Question

15. Health care workers who use latex gloves with glove powder on a daily basis are particularly susceptible to developing a latex allergy. Each in a sample of 46 hospital employees who were diagnosed with a latex allergy based on a​ skin-prick test reported on their exposure to latex gloves. Summary statistics for the number of latex gloves used per week are

    x overbar equals 19.3

    and

    s equals 12.1

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

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

    3. Give a practical interpretation of the interval you found in part b.

    4. Give the conditions required for the interval in part b to be valid.

Answer 1

Solution:

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

Answer: The point estimate for the average number of latex gloves used per week by all health care workers with a latex allergy is:

$\boldsymbol{\bar{x}=19.3}$

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

Answer: The 95% confidence interval is given below:

$\bar{x} \pm z_{\frac{0.05}{2}}\left ( \frac{s}{\sqrt{n}} \right )$

$z_{\frac{0.05}{2}}=1.96$ is the z critical value at 0.05 significance level

$\therefore$ 95% confidence interval is:

$19.3 \pm 1.96\left ( \frac{12.1}{\sqrt{46}} \right )$

$19.3 \pm3.497$

$\left ( 19.3-3.497,19.3+3.497 \right )$

$\boldsymbol{\left ( 15.803,22.797 \right )}$

Therefore, the 95% confidence interval for the average number of latex gloves used per week by all health care workers with a latex allergy is $\boldsymbol{\left ( 15.803,22.797 \right )}$

Give a practical interpretation of the interval you found in part b.

Answer: There is a 95% confidence that the confidence interval calculated $\boldsymbol{\left ( 15.803,22.797 \right )}$ contains the true
population mean.

Give the conditions required for the interval in part b to be valid

Answer: The conditions required for the interval in part b to be valid are:

1. The sample selected was randomly selected from the target population.

2. The sample size is sufficiently large, that is, n>30.