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

Question 1

A four-sided spinner is spun 80 times. A χ^{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.

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Question 2

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 λ from the sample. Determine the degrees of freedom for the chi-squared test.

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Question 3

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

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Question 4

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

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Question 5

This question assesses the student's ability to calculate the degrees of freedom for a χ^{2} goodness-of-fit test when parameters of the distribution are estimated from the data.

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 N B (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 χ^{2} goodness-of-fit test.

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Question 6

Chi-squared goodness-of-fit test

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

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Question 7

Observations are grouped into 7 bins and fitted to a gamma distribution by estimating both shape α and scale β parameters. Determine the degrees of freedom for the χ^{2} goodness-of-fit test.

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Question 8

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

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Question 9

Degrees of freedom in a chi-squared goodness-of-fit test for a binomial distribution.

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.

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Question 10

The question assesses the ability to determine the degrees of freedom for a Chi-squared goodness of fit test when parameters of the distribution are estimated from the data.

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.

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Question 11

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

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Question 12

This question assesses the student's ability to determine the degrees of freedom for a chi-squared goodness of fit test based on a given scenario.

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.

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IB Mathematics Applications & Interpretation Question Type 7: Determining the Degree of Freedom for Chi-squared Goodness of Fit Tests Exercises