13.1: Chi-Squared Goodness-of-fit tests

Question 1

What is the purpose of a Chi-Square Goodness-of-Fit test?

Answer 1

The Chi-Square Goodness-of-Fit test is used to determine if there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.

Question 2

What is a multinomial experiment?

Answer 2

A multinomial experiment is one that involves n identical trials with k possible outcomes on each trial, and the probabilities of the outcomes are constant throughout the trials.

Question 3

What are the assumptions for a multinomial experiment?

Answer 3

The trials must be independent, and the probabilities of the k outcomes must remain constant from trial to trial.

Question 4

How do you calculate the expected frequencies in a Chi-Square Goodness-of-Fit test?

Answer 4

Expected frequencies are calculated by multiplying the total number of observations by the expected probability of each category.

Question 5

What is the formula for the Chi-Square statistic?

Answer 5

The formula is

χ² = Σ[(Oᵢ - Eᵢ)² / Eᵢ] for i = 1 to k,

where Oᵢ is the observed frequency and Eᵢ is the expected frequency for each category.

Question 6

When do you reject the null hypothesis in a Chi-Square Goodness-of-Fit test?

Answer 6

You reject the null hypothesis if the Chi-Square statistic is greater than the critical value from the Chi-Square distribution at the desired level of significance.

Question 7

What indicates a significant result in a Chi-Square Goodness-of-Fit test?

Answer 7

A Chi-Square statistic that is larger than the critical value, or a p-value that is less than the level of significance, indicates a significant result, leading to the rejection of the null hypothesis.

Question 8

What does a Chi-Square test reveal about a set of observed and expected frequencies?

Answer 8

A Chi-Square test reveals whether the observed frequencies in each category differ significantly from what we would expect under the null hypothesis of no difference.

Question 9

How is the p-value interpreted in a Chi-Square Goodness-of-Fit test?

Answer 9

A p-value less than the chosen alpha level (e.g., 0.05) indicates that the observed frequencies are unlikely to have occurred by random chance, suggesting that the actual distribution differs from the expected distribution.

Question 10

What is the Chi-Square Goodness-of-Fit test used for in the context of multinomial probabilities?

Answer 10

It is used to test hypotheses about multinomial probabilities, assessing if the observed frequencies in categories differ significantly from the expected frequencies.

Question 11

How do you calculate the expected frequency for each category in a multinomial experiment?

Answer 11

The expected frequency, Ei, is calculated as Ei = n * pi, where n is the total number of observations and pi is the probability of an observation falling into category i.

Question 12

What is the formula for the Chi-Square statistic in a Goodness-of-Fit test?

Answer 12

The formula is χ² = Σ[(Oi - Ei)² / Ei] across all categories i, where Oi is the observed frequency and Ei is the expected frequency for category i.

Question 13

When do you reject the null hypothesis in a Chi-Square Goodness-of-Fit test?

Answer 13

You reject the null hypothesis if the Chi-Square statistic exceeds the critical value for the Chi-Square distribution at the desired significance level, or if the p-value is less than the significance level.

Question 14

What does the test for homogeneity in a Chi-Square Goodness-of-Fit test involve?

Answer 14

It involves testing the null hypothesis that all multinomial probabilities are equal, and rejecting this hypothesis suggests that not all categories have equal probabilities.

Question 15

What conditions must be met for the Chi-Square approximation to be valid in a Goodness-of-Fit test?

Answer 15

The sample size must be large, with all expected cell frequencies (Ei values) being at least 5, or if the number of categories (k) exceeds 4, the average of the Ei values is at least 5, and the smallest Ei value is at least 1

Question 16

What conclusion can be drawn when the entire confidence interval for a proportion is below an expected value?

Answer 16

If the entire confidence interval for a sample proportion is below the expected market share, it suggests that the actual market share for that category is smaller than expected, indicating a need to reassess market strategies or preferences.

Question 17

How is the confidence interval for a proportion calculated in the context of a Chi-Square test for multinomial probabilities?

