Question

A study was done to check whether there is a relationship betwen snoring and heart disease.

Group	Having heart disease (xx)	Total (nn)
Snorers	118	1175
Non-snorers	76	944

Let p1p1 and p2p2 represent population proportions of snorers and non-snorers respectively who are having heart disease.

1. What proportions of the snorers are having heart disease? [answer to 3 decimal places - answer in fraction, NOT in percentage]

2. What proportions of the non-snorers are having heart disease? [answer to 3 decimal places - answer in fraction, NOT in percentage]

3. What is the standard error of the difference of sample proportions, p̂ 1−p̂ 2p^1−p^2? [answer to 4 decimal places]

4. Which of the following formulas gives the 98% confidence interval for p1−p2p1−p2?
a. 0.02∓2.576×0.01250.02∓2.576×0.0125
b. 0.02∓1.96×0.01250.02∓1.96×0.0125
c. 0.02∓2.326×0.01250.02∓2.326×0.0125
d. 0.02∓1.645×0.01250.02∓1.645×0.0125

5. What is the margin of error of the 98% confidence interval of p1−p2p1−p2? [answer to 4 decimal places]

6. On the basis of the given information, can we conclude that the population proportion of snorers having heart disease is greater than the proportion of non-snorers with heart disease?
a. yes, because the entire 98% confidence interval for p1−p2p1−p2 is above zero
b. yes, because p1>p2p1>p2
c. yes, because p̂ 1>p̂ 2p^1>p^2
d. no, because the 98% confidence interval for p1−p2p1−p2 is not entirely above zero

Answer 1

1)

proportions of the snorers are having heart disease=118/1175 = 0.100

2)

proportions of the non-snorers are having heart disease=76/944=0.081

3)

pooled proportion , p =   (x1+x2)/(n1+n2)=   0.0916
      
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.0125

4)

level of significance, α =   0.02      
Z critical value =   Z α/2 =    2.326 [excel function: =normsinv(α/2)

confidence interval is (p1^ - p2^) ± Z*SE

(0.02 ± 2.326*0.0125)

5)

margin of error , E = Z*SE =    2.326   *   0.0125   =   0.0290

6)

confidence interval is (p1^ - p2^) ± Z*SE

(0.02 ± 2.326*0.0125)

so, confidence interval is (   -0.0091   < p1 - p2 <   0.0489   )  

answer is

d. no, because the 98% confidence interval for p1−p2p1−p2 is not entirely above zero