Question

Losses to Robbery. At the $5\%$significance level, do the data provide sufficient evidence to conclude that a difference in mean losses exists among the three types of robberies? Use one-way ANOVA to perform the required hypothesis test. (Note: $T_1=4899,T_2=7013,T_3=4567$and $5x^2=16,683,857.)$

Short answer

The data provided is adequate to determine that the three categories of robberies have different mean losses.

Step-by-step solution

Given information

The given data is

Explanation

Let us consider the test hypothesis

Null hypothesis

$H_0$: There is no evidence that difference exist three mean losses.

Alternative hypothesis:

$H_a$: There is a difference exist in the three losses.

Total number of samples

$n=\sum n$

$=4899+7013+4567$

$=16479$

Total sample size

$\sum x=\sum T$

$=4899+7013+4567$

$=16479$

The number of samples

$k=3$

$\sum x^2=16,683,857$

$SST=\sum x^2-\frac{{\left(\sum x\right)}^2}{n}$

$=16,683,857-\frac{{16479}^2}{17}$

$=09889.9$

$SSTR=\sum \frac{T_i^2}{n_i}-\frac{{\left(\sum x\right)}^2}{n}$

$=\frac{{4899}^2}{5}+\frac{{7013}^2}{5}+\frac{{4567}^2}{5}-\frac{{16479}^2}{17}$

$=499349.4157$

The ANOVA table is

The critical value at $5\%$significant level is $=3.7389$.

Critical value approach:

The level of significance $\alpha =0.05$

$F-statistic(16.60)>\alpha (0.05)$The null hypothesis is rejected

As a result, the results are statistically significant at the$5\%$level.

As a result, the data provided is adequate to determine that the three categories of robberies have different mean losses.