IHI insurance is interested in determining whether younger drivers (below the age of 25) are more prone to motor accidents than older drivers (25 or older). A random sample of 68 IHI policyholders below the age of 25 were selected and whether they had been in less than 2 (or 2 or more) motor accidents in the past year was recorded. This process was repeated for a random sample of 48 IHI policyholders aged 25 or older. The data from these samples is displayed in the table.

You may find this chi-square distribution table useful throughout the following questions.

a)The IHI statisticians would like to test whether the proportion of policyholders who have had at least 2 accidents in the past year is the same for those who are aged below 25 (<25) and those who are aged 25 or older (¡Ý25). They have constructed a hypothesis test with H0: <25 = ¡Ý25 and H1: <25 ¡Ù ¡Ý25. Calculate the test statistic (¦Ö2) that corresponds to this hypothesis test. Give your answer to 3 decimal places.

¦Ö2 =

Accidents Age Total
< 25 ¡Ý 25
< 2 47 32 79
¡Ý 2 21 16 37
Total 68 48 116

To calculate the test statistic (χ²) for this hypothesis test, we need to follow these steps:

Step 1: Create an observed frequency table.

Accidents Age Total
< 25 ≥ 25
< 2 47 32 79
≥ 2 21 16 37
Total 68 48 116

Step 2: Determine the expected frequencies for each cell. The expected frequency is calculated by the formula:

Expected Frequency = (Row Total * Column Total) / Grand Total

Calculate the expected frequencies for each cell:

Accidents Age Total
< 25 ≥ 25
< 2 40.379 38.621 79
≥ 2 27.621 9.379 37
Total 68 48 116

Step 3: Calculate the chi-square test statistic.

χ² = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency]

For each cell, calculate (Observed Frequency - Expected Frequency)² / Expected Frequency, and sum them up.

Accidents Age
< 25 ≥ 25
< 2 (47-40.379)²/40.379 + (32-38.621)²/38.621
≥ 2 (21-27.621)²/27.621 + (16-9.379)²/9.379

Step 4: Calculate the degrees of freedom. Degrees of freedom (df) is given by (number of rows - 1) * (number of columns - 1).

In this case, df = (2-1) * (2-1) = 1.

Step 5: Refer to the chi-square distribution table using the degrees of freedom to find the critical value at the chosen significance level (usually 0.05 or 0.01). Let's assume a significance level of 0.05.

For df = 1 and a significance level of 0.05, the critical value is approximately 3.841.

Step 6: Compare the calculated χ² value with the critical value.

If χ² > critical value, reject the null hypothesis (H0).
If χ² ≤ critical value, fail to reject the null hypothesis (H0).

In this case, if the calculated χ² value is greater than 3.841, we can reject the null hypothesis.