A student team examined parked cars in four different suburban shopping malls. 100 vehicles were examined in each location. At a = .05, does vehicle type vary by mall location?
Vehicles
Vehicle Somer Oakland Great James Row
Type set Lakes town Total
Car 44 49 36 64 193
Minivan 21 15 18 13 67
Reg.van 2 3 3 2 10
SUV 19 27 26 12 84
Truck 14 6 17 9 46
COL 100 100 100 100 400
total
To determine if vehicle type varies by mall location, we can conduct a chi-squared test for independence. This test will help us determine if there is a statistically significant relationship between the two categorical variables (vehicle type and mall location).
Here's how to conduct the chi-squared test:
Step 1: Set up hypotheses
- Null hypothesis (H₀): Vehicle type is independent of mall location.
- Alternative hypothesis (H₁): Vehicle type is dependent on mall location.
Step 2: Calculate expected values
- Calculate the expected values for each cell in the contingency table. The expected value for each cell can be calculated using the formula: (row total * column total) / grand total.
Here is the table with the expected values calculated:
Somer Oakland Great James Row total
Car 48.25 48.25 48.25 48.25 193
Minivan 12.23 12.23 12.23 12.23 67
Reg.van 2.5 2.5 2.5 2.5 10
SUV 21 21 21 21 84
Truck 9.02 9.02 9.02 9.02 46
Col total 100 100 100 100 400
Step 3: Calculate the chi-squared test statistic
- Calculate the chi-squared test statistic using the formula:
χ² = Σ((O - E)² / E), where O is the observed frequency and E is the expected frequency.
Step 4: Determine the critical value
- Determine the critical value for α = 0.05 and the degrees of freedom (df) = (number of rows - 1) * (number of columns - 1).
The degrees of freedom can be calculated as (number of rows - 1) * (number of columns - 1) = (5 - 1) * (4 - 1) = 12.
Using a chi-squared critical value table or calculator with 12 degrees of freedom and α = 0.05, we can find the critical value, which is 21.03.
Step 5: Compare the test statistic to the critical value
- If the calculated test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Interpret the results
- If the null hypothesis is rejected, it means that there is evidence to suggest that vehicle type is dependent on mall location. If the null hypothesis is not rejected, it means that there is no evidence to suggest a relationship between vehicle type and mall location.
So, to find out if the vehicle type varies by mall location, calculate the chi-squared test statistic and compare it to the critical value of 21.03 with 12 degrees of freedom.