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.

The observed counts of successes $k$ in $n = 4$ trials from $80$ independent experiments are recorded in the following table:

$\begin{array}{|l|c|c|c|c|c|} \hline k & 0 & 1 & 2 & 3 & 4 \\ \hline \text{Observed Frequency} & 5 & 20 & 30 & 20 & 5 \\ \hline \end{array}$

Use a $\chi^2$ goodness of fit test at the $5\%$ significance level to determine if the data follows a binomial distribution $\text{B}(4, 0.5)$.

Five drink brands are expected to have equal popularity. In a taste test of 250 people, the observed counts for each brand are 50, 60, 55, 45, and 40.

Test, at a 5% significance level, whether the popularity of the brands is equal.

An experiment on Mendelian inheritance yields counts for four phenotypes in an expected ratio of $9:3:3:1$. The observed counts are $90$, $28$, $32$, and $10$. Test the goodness of fit at the $5\%$ significance level.

The observed frequencies of blood groups O, A, B, and AB in a sample of 200 people are 88, 64, 36, and 12, respectively. The expected proportions for these groups are 0.44, 0.42, 0.10, and 0.04.

Perform a chi-squared goodness of fit test at the 5% significance level to determine if the distribution fits the expected proportions.

The following table shows the number of defects found in 100 randomly selected items produced by a manufacturing process.

$\begin{array}{|l|c|c|c|c|} \hline \text{Number of defects} & 0 & 1 & 2 & 3+ \\ \hline \text{Count (observed)} & 30 & 45 & 20 & 5 \\ \hline \end{array}$

(a) Show that the estimated value of the parameter $\lambda$ for the Poisson distribution is $1.00$.

(b) Conduct a $\chi^2$ goodness of fit test at the $5\%$ significance level to determine whether the data follows a Poisson distribution. Use the $3+$ category as a single class for the test.

A coin is flipped $200$ times, resulting in $130$ heads and $70$ tails. Test, at the $5\%$ significance level, whether the coin is fair.

A survey of 100 people gives the following counts for four brands A, B, C, and D: 25, 30, 20, and 25 respectively.

Test, at the $1\%$ significance level, whether the market share is equally distributed among the four brands.

A researcher conducts a $\chi^2$ goodness of fit test to determine if seven categories are equally likely to occur. The observed counts from 100 trials are recorded in the table below:

Perform the test at a 5% significance level to determine if the data is consistent with the expected distribution.

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.

A spinner has four colors with expected probabilities $0.2, 0.3, 0.25,$ and $0.25$. After $120$ spins, the observed frequencies are $28, 36, 32,$ and $24$ respectively.

Test, at the $5\%$ level of significance, whether the observed frequencies fit the expected distribution.