5,20,30,20,5]. Test fit to Binomial(4,0.5) at 5%.

A spinner has four colors with expected probabilities 0.2, 0.3, 0.25, 0.25. After 120 spins, observed frequencies are 28, 36, 32, 24. Test the fit at 5%.

A survey of 100 people gives the following counts for four brands A, B, C, D: 25, 30, 20, 25. Test at 1% whether the market share is equally distributed.

Five drink brands have equal expected popularity. In a taste test of 250 people, counts are [50,60,55,45,40]. Test at 5% whether popularity is equal.

Seven categories each expected equally likely, observed counts: [15,12,18,20,10,14,11] for 100 trials. Perform a chi-squared goodness of fit test at 5%.

A die is rolled 60 times with observed face counts 8, 12, 10, 9, 11, 10. Test at the 5% level whether the die is fair.

An experiment on Mendelian inheritance yields counts for four phenotypes in ratio $9:3:3:1$ as observed [90, 28, 32, 10]. Test the fit at the 5% level.

Observed blood groups O, A, B, AB in 200 people are 88, 64, 36, 12. Expected proportions are 0.44, 0.42, 0.10, 0.04. Test the fit at 5%.

Data on counts of 0,1,2,3+ defects per item: [30,45,20,5] for 100 items. Test fit to Poisson distribution by estimating $\lambda$ from data and using 3+ as one category at 5% significance.