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?

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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 > critical value, reject H₀ and conclude that the digits are not equally likely. Otherwise, fail to reject H₀.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. In the context of the lottery, it shows the number of times each digit from 0 to 9 was drawn. This distribution helps identify patterns or anomalies in the data, such as whether certain digits are drawn more frequently than others.

Chi-Square Goodness of Fit Test

The Chi-Square Goodness of Fit Test is a statistical method used to determine if a sample distribution matches an expected distribution. In this case, it tests the hypothesis that all digits in the lottery are equally likely to be drawn. A significant result would indicate that the observed frequencies differ from what would be expected under the assumption of equal likelihood.

Random Sampling

Random sampling is a technique where each member of a population has an equal chance of being selected. In the lottery context, it implies that each digit should have an equal probability of being drawn in each drawing. If the sampling is not random, it could lead to biased results, affecting the fairness of the lottery.