a sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way classification:

speed Never Speed totals
male 222 78 300
female 128 72 200
totals 350 150 500
We wish to test whether gender and speeding are related at the 1% significant level.
a. state the null and alternative hypothesis for this test.
b. find the critical value for this test.
c. calculate the test statistic.
d. should we reject or fail to reject the null hypothesis?
e. based on your answers to a-d, are gender and speeding likely to be independent?

Use the Chi-square (X^2) method.

X^2 = ∑ (O-E)^2/E, where O = observed frequency and E = expected frequency.

∑ = sum of all the cells.

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

df = n - 1, where n = number of cells

Look up value in X^2 table in the back of your textbook.

a. The null hypothesis (H0) for this test is that gender and speeding are independent, meaning that there is no relationship between gender and speeding. The alternative hypothesis (H1) is that there is a relationship between gender and speeding.

b. To find the critical value for this test, we need to determine the degrees of freedom and consult the chi-square distribution table. The degrees of freedom (df) can be calculated as (number of rows - 1) x (number of columns - 1). In this case, df = (2-1) x (2-1) = 1.

At a 1% significant level, the critical value is found by looking up the value in the chi-square distribution table for df = 1 and a significance level of 0.01. From the table, the critical value is approximately 6.63.

c. To calculate the test statistic, we can use the chi-square test statistic formula:

χ^2 = Σ [(Observed - Expected)^2 / Expected]

First, we need to calculate the expected frequencies for each cell in the table under the assumption of independence. The expected frequency for a cell is calculated by multiplying the row total and column total for that cell and dividing by the grand total. For example, the expected frequency for the cell in the top left corner is (300*350) / 500 = 210.

Using the formula, we calculate the test statistic as follows:

χ^2 = [(222-210)^2 / 210] + [(78-90)^2 / 90] + [(128-140)^2 / 140] + [(72-60)^2 / 60]
= [(12^2 / 210) + (12^2 / 90) + (12^2 / 140) + (12^2 / 60)]
= (144 / 210) + (144 / 90) + (144 / 140) + (144 / 60)
= 0.686 + 1.6 + 1.028 + 4.8
= 8.114

d. To determine whether to reject or fail to reject the null hypothesis, we compare the test statistic to the critical value. If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the test statistic (8.114) is greater than the critical value (6.63). Therefore, we reject the null hypothesis.

e. Based on the answers to a-d, we can conclude that gender and speeding are likely to be dependent (not independent). The data suggests that there is a relationship between gender and speeding.