6. Independence Test Researchers conducted a study4 investigating the relationsh

Question

6. Independence Test Researchers conducted a study4 investigating the relationship between caffeinated coffee consumption and risk of depression in women. They collected data on 50,739 women free of depression symptoms at the start of the study in the year 1996, and these women were followed through 2006. The researchers used questionnaires to collect data on caffeinated coffee consumption, asked each individual about physician-diagnosed depression, and also asked about the use of antidepressants. The table below shows the distribution of incidences of depression by amount of caffeinated coffee consumption. (a) Is there sufficient evidence of a relationship between coffee intake and depression? Perform hypothesis test; include expected values, hypotheses, test statistic, result, and conclusion in context (b) Do part (a) again in R On the table, c/w means cups of caffeinated coffee per week, c/d means cups of caffeinated coffee per day. caff c('-1c/w', '2-6c/w,'1c/d','2-3c/d', '>-4c/d') depr(' Yes' ,' No' ) coffee-as.table(rbind(c (670,373,905,564,95),c(11545,6244,16329,11726,2288))) dimnames (coffee)

Explanation / Answer

In R

a)

Null Hypothesis :  there is no sufficient evidence of coffee intake and depression.

Alternative Hypothesis :   there is sufficient evidence of coffe intake and depression.

Computation of expected values

Formula: X2 = [ (Oi - Ei)2 / Ei ]

Where Oi is observed value and Ei is the expected valuehence test statistic chi-square = 20.93

Degree of freedom =(r-1)*(c-1) = (2-1)*(5-1) = 4

The critical value at 95% CI and 4 DF is 9.88, Hence we reject nullhypothesis and can conclude that yes there is sufficient evidence of coffe intake and depression.

Conclusion : Since the calculated value 20.93 is greater than 9.88 ,

b) > chisq.test(coffee)Chi-square is given by X2

Pearson's Chi-squared test

data: coffee

X-squared = 20.932, df = 4, p-value = 0.0003267

Expected Value:

Caffeinated.Coffee
Depression <=1c/w 2-6c/w 1c/d 2-3c/d >=c/d
Yes 627.614 339.9854 885.4932 631.4675 122.44
No 11587.39 6277.015 16348.51 11658.53 2260.56

Since the p-value is 0.00032 which is less than 0.05, Hence we reject nullhypothesis and can conclude that yes there is sufficient evidence of coffee intake and depression.

<=1c/w 2-6c/w 1c/d 2-3c/d >=c/d Yes 670 373 905 564 95 2607 No 11545 6244 16329 11726 2288 48132 12215 6617 17234 12290 2383 50739

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