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?
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To determine if the frequencies of occurrence of the digits 1, 2, and 3 (represented as 35, 35, 30 respectively) over 100 consecutive days can be considered as coming from a uniform population, you can perform a hypothesis test using the Chi-Square test.
Here's how you can approach it step by step:
Step 1: Formulate the null (H0) and alternative (Ha) hypotheses:
- Null hypothesis (H0): The frequencies of occurrence of the digits 1, 2, and 3 are from a uniform population.
- Alternative hypothesis (Ha): The frequencies of occurrence of the digits 1, 2, and 3 are not from a uniform population.
Step 2: Determine the expected frequencies:
Under the null hypothesis of a uniform distribution, we would expect each digit 1, 2, and 3 to occur with equal frequency over 100 days. Since there are three digits, the expected frequency for each digit would be 100 / 3 = 33.33.
Step 3: Calculate the Chi-Square test statistic:
- Calculate the difference between the observed frequencies (O) and the expected frequencies (E) for each digit.
- Square each difference.
- Divide each squared difference by the corresponding expected frequency (E).
- Sum up all the resulting values.
Applying this calculation for each digit:
Digit 1:
(Observed frequency - Expected frequency)² / Expected frequency = (35 - 33.33)² / 33.33
Digit 2:
(Observed frequency - Expected frequency)² / Expected frequency = (35 - 33.33)² / 33.33
Digit 3:
(Observed frequency - Expected frequency)² / Expected frequency = (30 - 33.33)² / 33.33
Sum up all the resulting values to obtain the Chi-Square test statistic.
Step 4: Determine the critical value:
The critical value can be obtained from the Chi-Square table with (k - 1) degrees of freedom, where k is the number of categories (in this case, k = 3). The significance level is given as 0.05.
Step 5: Compare the test statistic with the critical value:
- If the test statistic is greater than the critical value, you can reject the null hypothesis at the 0.05 significance level, concluding that the frequencies are not from a uniform population.
- If the test statistic is less than or equal to the critical value, you fail to reject the null hypothesis, meaning there is no evidence to conclude that the frequencies are not from a uniform population.
By following these steps and calculating the Chi-Square test statistic, you will be able to determine whether or not you can reject the hypothesis that the frequencies of occurrence of the digits are from a uniform population.