Exercises 1–5 refer to the sample data in the following table, which summarizes the frequencies of 500 digits randomly generated by Statdisk. Assume that we want to use a 0.05 significance level to test the claim that Statdisk generates the digits in a way that they are equally likely.

Is the hypothesis test left-tailed, right-tailed, or two-tailed?

검증된 단계별 안내

Step 1: Understand the hypothesis test. The null hypothesis (H₀) states that the digits are equally likely to occur, meaning each digit has the same probability of being generated. The alternative hypothesis (H₁) states that the digits are not equally likely to occur.

Step 2: Determine the type of test. Since we are testing whether the digits are equally likely (H₀) versus not equally likely (H₁), we are looking for deviations in either direction. This makes the test a two-tailed test.

Step 3: Review the significance level. The significance level is given as 0.05, which is the threshold for determining whether the observed frequencies differ significantly from the expected frequencies.

Step 4: Calculate the expected frequency for each digit. Since there are 500 total digits and 10 possible digits, the expected frequency for each digit is calculated as:

\frac{500}{10}

= 50.

Step 5: Use a chi-square goodness-of-fit test to compare the observed frequencies (from the table) to the expected frequencies (calculated in Step 4). The chi-square test statistic is calculated using the formula:

(\frac{(^{O - E) 2}}{E})

, where O represents the observed frequency and E represents the expected frequency for each digit.

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주요 개념

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Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) that represents no effect or no difference, and an alternative hypothesis (H1) that indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis at a specified significance level.

Significance Level

The significance level, denoted as alpha (α), is the threshold used to determine whether to reject the null hypothesis. In this case, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none. It helps to control the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected.

Types of Hypothesis Tests

Hypothesis tests can be classified as left-tailed, right-tailed, or two-tailed based on the nature of the alternative hypothesis. A left-tailed test checks for a decrease in the parameter, a right-tailed test checks for an increase, and a two-tailed test checks for any difference in either direction. In this scenario, since we are testing if the digits are equally likely, we would use a two-tailed test to assess deviations in both directions.