• Assumptions
  • random sample
  • categorical variable
  • frequencies ≥ 1
  • no more than 20% of the frequencies < 5
• The chi-squared statistic \(X^2\) is calculated as:

\[X^2 = \sum \frac{(Observed - Expected)^2}{Expected}\]

Is there a uniform distribution of zodiac signs among successful individuals? Let’s analyze with a significance level of α = 0.05.

• Null Hypothesis(\({H_0}\)): Births are uniformly distributed over zodiac signs.
• Alternative Hypothesis(\({H_a}\)): Births are not uniformly distributed over zodiac signs.

Ho: Births are uniformly distributed over zodiac signs Ha: Births are not uniformly distributed over zodiac signs

Test statistic calculation: \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency.

Substituting the observed and expected frequencies: \[ \chi^2 = \frac{(18 - 20.833)^2}{20.833} + \frac{(19 - 20.833)^2}{20.833} + \cdots + \frac{(28 - 20.833)^2}{20.833} \] \[ \chi^2 = 5.64800 \]

chi_square_statistic <- sum((observed - expected)^2 / expected)
chi_square_statistic

## [1] 5.648

critical_value <- qchisq(p = 0.95, df = 11)
c(chi_square_statistic, critical_value)

## [1]  5.64800 19.67514

• From the Chi-Square distribution table, with df=11, the critical value is 19.675 and the test statistic (5.64800) is outside the rejection region
• Fail to reject the null hypothesis. There is not enough evidence to suggest that births are not uniformly distributed over zodiac signs