Answer 17

The confidence interval is calculated as p-hat plus or minus z times the square root of (p-hat times (1 - p-hat) divided by n), where p-hat is the sample proportion, z is the critical value from the standard normal distribution, and n is the sample size.

Question 18

What is the assumption behind many statistical methods?

Answer 18

The assumption is that a random sample has been selected from a normally distributed population.

Question 19

How can the validity of the normality assumption be checked?

Answer 19

It can be checked using frequency distributions, stem-and-leaf displays, histograms, and normal plots. Another method is the chi-square goodness-of-fit test.

Question 20

What is the chi-square goodness-of-fit test used for?

Answer 20

It is used to test the normality assumption of a distribution.

Question 21

How are expected frequencies in a chi-square goodness-of-fit test calculated?

Answer 21

Expected frequencies are calculated using the sample mean and sample standard deviation as point estimates for the population mean and standard deviation, and then determining the probability for each interval under the normal distribution assumption.

Question 22

What is the formula to estimate the probability (p_i) in a chi-square goodness-of-fit test?

Answer 22

p_i = P(lower bound <= X < upper bound) using the standard normal distribution.

Question 23

How is the expected frequency (E_i) for an interval calculated in a chi-square goodness-of-fit test?

Answer 23

E_i = n * p_i, where n is the sample size and p_i is the probability of an observation falling in the i-th interval under a normal distribution.

Question 24

What is the chi-square statistic formula for testing goodness-of-fit?

Answer 24

chi-square = sum from i=1 to k of ((f_i - E_i)^2 / E_i), where f_i is the observed frequency and E_i is the expected frequency for the i-th interval.

Question 25

How do you decide to reject the null hypothesis (H_0) in a chi-square goodness-of-fit test?

Answer 25

If the calculated chi-square statistic is greater than the critical value from the chi-square distribution table, the null hypothesis is rejected.

Question 26

What does the null hypothesis (H_0) typically state in a chi-square goodness-of-fit test for normality?

Answer 26

H_0: The population is normally distributed.

Question 27

What does the alternative hypothesis (H_a) suggest in a chi-square goodness-of-fit test for normality?

Answer 27

H_a: The population is not normally distributed.

Question 28

How are degrees of freedom calculated in a chi-square goodness-of-fit test?

Answer 28

Degrees of freedom are calculated as k - 1 - m, where k is the number of categories and m is the number of estimated parameters.

Question 29

When is it acceptable to combine cell frequencies in a chi-square goodness-of-fit test?

Answer 29

When the expected cell frequencies are less than 5, adjacent cell frequencies can be combined to ensure each expected frequency is at least 5.

Question 30

What is the p-value in the context of a chi-square goodness-of-fit test?

Answer 30

The p-value represents the probability of observing a chi-square statistic as extreme as, or more extreme than, the one calculated from the data, assuming the null hypothesis is true.

Question 31

What does a p-value greater than the significance level (e.g., 0.05) indicate in a chi-square goodness-of-fit test?

Answer 31

It indicates that there is not enough evidence to reject the null hypothesis; thus, the distribution may be considered normal.

Question 32

What is the null hypothesis for testing a normal distribution using chi-square test?

Answer 32

The null hypothesis H_0 is that the population has a normal distribution.

Question 33

What is the alternative hypothesis in a chi-square test for normal distribution?

Answer 33

The alternative hypothesis Ha is that the population does not have a normal distribution.

Question 34

How do you select a sample for a chi-square test for normality?

Answer 34

Select a random sample of size n and compute the sample mean ̅x and sample standard deviation s.

Question 35

What are k intervals in a chi-square test?

Answer 35

k intervals are defined categories for the test, based on the classes of a histogram of the data or using intervals based on the Empirical Rule.

Question 36

How to calculate expected frequency in a chi-square test?

Answer 36

Calculate the probability that a normal variable with mean ̅x and s is within the interval and multiply by n. Combine intervals if necessary for adequate expected frequency size.

Question 37

What formula is used to calculate the chi-square statistic?

Answer 37

The chi-square statistic is calculated using χ² = ∑{i=1}^{k} ²((fi - Ei)^2)/Ei), where fi is the observed frequency and Ei is the expected frequency.