For 100 consecutive day frequencies of occurrence of the digits 35,35,30 for the digits 1,2,3 at .05, can you reject the hypothesis that the digits are from a uniform population?
To answer this question, we need to conduct a chi-square test for goodness of fit. This test helps us determine if observed frequencies from a sample differ significantly from the expected frequencies.
Here's how we can perform the chi-square test to test the hypothesis that the digits are from a uniform population:
1. Formulate the null and alternative hypotheses:
- Null hypothesis (H₀): The digits are from a uniform population.
- Alternative hypothesis (H₁): The digits are not from a uniform population.
2. Determine the expected frequencies:
- Since we are testing for uniformity, we expect equal frequencies for each digit. Therefore, the expected frequency for each digit should be 100 / 3 = 33.33.
3. Calculate the chi-square test statistic:
- The chi-square test statistic formula is: χ² = ∑((O - E)² / E), where O is the observed frequency and E is the expected frequency.
- For each digit, calculate the squared difference between the observed frequency and expected frequency, divided by the expected frequency, and sum these values.
For the digit 1: χ²₁ = ((35 - 33.33)^2 / 33.33) = 0.1818
For the digit 2: χ²₂ = ((35 - 33.33)^2 / 33.33) = 0.1818
For the digit 3: χ²₃ = ((30 - 33.33)^2 / 33.33) = 0.3318
- Add up these three chi-square values to get the overall chi-square test statistic: χ² = χ²₁ + χ²₂ + χ²₃ = 0.1818 + 0.1818 + 0.3318 = 0.6954.
4. Determine the degrees of freedom:
- For a goodness-of-fit test, the degrees of freedom (df) is calculated as (number of categories - 1). In this case, since we have three categories (digits 1, 2, and 3), df = 3 - 1 = 2.
5. Find the critical chi-square value:
- The critical chi-square value depends on the significance level and the degrees of freedom. In this case, for a significance level of 0.05 and df = 2, the critical chi-square value is 5.991 (obtained from the chi-square distribution table).
6. Compare the calculated chi-square statistic with the critical chi-square value:
- If the calculated chi-square statistic is greater than the critical chi-square value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, χ² = 0.6954, which is less than 5.991. Therefore, we fail to reject the null hypothesis.
So, based on the chi-square test for goodness of fit, we do not have sufficient evidence to reject the hypothesis that the digits are from a uniform population.