The author recorded all digits selected in California's...

The author recorded all digits selected in California's Daily 4 Lottery for the 60 days preceding the time that this exercise was created. The frequencies of the digits from 0 through 9 are $21,30,31,33,19,23,21,16,24$, and 22. Use a $0.05$ significance level to test the claim of lottery officials that the digits are selected in a way that they are equally likely.

Key Concepts

Chi-Square Goodness-of-Fit Test

This test is a statistical method used to compare observed frequencies of events with expected frequencies derived from a specific hypothesis. It determines whether any deviations between observed and expected counts are due to random chance or indicate a statistically significant difference.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis represents the default assumption that there is no effect or difference—in this case, that each outcome has an equal chance of occurring. The alternative hypothesis challenges this by suggesting that the outcomes do not occur with equal likelihood.

Significance Level

The significance level is the predetermined threshold for rejecting the null hypothesis. Commonly set at 0.05, it reflects a 5% risk of concluding that a difference exists when, in fact, there is none, thereby controlling the probability of a Type I error.

Expected Frequency

Expected frequency refers to the hypothetical count for each category if the null hypothesis were true. When outcomes are presumed equally likely, each expected frequency is calculated by dividing the total number of observations by the number of categories.

Degrees of Freedom

Degrees of freedom in the context of a chi-square test are calculated as the number of categories minus one. This value is crucial for determining the appropriate chi-square distribution against which the test statistic is compared.

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