Question

A study is being conducted to determine whether there is a relationship between jogging and blood pressure. A random sample of 250 subjects is selected, and they are classified as shown in the following table. At α = 0.10, test the claim that jogging and blood pressure are not related. (apply Chi-square independent test) Blood Pressure Jogging Status Low Moderate High Joggers 44 77 31 Non-joggers 15 63 20 (a) State the null and alternative hypothesis. (b) Calculate the test statistic. State your conclusion about the hypothesis based on the test statistic and critical value.

Answer 1

To analyze the relationship between jogging and blood pressure, we perform a chi-square test for independence. The test involves comparing calculated chi-square statistic with a critical value, using given observed frequencies, expected frequencies, and an alpha level of 0.10 to either reject or accept the null hypothesis.

Step-by-step explanation:

Chi-Square Independence Test for Jogging and Blood Pressure

To determine if there is a relationship between jogging and blood pressure, we use a chi-square test for independence. The null hypothesis (H0) is that jogging and blood pressure are not related. The alternative hypothesis (H1) is that there is a relationship between jogging and blood pressure.

Calculation of Test Statistic

The test statistic for a chi-square test is calculated using the formula:

χ² = Σ((O - E)² / E)

Where O is the observed frequency and E is the expected frequency. To find the expected frequencies, we use the following formula for each cell of the table:

E = (row total * column total) / overall total

After calculating the expected frequencies and the chi-square statistic, we then compare the calculated chi-square statistic to the critical value from the chi-square distribution table based on the degrees of freedom (df) and the given alpha level (0.10). If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis, indicating a relationship between jogging and blood pressure.

Answer 2

(a) The null hypothesis is that jogging and blood pressure are not related and alternative hypothesis is that jogging and blood pressure are related. (b) The chi-square test statistic is 0.47 and the critical value is 5.99.

The null hypothesis
`(\(H_0\))` is that jogging and blood pressure are not related. The alternative hypothesis
`(\(H_1\))` is that jogging and blood pressure are related.

The formula for expected frequency is
`\(\frac{{\text{{row total}} * \text{{column total}}}}{{\text{{total sample size}}}}\)`.

The expected frequencies for the cells are: low joggers - 25.6, moderate joggers - 42.7, high joggers - 16.7, low non-joggers - 23.4, moderate non-joggers - 39.1, high non-joggers - 15.4.

The chi-square test statistic using the formula
`\(\chi^2 = \sum \frac{{(O_i - E_i)^2}}{{E_i}}\), where \(O_i\)` is the observed frequency and
`\(E_i\)` is the expected frequency for each cell. The critical value for the chi-square distribution with degrees of freedom equal to
`\((\text{{number of rows}} - 1) * (\text{{number of columns}} - 1)\)` at the given significance level.

The chi-square test statistic is 0.47 and the critical value is 5.99.

Since 0.47 is less than 5.99, we fail to reject the null hypothesis. There is not enough evidence to conclude that jogging and blood pressure are related.