Question

Researchers at the National Cancer Institute re- leased the results of a study that investigated the effect of weed-killing herbicides on house pets. They examined 827 dogs from homes where an herbicide was used on a regular basis, diagnosing malignant lymphoma in 473 of them. Of the 130 dogs from homes where no herbicides were used, only 19 were found to have lymphoma.a) What’s the standard error of the difference in the two proportions? b) Construct a 95% confidence interval for this difference. c) State an appropriate conclusion.

Answer 1

A)The Standard error for difference in proportion is 0.0354

B)The Confidence interval with 95 % for difference in proportion is (0.356 ,0.495)

C)The conclusion is that proportion of herbicides and no herbicides and having them diagnosed with malignant lymphoma is between 35.6 % and 49.5 % And higher than the proportion to pets in homes where no pesticides were used.

Explanation:

Given:

Total dogs which are herbicide =827=n1

diagnosing malignant lymphoma Dogs=473

Total no herbicides Dogs=130=n2

19 were found to have lymphoma.

To Find:

a)What’s the standard error of the difference in the two proportions?

b) Construct a 95% confidence interval for this difference.

c) State an appropriate conclusion

Solution:

The standard error of difference is given by ,

Consider S1 for case 1 (herbicides) and S2 for case 2 (no herbicides)

So

`S1-S2=Sqrt[p1(1-p1)/n1 +p2(1-P2)/n2)]`

Here
`p1=473/827` =0.578 and
`p2=19/130`=0.146

`S1-S2=Sqrt[0.572(0.428)/827 +0.146(0.854)/130]`

`S1-S2=Sqrt[2.96e-4 +9.59e-4]`

`S1-S2=Sqrt[0.001255]`

`S1-S2=0.0354`

This gives the difference between two proportions 0.0354

Now,

For 95 % C.I it is given by for above difference at Z=1.96

`C.I.=[p1-p2-Z*Sqrt[p1(1-p1)/n1 +p2(1-P2)/n2) ,p1-p2+Z*Sqrt[p1(1-p1)/n1 +p2(1-P2)/n2)]`

`C.I=0.426-1.96*Sqrt[0.57(0.42)/827 +0.146(0.85)/130],0.426+1.96*Sqrt[0.57(0.42)/827 +0.146(0.85)/130]`

`C.I.=(0.356,0.495)`

The conclusion is that proportion of herbicides and no herbicides and having them diagnosed with malignant lymphoma is between 35.6 % and 49.5 % And higher than the proportion to pets in homes where no pesticides were used.