Question

Win 4 Lottery Shown below is a histogram of digits selected in California’s Win 4 lottery. Each drawing involves the random selection (with replacement) of four digits between 0 and 9 inclusive.

c. Identify the frequencies, then test the claim that the digits are selected from a population in which the digits are all equally likely. Is there a problem with the lottery?

Accepted Answer

Step 1: Extract the frequencies of each digit from the histogram. The frequencies are as follows: 0 (18), 1 (22), 2 (23), 3 (21), 4 (19), 5 (24), 6 (16), 7 (15), 8 (23), 9 (23). Step 2: Formulate the null hypothesis (H₀) and alternative hypothesis (H₁). H₀: The digits are equally likely to be selected (uniform distribution). H₁: The digits are not equally likely to be selected. Step 3: Calculate the expected frequency for each digit under the assumption of uniform distribution. Since there are 10 digits (0-9) and the total frequency is the sum of all observed frequencies, divide the total frequency by 10 to get the expected frequency for each digit. Step 4: Use the Chi-Square Goodness-of-Fit Test formula: χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ is the observed frequency and Eᵢ is the expected frequency for each digit. Compute the χ² statistic using the observed and expected frequencies. Step 5: Compare the calculated χ² statistic to the critical value from the Chi-Square distribution table at the appropriate degrees of freedom (df = number of categories - 1) and significance level (e.g., α = 0.05). If χ² \u003e critical value, reject H₀ and conclude that the digits are not equally likely. Otherwise, fail to reject H₀.

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Key Concepts

Frequency Distribution

Chi-Square Goodness of Fit Test

Random Sampling