8.2 The Chi-Square Test for Goodness of Fit

• Match the hypothesis with its definition:

  Null Hypothesis (H₀) ↔️ The observed frequencies match the expected frequencies

  Alternative Hypothesis (H₁) ↔️ The observed frequencies do not match the expected frequencies
• Expected frequencies are calculated by multiplying the total sample size by the hypothesized proportions.


  True
• Steps to compute the Chi-Square statistic for Goodness of Fit

  1️⃣ Calculate expected frequencies

  2️⃣ Compute the Chi-Square statistic

  3️⃣ Determine the degrees of freedom

  4️⃣ Find the p-value using a Chi-Square distribution

  5️⃣ Make a conclusion based on the p-value
• What is the chi-square statistic compared to in order to determine the p-value?

  Critical value
• The chi-square statistic is used to determine if observed frequencies match expected frequencies.

  True
• The Chi-Square Test for Goodness of Fit is used for categorical variables.


  True
• The expected frequency is calculated using the formula: (Total Sample Size) × (Hypothesized Proportion)
• A chi-square statistic of 5.17 with appropriate degrees of freedom can be used to determine the p-value for hypothesis testing.


  True
• The degrees of freedom are calculated using the formula df = k - 1, where k is the number of categories.
• If the p-value is less than or equal to the significance level α, we reject the null hypothesis.
• In the Chi-Square Test for Goodness of Fit, the null hypothesis states that the observed frequencies match the expected frequencies.


  True
• What is the formula to compute the chi-square statistic?

  $\chi^{2} =$$\sum\frac{(Observed - Expected)^{2}}{Expected}$
• The Chi-Square Test for Goodness of Fit is used to determine if observed frequencies match expected frequencies.


  True
• What is the formula to calculate the degrees of freedom for the Chi-Square Test for Goodness of Fit?

  $df =$$k - 1$
• What is the first step to find the p-value using the Chi-Square distribution?

  Calculate the Chi-Square statistic
• What is the p-value if the Chi-Square statistic is 6.25 and the degrees of freedom are 3?

  0.099
• If the p-value is less than or equal to the significance level, we reject the null hypothesis.
• The null hypothesis in the Chi-Square Test for Goodness of Fit states that observed frequencies match expected frequencies.


  True
• What is the decision based on the p-value in the Chi-Square Test for Goodness of Fit?

  Reject or fail to reject
• What is the formula for the Chi-Square statistic?

  \chi^{2} = \sum\frac{(Observed - Expected)^{2}}{Expected}</latex>
• The Chi-Square statistic for the example data is \frac{(50 - 60)^{2}}{60} + \frac{(30 - 40)^{2}}{40} + \frac{(20 - 20)^{2}}{20}</latex>
• Steps in the chi-square test decision-making process

  1️⃣ Calculate the chi-square statistic

  2️⃣ Compare the statistic to a critical value

  3️⃣ Determine the p-value

  4️⃣ Decide whether to reject or fail to reject the null hypothesis
• The chi-square statistic for the die example is 1.0
• What is the Chi-Square Test for Goodness of Fit used to determine?

  Hypothesized probability distribution
• The alternative hypothesis states that observed frequencies match expected frequencies.

  False
• The chi-square statistic is compared to a critical value to determine the p-value
• What do the degrees of freedom represent in the Chi-Square Test for Goodness of Fit?

  Independent categories
• The Chi-Square statistic is compared to a critical value to determine the p-value
• A small p-value in the Chi-Square Test for Goodness of Fit indicates evidence to reject the null hypothesis.


  True
• The Chi-Square Test for Goodness of Fit is used to assess if a categorical variable follows a hypothesized probability distribution.
• Expected frequencies are calculated by multiplying the total sample size by the hypothesized proportions.
• Expected Frequency = (Total Sample Size) × (Hypothesized Proportion
• The formula for the Chi-Square statistic is \chi^{2}
• The Chi-Square statistic measures the difference between observed and expected frequencies.

  True
• The degrees of freedom for the Chi-Square Test for Goodness of Fit are calculated using the number of categories
• The chi-square statistic is calculated using the formula: \chi^{2} = \sum\frac{(Observed - Expected)^{2}}{Expected}
• Match the number of categories with the correct degrees of freedom:

  k = 4 ↔️ df = 3

  k = 6 ↔️ df = 5

  k = 10 ↔️ df = 9
• The chi-square statistic in the example is 5.17
• The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the calculated one, assuming the null hypothesis is true.


  True
• Match the hypothesis with its definition:

  Null Hypothesis (H₀) ↔️ Observed frequencies match expected frequencies

  Alternative Hypothesis (H₁) ↔️ Observed frequencies do not match expected frequencies