Question Type 7: Determining the degree of freedom for chi-squared goodness of fit tests Exercises

A four-sided spinner is spun 80 times. A $\chi^2$ goodness-of-fit test is performed to determine whether the spinner is uniform across its four outcomes.

Determine the degrees of freedom for this test.

A scientist collects counts of rare events per hour over many hours and groups the data into 6 count-categories. She fits a Poisson distribution and estimates the mean $\lambda$ from the sample. Determine the degrees of freedom for the chi-squared test.

Data are grouped into 5 bins and fitted to an exponential distribution by estimating the rate parameter $\lambda$. Determine the degrees of freedom for the chi-squared goodness-of-fit test.

Data are binned into 6 intervals and fitted to a log-normal distribution by estimating both parameters $\mu$ and $\sigma$. Determine the degrees of freedom for the chi-squared test.

The number of failures observed in a series of independent trials is recorded and grouped into 9 categories. The data is then fitted to a negative binomial distribution, $X \sim NB(r, p)$, where $r$ is the number of successes and $p$ is the probability of success. Both parameters $r$ and $p$ are estimated from the sample data.

Determine the number of degrees of freedom for the $\chi^2$ goodness-of-fit test.

A researcher tests whether observed counts fall uniformly across 10 categories. Determine the degrees of freedom for this $\chi^2$ goodness-of-fit test.

Observations are grouped into $7$ bins and fitted to a gamma distribution by estimating both shape $\alpha$ and scale $\beta$ parameters. Determine the degrees of freedom for the $\chi^2$ goodness-of-fit test.

Determine the degrees of freedom for a $\chi^2$ goodness-of-fit test where a six-sided die is rolled 60 times and the null hypothesis states that the die is fair.

Data on the number of successes in 5 trials are observed and grouped into all 6 possible outcomes (0 through 5 successes). A binomial distribution with unknown probability $p$ is fitted by estimating $p$ from the data. Determine the degrees of freedom for the corresponding chi-squared goodness-of-fit test.

Observations of the count of trials until the first success are grouped into 7 categories and fitted to a geometric distribution with unknown success probability $p$. Determine the degrees of freedom.

A dataset is binned into 8 intervals and fitted to a normal distribution, with both the mean $\mu$ and standard deviation $\sigma$ estimated from the data. Determine the degrees of freedom for the chi-squared test.

Two fair dice are rolled repeatedly and the sums (from 2 to 12) are recorded. To test the theoretical distribution of sums without estimating any parameters, determine the degrees of freedom for the chi-squared test.