Chi-square test results in p-value of 0.324, failing to reject the null hypothesis. No evidence to suggest unequal selection likelihood among 5 categories.
Chi-square test statistic:
To calculate the chi-square test statistic, we first need to calculate the expected frequencies for each category. These are calculated by multiplying the sample size by the hypothesized probabilities for each category. In this case, the hypothesized probabilities are all equal, so the expected frequency for each category is simply 100/5 = 20.
Next, we calculate the Pearson residuals for each category. These are calculated by subtracting the observed frequency from the expected frequency for each category. The Pearson residuals are then squared and divided by the expected frequency to get the chi-squared contribution for each category.
Finally, we sum the chi-squared contributions for each category to get the overall chi-square test statistic.
Chi-square test statistic = (19 - 20)^2 / 20 + (21 - 20)^2 / 20 + (25 - 20)^2 / 20 + (19 - 20)^2 / 20 + (16 - 20)^2 / 20
= 0.25 + 0.05 + 2.50 + 0.25 + 1.60
= 4.65
Degrees of freedom:
The degrees of freedom for the chi-square test are equal to the number of categories minus one. In this case, there are 5 categories, so the degrees of freedom are 5 - 1 = 4.
P-value:
To calculate the p-value, we need to look up the chi-square test statistic in a chi-square distribution table with 4 degrees of freedom. The p-value is the probability of obtaining a chi-square test statistic as large as or larger than the observed value, assuming that the null hypothesis is true.
In this case, the p-value is 0.324. This means that there is a 32.4% chance of obtaining a chi-square test statistic as large as or larger than 4.65, assuming that the null hypothesis is true.