91Ó°ÊÓ

Avobenzone is one of the active ingredients in several commercially available sunscreens. It can be absorbed into the bloodstream when sunscreen is applied to the skin. The Food and Drug Administration has expressed concern about the safety of absorbing too much avobenzone. Amounts less than or equal to \(0.5 \mathrm{ng} / \mathrm{mL}\) (nanograms absorbed per milliliter applied) are considered acceptable. Researchers recruited 12 healthy volunteers to investigate avobenzone absorption for two different commercially available sunscreens: a spray and a cream. Subjects were randomly assigned to one of the two sunscreens, with 6 subjects for each. Subjects had 2 milligrams of sunscreen per \(1 \mathrm{~cm}^{2}\) applied to \(75 \%\) of their body surface area (area outside of normal swimwear). The amount of avobenzone absorbed into the bloodstream after 6 hours was then measured for each subject. Four of the six subjects receiving the spray had avobenzone levels exceeding \(0.5 \mathrm{ng} / \mathrm{mL}\), and one of the six subjects receiving the cream had levels exceeding \(0.5 \mathrm{ng} / \mathrm{mL} .16\) The \(z\) test for "no difference" in the two proportions exceeding \(0.5 \mathrm{ng} / \mathrm{mL}\) against "the two proportions differ" has a. \(z=1.76, P<0.05\). b. \(z=1.84, P<0.055\). c. \(z=1.76,0.05

Step by step solution

01

Understanding the Problem

We have two groups of data from two types of sunscreen: a spray and a cream. We want to compare the proportions of subjects in each group with avobenzone levels exceeding 0.5 ng/mL.

02

Identify the Proportions

For the spray, 4 out of 6 subjects exceeded 0.5 ng/mL, so the proportion is \( \frac{4}{6} \). For the cream, 1 out of 6 subjects exceeded 0.5 ng/mL, so the proportion is \( \frac{1}{6} \).

03

Calculate Pooled Proportion

The pooled proportion \( p \) is calculated from both samples:\[p = \frac{x_1 + x_2}{n_1 + n_2} = \frac{4 + 1}{6 + 6} = \frac{5}{12}\]

04

Compute Standard Error

The standard error (SE) for the difference in proportions is calculated as:\[SE = \sqrt{p(1-p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{\frac{5}{12}\cdot\frac{7}{12}\left(\frac{1}{6} + \frac{1}{6}\right)}\]

05

Calculate the Z Score

The Z-score is calculated with the formula:\[z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{\frac{4}{6} - \frac{1}{6}}{SE}\] Substitute \( SE \) from the previous step to find \( z \approx 1.76 \).

06

Interpret the Results

The question provides options for interpretation of \( z \) and \( p \)-values. The calculated \( z = 1.76 \), and since \( z = 1.76 \) with a \( p \)-value between 0.05 and 0.10 matches option (c), choose this option.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sunscreen Absorption Study

Sunscreen absorption studies aim to understand how ingredients like avobenzone are absorbed through the skin and enter the bloodstream. The concern around avobenzone absorption is due to potential health implications when levels surpass acceptable thresholds. In this study, researchers were interested in the absorption levels of avobenzone from two different sunscreens: a spray and a cream. Twelve healthy volunteers participated, with half using the spray and the other half using the cream. The principal goal was to track whether the avobenzone level in the bloodstream exceeded 0.5 nanograms per milliliter. To ensure that the study was fair and unbiased, subjects were randomly assigned to either the spray or the cream sunscreen group. This random assignment helps to reduce potential biases and balances out other variables that could affect absorption rates. Each subject applied a significant amount of sunscreen, specifically two milligrams per square centimeter, over most of their body—75% to be exact—outside the area that typical swimwear would cover. This thorough application is aimed at closely mimicking real-world scenarios where sunscreen is applied generously over large skin areas.

Proportions Comparison

When comparing the effects of two different sunscreens, proportions of subjects with high levels of avobenzone absorption in their bloodstream were calculated. This means looking at how many subjects in each group had absorption levels that climbed over the 0.5 ng/mL mark. In the study, you have two groups: one group using the spray sunscreen, and the other using the cream. The proportions of those exceeding 0.5 ng/mL were:

• Spray sunscreen: 4 out of 6 subjects
• Cream sunscreen: 1 out of 6 subjects

These proportions translate to about 67% for the spray group and about 17% for the cream group. By examining these proportions, researchers try to determine if there's a significant difference in absorption rates between the two types of sunscreens. The comparison is crucial because it highlights whether one type of sunscreen may lead to higher levels of avobenzone absorption than the other. This information is essential for assessing the safety and effectiveness of these products.

Avobenzone Safety

Avobenzone is a common active ingredient used in sunscreens to provide broad-spectrum UV protection. However, the safety of avobenzone is contingent on how much is absorbed into the bloodstream. The threshold that delineates safe levels from potentially concerning levels is 0.5 ng/mL. When levels rise above this mark, it triggers safety concerns as chronic exposure could pose health risks. The study aims to quantify how different types of sunscreen formulations—spray versus cream—contribute to the levels of avobenzone absorbed. Understanding these differences is crucial, as products that result in higher absorption can pose health risks if they frequently push absorption levels above acceptable limits. By studying avobenzone absorption, researchers, regulatory bodies, and consumers can make better-informed decisions on sunscreen use. If a particular formulation consistently contributes to excessive absorption levels, it may prompt reformulation or indicate a need for restrictions on its use.

Statistical Hypothesis Testing

Statistical hypothesis testing is the backbone of such studies, allowing researchers to determine if the differences in avobenzone absorption between the two sunscreen groups are significant. In this particular case, they are interested in comparing the proportions of people exceeding 0.5 ng/mL of avobenzone in the bloodstream between the two types. The null hypothesis (H0) claims that there is no difference in proportions between the groups using spray and cream sunscreens. In contrast, the alternative hypothesis (H1) posits that a difference does exist. By calculating a z-score (approximately 1.76 in this instance), researchers quantify the difference in standard deviations between the observed and expected results. The calculated p-value reflects the probability of observing this difference under the null hypothesis. If the p-value is small (usually less than 0.05), the null hypothesis can be rejected. However, in this context, the p-value is between 0.05 and 0.10, indicating that the data is suggestive but not definitive. Therefore, while there may be a tendency towards higher absorption in the spray group, the evidence isn't strong enough to conclude a significant difference.