In: Statistics and Probability
A team of researchers wanted to assess the effectiveness of the 4 components of a weight loss program by comparing the distribution of weight status across 4 different treatment groups. The data is summarized in the table below:
Weight Status |
|||
Intervention Component |
Normal Weight |
Overweight |
Obese |
Control Group (n=100) |
50 |
35 |
15 |
Calorie Counting (n=100) |
70 |
20 |
10 |
Physical Activity (n=100) |
73 |
15 |
12 |
Combination (n=100) |
80 |
15 |
5 |
Is there a difference in weight status by intervention component? (Hint: Are intervention component and weight status independent?) Run the appropriate test at a 5% level of significance.
Indicate the correct competing hypotheses: (1pt)
H0: Intervention component and weight status are not independent.
H1: The null hypothesis is not false.
H0: Intervention component and weight status are independent.
H1: The null hypothesis is false.
H0: Intervention component and weight status are independent.
H1: The null hypothesis is not false.
H0: Intervention component and weight status are not independent.
H1: The null hypothesis is false.
The null and alternative hypothesis is
H0: intervention component and weight status independent.
H1: intervention component and weight status not independent.
Test statistic is
O: Observed frequency
E: Expected frequency.
E = ( Row total*Column total) / Grand total
O | E | (O-E) | (O-E)^2 | (O-E)^2/E |
50 | 68.25 | -18.25 | 333.0625 | 4.880037 |
35 | 21.25 | 13.75 | 189.0625 | 8.897059 |
15 | 10.5 | 4.5 | 20.25 | 1.928571 |
70 | 68.25 | 1.75 | 3.0625 | 0.044872 |
20 | 21.25 | -1.25 | 1.5625 | 0.073529 |
10 | 10.5 | -0.5 | 0.25 | 0.02381 |
73 | 68.25 | 4.75 | 22.5625 | 0.330586 |
15 | 21.25 | -6.25 | 39.0625 | 1.838235 |
12 | 10.5 | 1.5 | 2.25 | 0.214286 |
80 | 68.25 | 11.75 | 138.0625 | 2.022894 |
15 | 21.25 | -6.25 | 39.0625 | 1.838235 |
5 | 10.5 | -5.5 | 30.25 | 2.880952 |
Total | 24.973 |
Degrees of freedom = ( Number of rows - 1 ) * ( Number of column - 1) = ( 4 - 1) * (3 - 1) = 3 * 2 = 6
Level of signfiicance = 0.05
Critical value = 12.592
Test statistic > critical value we reject null hypothesis.
Conclusion: Intervention component and weight status are not independent.