Pigmented makeup products like mascara and eye shadow may contain metal (e.g., nickel) allergens. Is a nickel allergy more likely to occur in women who report cosmetic dermatitis from using eye shadow or mascara? This was the question of interest in a paper published in the Journal of the European Academy of Dermatology and Venereology (June 2010). In a sample of 131 women with cosmetic dermatitis from using eye shadow, 12 were diagnosed with a nickel allergy. In a sample of 250 women with cosmetic dermatitis from using mascara, 25 were diagnosed with a nickel allergy. Suppose you are informed that the true proportion with a nickel allergy for one of the two groups (eye shadow or mascara) is .12. Can you determine which group is referenced? Explain.

For functions f(x, y, z) = xyz and $g(x, y, z)=x^{2}+2 y^{2}+3 z^{2}-1$, write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes f subject to the constraint g(x, y) = 0.

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 $$ \overline { 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 b . d. Give the conditions required for the interval in part b to be valid.

Under ultraviolet radiation, reactant $\mathrm{A}$ of $C_{\mathrm{A} 0}=10 \mathrm{kmol} / \mathrm{m}^{3}$ in a process stream $\left(v=1 \mathrm{m}^{3} / \mathrm{min}\right)$ decomposes as follows.

$$\begin{matrix} \quad & \quad & \text{R,} & r_{\mathrm{R}}=16 C_{\mathrm{A}}^{0.5} \quad \mathrm{kmol} / \mathrm{m}^{3} \cdot \min\\ \quad & \nearrow\\ \text{A} & \rightarrow & \text{S,} & r_{\mathrm{S}}=12 C_{\mathrm{A}} \quad \mathrm{kmol} / \mathrm{m}^{3} \cdot \min\\ \quad & \searrow\\ \quad & \quad & \text{T,} & r_{\mathrm{T}}=C_{\mathrm{A}}^{2} \quad \mathrm{kmol} / \mathrm{m}^{3} \cdot \min\\ \end{matrix}$$

We wish to design a reactor setup for a specific duty. Sketch the scheme selected, and calculate the fraction of feed transformed into desired product as well as the volume of reactor needed. Product R is the desired material.

In a test of the hypothesis

$$H _ { 0 } : \mu = 10 \text { versus } H _ { a } : \mu \neq 10$$

a sample of n = 50 observations possessed mean

$$\overline { x } = 10.7$$

and standard deviation s = 3.1. Find and interpret the p -value for this test.

Determine the location and value of the absolute extreme values of f on the given interval, if they exist. $f(x)=3 x^{2 / 3}-x \text { on }[0,27]$

To use the t -statistic to test for a difference between the means of two populations, what assumptions must be made about the two populations? About the two samples?

Toe International Nanny Association reports that in a sample of 528 in-home child care providers (nannies), 20 work for either a nationally known, a locally known, or an internationally known celebrity (2011 International Nanny Association Salary and Benefits Survey). Use Wilson's adjustment to find a 95% confidence interval for the true proportion of all nannies who work for a celebrity. Interpret the resulting interval.

What conditions are required for valid large-sample inferences about

$$\mu _ { d } ?$$

small-sample inferences?

According to the National Highway Traffic and Safety Administration’s National Center for Statistics and Analysis (NCSA), “Speeding is one of the most prevalent factors contributing to fatal traffic crashes” ( NHTSA Technical Report , Aug. 2005). The probability that speeding is a cause of a fatal crash is .3. Furthermore, the probability that speeding and missing a curve are causes of a fatal crash is .12. Given that speeding is a cause of a fatal crash, what is the probability that the crash occurred on a curve?

A computer intrusion detection system (IDS) is designed to provide an alarm whenever someone intrudes (e.g., through nauthorized access) into a computer system. A robabilistic evaluation of a system with two independently operating intrusion detection systems (a double IDS) was published in the Journal of Research of the National Institute of Standards and Technology (Nov.–Dec. 2003). Consider a double IDS with system A and system B. If there is an intruder, system A sounds an alarm with probability .9 and system B sounds an alarm with probability .95. If there is no intruder, the probability that system A sounds an alarm (i.e., a false alarm) is .2 and the probability that system B sounds an alarm is .1. a. Use symbols to express the four probabilities just given. b. If there is an intruder, what is the robability that both systems sound an alarm? c. If there is no intruder, what is the probability that both systems sound an alarm? d. Given that there is an intruder, what is the probability that at least one of the systems sounds an alarm?

The heart rate variability (HRV) of police officers was the subject of research published in the American Journal of Human Biology (Jan. 2014). HRV is defined as the variation in the time intervals between heartbeats. A measure of HRV was obtained for each in a sample of 355 Buffalo, NY, police officers. (The lower the measure of HRV, the more susceptible the officer is to cardiovascular disease.) For the 73 officers diagnosed with hypertension, a 95% confidence interval for the mean HRV was (4.1, 124.5). For the 282 officers that are not hypertensive, a 95% confidence interval for the mean HRV was (148.0, 192.6). If you want to reduce the width of each confidence inter val, should you use a smaller or larger confidence coefficient? Explain.

Due to the popularity of instant messaging and social networking sites among American teenagers, informal elements such as emoticons (e.g., the symbol “:)” to represent a smile) and abbreviations (e.g., “LOL” for “laughing out loud”) have worked their way into teenagers’ school writing assignments. In a survey cosponsored by the National Commission on Writing at the College Board, the Pew Internet and American Life Project (April 2008) interviewed 700 randomly selected U.S. teenagers by telephone on their writing habits. Overall, 448 of the teenagers admitted using at least one informal element in school writing assignments. Based on the survey results, construct a 99% confidence interval for the proportion of all U.S. teenagers who have used at least one informal element in school writing assignments. Give a practical interpretation of the interval, one that is relevant to the National Commission on Writing at the College Board.

The days to maturity for a sample of five money market funds are shown here. The dollar amounts invested in the funds are provided. Use the weighted mean to determine the mean number of days to maturity for dollars invested in these five money market funds.

$$\begin{matrix} \text{Days to Maturity} & \text{Dollar Value (\$millions)}\\ \text{20} & \text{20}\\ \text{12} & \text{30}\\ \text{7} & \text{10}\\ \text{5} & \text{15}\\ \text{6} & \text{10}\\ \end{matrix}$$