3. Nationally, the distribution of weapons used in robberies is as shown in the table along with the weapon used in 1652 robberies on school property.
Weapon Overall 1652 Robberies
Gun 0.42 329
Knife 0.09 122
Strong-arm 0.40 857
Other 0.09 344
Does the distribution of weapons choice in robberies in schools follow the national distribution? Use á = 0.05 level of significance.
To determine if the distribution of weapons choice in robberies on school property follows the national distribution, we can perform a chi-square test of independence.
Let's set up the hypotheses:
Null Hypothesis (H0): The distribution of weapons choice in robberies on school property follows the national distribution.
Alternative Hypothesis (H1): The distribution of weapons choice in robberies on school property does not follow the national distribution.
To calculate the expected frequencies for each weapon category, we need to multiply the overall proportion of each weapon by the total number of robberies on school property (1652).
First, let's calculate the expected frequencies:
Weapon Overall 1652 Robberies Expected Frequency(Gun) Expected Frequency(Knife) Expected Frequency(Strong-arm) Expected Frequency(Other)
Gun 0.42 329 (0.42 * 1652) (0.09 * 1652) (0.40 * 1652) (0.09 * 1652)
Knife 0.09 122 (0.42 * 1652) (0.09 * 1652) (0.40 * 1652) (0.09 * 1652)
Strong-arm 0.40 857 (0.42 * 1652) (0.09 * 1652) (0.40 * 1652) (0.09 * 1652)
Other 0.09 344 (0.42 * 1652) (0.09 * 1652) (0.40 * 1652) (0.09 * 1652)
Next, we calculate the chi-square test statistic:
χ2 = Σ ((Observed Frequency - Expected Frequency)^2 / Expected Frequency)
We need to calculate this for each weapon category and sum up the values.
Finally, we compare the calculated chi-square test statistic with the critical value from the chi-square distribution table at the given level of significance (α = 0.05). If the calculated chi-square value is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
I hope this helps you understand how to determine if the distribution of weapon choices in robberies on school property follows the national distribution.