Question

Latex allergy in health care workers. Refer to the Current Allergy & Clinical Immunology (March 2004) study of health care workers who use latex gloves, Exercise 6.112 (p. 375). In addition to the 46 hospital employees who were diagnosed with a latex allergy based on a skin-prick test, another 37 health care workers were diagnosed with the allergy using a latex-specific serum test. Of these 83 workers with confirmed latex allergy, only 36 suspected that they had the allergy when asked on a questionnaire. Make a statement about the likelihood that a health care worker with latex allergy suspects he or she actually has the allergy. Attach a measure of reliability to your inference.

Short answer

It is 95% confident that the proportion of health care workers with a latex allergy suspect their allergy is between 0.32712 and 0.54011.

Step-by-step solution

Given information

Let x be the number of workers who had allergies they suspected on the questionnaire.

So, x = 36

Also, the total numbers of workers are 83. That is, n = 83

Calculating the confidence interval

Since, out of 83 workers with confirmed latex allergy, only 36 suspected that they had the allergy.

Therefore, the proportion is given as,

$$\begin{gathered} p=\frac{36}{83} \\ =0.4337 \end{gathered}$$

Confidence Interval

The confidence interval for proportion is given by,

$\mathit{p}\mathbf{\pm }{\mathit{z}}_{\frac{\mathbf{\alpha }}{\mathbf{2}}}\mathbf{\times }\sqrt{\frac{\mathbf{p}\left(1-p\right)}{\mathbf{n}}}$

Now, from the standard normal table, the value $z_{\frac{\alpha }{2}}$for a 95% confidence level is 1.96.

Therefore,

$$\begin{gathered} p\pm z_{\frac{\alpha }{2}}\times \sqrt{\frac{p\left(1-p\right)}{n}}=0.4337\pm 1.96\times \sqrt{\frac{0.4337\left(1-0.4337\right)}{83}} \\ =0.4337\pm 1.96\times \sqrt{\frac{0.4337\times 0.5667}{83}} \\ =0.4337\pm 0.10662 \end{gathered}$$

$$\begin{gathered} =\left(0.4337-0.10662,0.4337+0.10662\right) \\ =\left(0.32712,0.54054\right) \end{gathered}$$

Therefore, it is 95% confident that the proportion of health care workers with a latex allergy suspects their allergy is between 0.32712 and 0.54011.