Latex allergy in health care workers. Refer to the Current Allergy & Clinical Immunology (March 2004) study of n = 46 hospital employees who were diagnosed with a latex allergy from exposure to the powder on latex gloves, Exercise 6.112 (p. 375). The number of latex gloves used per week by the sampled workers is summarized as follows: \(\bar x = 19.3\) and s = 11.9. Let \(\mu \) represent the mean number of latex gloves used per week by all hospital employees. Consider testing \({H_0}:\mu = 20\) against \({H_a}:\mu < 20.\)

a. Give the rejection region for the test at a significance level of \(\alpha = 0.01.\)

The number of hospital employees who were diagnosed with a latex allergy, n=46.

The sample mean is, \(\bar x = 19.3.\)

The sample standard deviation is,\(s = 11.9.\)

The test hypothesis is \({H_0}:\mu = 20\) against \({H_a}:\mu < 20.\)

The elements of the one-tailed hypothesis test are the null hypothesis\(\left( {{H_0}} \right)\), alternative hypothesis\(\left( {{H_a}} \right)\), and test statistic. The sample statistic is used to decide whether to reject the null hypothesis.

A one-tailed test of a hypothesis is one in which the alternative hypothesis is directional and includes the symbol “<” or “>”.

The one-tailed hypothesis are:

a.

Let \(\mu \)represent the mean number of latex gloves used per week by all hospital employees.

The null and alternative hypothesis are:

\(\begin{aligned}{l}{H_0}:\mu = 20\\{H_a}:\mu < 20\end{aligned}\)

The significance level, \(\alpha = 0.01.\)

The rejection region for the one-tailed test is \({Z_c} < - {Z_\alpha }\).

The \( - {Z_\alpha }\) corresponding significance level \(\alpha = 0.01\) from the standard normal table is -2.33.

Therefore,

The rejection region can written as:

\({Z_c} < - {Z_\alpha } \Rightarrow {Z_c} < - 2.33.\)