Question

7. The following data summarizes the incidence of Coronary Heart Disease (CHD) for people who smoked regularly and for people who did not smoke regularly: CHD No CHD Smoked 84 87 Did not Smoke 2916 4913 (a) State the null and alternative hypothesis for testing if CHD status is independent of smoking status. (b) Calculate the Likelihood-Ratio test-statistic and its corresponding p-value. (c) State your conclusion in terms of the problem if α = 0.05. (d) Describe what “more extreme” would mean in terms of this null. What would it mean for our data to be “different” than the null. (e) Can we say with statistical evidence that smokers have a higher chance of having CHD based on this test? Explain.

Answer 1

Here we have to test the hypothesis that,

H0 : CHD status is independent of smoking status.

H1 : CHD status is dependent of smoking status.

Assume alpha = level of significance = 0.05

This is the test of independence.

The test statistic follows X^2-distribution with (R-1)(C-1) degrees of freedoms.

where R is number of rows

C is number of columns

The test statistic is,

$X^2 = \frac{\sum (O-E)^{2}}{E}$

where O is observed frequency and

E is expected frequency

We can do this test in MINITAB.

steps :

ENTER data into MINITAB sheet --> Stat --> Tables --> Chi-Square test for association --> CLick on summarized data in a two way table --> COlumns containing the table: Select all the data columns together --> ok

————— 30-01-2019 18:54:12 ————————————————————

Welcome to Minitab, press F1 for help.

Chi-Square Test for Association: Worksheet rows, Worksheet columns

Rows: Worksheet rows Columns: Worksheet columns

C1 C2 All

1 84 87 171
64.1 106.9

2 2916 4913 7829
2935.9 4893.1

All 3000 5000 8000

Cell Contents: Count
Expected count

Pearson Chi-Square = 10.071, DF = 1, P-Value = 0.002
Likelihood Ratio Chi-Square = 9.772, DF = 1, P-Value = 0.002

Likelihood ratio test statistic = 9.772

P-value = 0.002

P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : CHD status is dependent of smoking status.

There is statistical evidence that smokers have a higher chance of having CHD based on this test