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What conditions are required to form a valid large-sample confidence interval for \(\mu ?\)

What is the confidence level of each of the following confidence intervals for \(\mu\) ? a. \(\bar{x} \pm 1.96\left(\frac{\sigma}{\sqrt{n}}\right)\) b. \(\bar{x} \pm 1.645\left(\frac{\sigma}{\sqrt{n}}\right)\) c. \(\bar{x} \pm 2.575\left(\frac{\sigma}{\sqrt{n}}\right)\) d. \(\bar{x} \pm 1.28\left(\frac{\sigma}{\sqrt{n}}\right)\) e. \(\bar{x} \pm .99\left(\frac{\sigma}{\sqrt{n}}\right)\)

Health care workers who use latex gloves with glove powder on a daily basis are particularly susceptible to developing a latex allergy. Symptoms of a latex allergy include conjunctivitis, hand eczema, nasal congestion, a skin rash, and shortness of breath. Each in a sample of 46 hospital employees who were diagnosed with latex allergy based on a skin-prick test reported on their exposure to latex gloves (Current Allergy \& Clinical Immunology, March 2004). Summary statistics for the number of latex gloves used per week are \(\bar{x}=19.3\) and \(s=11.9\) 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 you found in part \(\mathbf{b}\). d. Give the conditions required for the interval in part b to be valid.

Suppose you have selected a random sample of \(n=7\) measurements from a normal distribution. Compare the standard normal \(z\) -values with the corresponding \(t\) -values if you were forming the following confidence intervals: a. \(80 \%\) confidence interval b. \(90 \%\) confidence interval c. \(95 \%\) confidence interval d. \(98 \%\) confidence interval e. \(99 \%\) confidence interval fo Use the table values you obtained in parts a-e to sketch the \(z\) - and \(t\) -distributions. What are the similarities and differences?

What conditions are required for a valid confidence interval for \(\sigma^{2}\) ?