In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.


Heights Measured or Reported? Repeat the preceding exercise using the frequencies in the following table, which summarizes all of the 2784 male heights listed in Data Set 4 “Measured and Reported.” Does the larger data set have much of an effect on the results from Exercise 5?

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that the last digits of the heights are uniformly distributed, meaning each digit (0 through 9) has an equal probability of occurrence. The alternative hypothesis (H₁) states that the last digits are not uniformly distributed.
2. Step 2: Calculate the expected frequency for each digit under the assumption of uniform distribution. Since there are 2784 total heights, divide this total by 10 (the number of digits) to find the expected frequency for each digit: Expected Frequency = 2784 ÷ 10 = 278.4.
3. Step 3: Use the observed frequencies from the table and the expected frequencies to calculate the test statistic. The test statistic for a chi-square goodness-of-fit test is given by: ${\chi }^2=\; \Sigma ((Oᵢ\; -\; Eᵢ)²\; /\; Eᵢ),\; where\; Oᵢ\; represents\; the\; observed\; frequency\; and\; Eᵢ\; represents\; the\; expected\; frequency\; for\; each\; digit.$
4. Step 4: Determine the degrees of freedom (df) for the chi-square test. The degrees of freedom are calculated as df = k - 1, where k is the number of categories (digits). In this case, df = 10 - 1 = 9.
5. Step 5: Compare the calculated test statistic to the critical value from the chi-square distribution table at the chosen significance level (e.g., α = 0.05) or use the P-value to make a decision. If the test statistic exceeds the critical value or if the P-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Key Concepts

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) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. The process includes calculating a test statistic and comparing it to a critical value or using a P-value to assess the strength of the evidence against H0.

Test Statistic

A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It measures how far the sample statistic is from the null hypothesis, expressed in terms of standard errors. Depending on the type of test (e.g., z-test, t-test), the test statistic helps determine whether to reject the null hypothesis by comparing it to a critical value or using it to find the P-value.

P-value

The P-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller P-value indicates stronger evidence. In hypothesis testing, if the P-value is less than the significance level (commonly 0.05), the null hypothesis is rejected, suggesting that the observed data is unlikely under H0.