Question

Use the research described below to answer the questions that follow. Shargorodsky, Curhan, Curhan, and Eavey (2010) examined hearing loss data for participants from the National Health and Nutrition Examination Survey (NHANES), aged 12-19 years. (NHANES provides nationally representative cross-sectional data on the health status of the civilian, non-institutionalized U.S. population.) The researchers compared the more recent hearing loss rate among teens (12-19 years) in 2005 to previous levels in 1988. Their goal was to see whether teen hearing loss is increasing, possibly due to heavier use of ear buds. They examined data from 1771 participants in the NHANES 2005 study (333 with some level of hearing loss), as well as data on 2928 teens from the NHANES 1988 study, with 480 in this group showing some level of hearing loss.

QUESTION 1: Calculate the p-value

QUESTION 2: Calculate an appropriate confidence interval for the change (if it exists) in hearing loss.

Answer 1

Q1) P-value =0.04252

Q2) 95% confidence interval for the change =(0.0018 ,0.0464)

Solution:

For 2005 x1 = 333 n1=1771 p1=333/1771=0.188029

For 1988 x2=480 n2=2928, p2= 480/2928=0.163934

p-hat = ( x1 + x2) / ( n1 + n2). = (333+480)/(1771+2928) =0.173016

Here Null and alternative hypothesis are below:

H0 : p1=p2

H1: p1 > p2

To calculate we first need to calculate test statistics z and then calculate p(Z>z)

Test statistics z=(p1-p2)/sqrt(p-hat*(1-p-hat)*(1/n1+1/n2

z=(0.188029- 0.163934)/sqrt(0.173016*(1-0.173016)*((1/1771)+(1/2928)))

=2.1160

P-Value

For Ha: p1 - p2 > 0, we calculate the proportion of the normal distribution that is greater than Z.

i.e p(Z> 2.1160)=0.04252

P-value =0.04252

appropriate confidence interval

95% Confidence interval

((p1-p2) - Z_alpha*sqrt(p-hat*(1-p-hat)*(1/n1+1/n2,(p1-p2) + Z_alpha*sqrt(p-hat*(1-p-hat)*(1/n1+1/n2)

at alpha=0.05  Z_alpha=1.96

CI= ((0.188029- 0.163934)-1.96*sqrt(0.173016*(1-0.173016)*((1/1771)+(1/2928))),(0.188029- 0.163934)+1.96*sqrt(0.173016*(1-0.173016)*((1/1771)+(1/2928))))

=(0.0018 ,0.0464)

95% confidence interval for the change =(0.0018 ,0.0464)