Question

A particular paper described a study of children who were underweight or normal weight at age 2. Children in the sample were classified according to the number of sweet drinks consumed per day and whether or not the child was overweight one year after the study began. Is there evidence of an association between whether or not children are overweight after one year and the number of sweet drinks consumed? Assume that it is reasonable to regard the sample of children in this study as representative of 2 to 3 years old children and then test the appropriate hypotheses using a .05 significance level. (Round your answer to two decimal places.)

Answer 1

Conclusion: Children are overweight after one year are not dependent on the number of sweet drinks consumed

Explanation:

Solution:-

- The hypothesis is to whether or not the child was overweight after the consumption of sweet drink. The sample data is given in the form of a table:

Yes. No

0 22. 930

1. 73. 2071

2. 56. 1681

3<. 102 3386

- The above given is a two way table, comparing the number of sweet drink consumed vs. (yes or no) the children are found to be overweight (dependence).

- State the hypothesis:

Null Hypothesis : Children are overweight after one year are not dependent on the number of sweet drinks consumed.

Alternate Hypothesis : Children are overweight after one year are dependent on the number of sweet drinks consumed.

- Determine the expected table for the given data:

Yes. No Total of row

0 22 930 952

1 73 2071 2144

2 56 1681 1737

3< 102 3386 3488

Total Column 253 8068 Total : 8321

- The expected table would be calculated by using each element of given table:

Element = (Sum of column)*(Sum of Row) / Total

E (X) Yes. No

0 253*952/8321 8068*952/8321

1 253*2144/8321 8068*2144/8321

2 253*1737/8321 8068*1737/8321

3< 253*3488/8321 8068*3488/8321

- Resulting values are tabulated:

E (X) Yes. No

0 28.95 923.05

1 65.188 2078.812

2 52.81 1684.19

3< 106.05 3381.947

- Now determine the Chi-square test statistic:

`X^2 = Sum ( Xi - E(Xi) )^2 / E(X) \\\\X^2 = ( 22 - 28.95 )^2 / 28.95 + ( 930 - 923.05 )^2 / 923.05 + ( 73 - 65.188 )^2 / 65.188\\\\ + ( 2071 - 2078.812 )^2 / 2078.812 + ( 56 - 52.81 )^2 / 52.81 + ( 1681 - 1684.19 )^2 / 1684.19\\\\ + ( 102 - 106.05 )^2 / 106.05 + ( 3386 - 3381.947 )^2 / 3381.947 \\\\X^2 = 1.666 + .052 + .93 + .029 + .192 + .006 + .15 + .004 =3.04\\`

- Then use chi-square look up table t -test and determine the p-value for the rejection of null hypothesis:

Significance level ( α ) = 0.05

Degree of freedom = ( 4 - 1 ) = 3

p-value = 0.25

- The X^2 > 0.25 : Fail to reject the Null hypothesis.

Conclusion: Children are overweight after one year are not dependent on the number of sweet drinks consumed