Question

Latex allergy in health care workers. Health care workers who use latex gloves with glove powder on a daily basis are particularly susceptible to developing a latex allergy. Each sample of 46 hospital employees who were diagnosed with a latex allergy based on a skin-prick test reported their exposure to latex gloves (Current Allergy& Clinical Immunology, March 2004). Summary statistics for the number of latex gloves used per week is ,$\overline{x}=19.3$

s = 11.9.

a. Give a point estimate for the average number of latex gloves used per week by all health care workers with a latex allergy.

b. Form a 95% confidence interval for the average number of latex gloves used per week by all health care workers with a latex allergy.

c. Give a practical interpretation of the interval, part b.

d. Give the conditions required for the interval, part b, to be valid.

Short answer

a. The point estimate for average number of latex gloves used per week by all health care workers with a latex allergy is$\overline{x}=19.3$

b. The 95% confidence interval is (15.861, 22.738)

c. This means that a 95% confidence interval of the mean isa range with an upper and lower number calculated from a sample.

d. The assumptions that should be met to build a confidence interval are that the sample should be derived by random sampling, and the variables in the sample should have Independence.

Step-by-step solution

Given Information

It is given that the sample size consists of 46 hospital employees, therefore, n=46.

The summary statistics for the number of latex gloves used per week are $\overline{x}=19.3,s=11.9$, that is sample mean and sample standard deviation respectively.

Find the point estimate for average number of latex gloves.

a.

To find the point estimate for an average number of latex gloves used per week by all health care workers with a latex allergy we need to substitute it with the sample mean value that is already in the given information.

Therefore,$\overline{x}=19.3$ , is the point estimate.

Find the confidence interval

b.

The 95% confidence interval for average number of latex gloves used per week by all health care workers with a latex allergy is denoted by the formula:

$CI=\overline{x}\pm z\frac{s}{\sqrt{n}}$

Now, $\overline{x}=19.3,s=11.9$and n=46. Since n=46, which is greater than 30, we will use the normal distribution. Therefore, we will use the Z confidence interval.

Also, z value for 95% level of significance is$\pm 1.96$

Substituting the values we get,

$\begin{array}{rcl}CI& =& 19.3\pm 1.96\times \frac{11.9}{\sqrt{46}}\\ & =& 19.3\pm 3.4389\\ & =& 15.861,22.738\\ & & \end{array}$

Therefore the 95% confidence interval is (15.861, 22.738).

Interpreting the confidence interval

c.

The interval (15.861, 22.738) encloses an unknown population parameter with a certain level of confidence that is 95%. This means that one is 95% confident that the true mean lies in the given interval.

Specifying the assumptions of the confidence interval

c.

The assumptions that should be met to build a confidence interval are that the sample should be derived by random sampling, and the variables in the sample should have Independence. The Central Limit Theorem should be applicable.