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. Researchers recruited 24 healthy volunteers to investigate avobenzone absorption for four different commercially available sunscreens: two sprays, a lotion, and a cream. Do the types of sunscreens (spray, lotion, and cream) differ in terms of how much avobenzone is absorbed into the blood? Subjects were randomly assigned to one of the four sunscreens, with six subjects for each. Subjects had 2 milligrams of sunscreen per 1 cm? applied to 75% of their body surface area (area outside of normal swimwear) four times per day for four days. The amount of avobenzone absorbed into the bloodstream (in nanograms per milliliter ap-plied, ng/mL) after 150 hours was then measured for each subject. For the purpose of analyzing the data us ing ANOVA, the logarithms of the measurements were used. Thus, for example, a value of 0 corresponds to an absorption of 1 ng/ mL and negative values correspond to absorptions less than 1 ng/ mL. Larger values of the logarithms correspond to larger levels of absorption.Figure 27.13 shows JMP ANOVA output for these data. Show the work on how the values are acquired.a. Do these data satisfy the conditions for ANOVA?b. State Ho and H. for the ANOVA F test and explain in words what ANOVA tests in this setting.c. Report your overall conclusions about whether the types of sunscreen differ in the mean amount of avobenzone absorbed into the bloodstream after 150 hours. The table displays comparative data for different topical treatments over several entries. Each treatment option—Spray 1, Spray 2, Lotion, and Cream—is associated with a set of numerical values that likely represent measurements or evaluations under various conditions.### Table: Comparative Analysis of Topical Treatments| Treatment | Entry 1 | Entry 2 | Entry 3 | Entry 4 | Entry 5 | Entry 6 ||-----------|---------|---------|---------|---------|---------|---------|| Spray 1 | 0.693 | 0.405 | 0.405 | -0.511 | -0.916 || Spray 2 | 0.588 | 0.000 | 0.000 | -0.511 | -0.693 | -1.204 || Lotion | 1.386 | 0.693 | 0.182 | 0.095 | -0.511 | 0.511 || Cream | -0.357 | -0.357 | -0.357 | -1.204 | -1.609 | -1.609 |### Explanation of Data:- **Spray 1:** Positive initial values, decreasing in later entries.- **Spray 2:** Starts with positive and zero values, gradually decreases into negative territory.- **Lotion:** Begins with a high positive value, fluctuates with both positive and negative entries.- **Cream:** Consistently negative, with increasingly negative values in later entries.This table could be used for educational purposes to illustrate statistical evaluation or comparative studies in dermatological research or product analysis. **Oneway Anova Analysis**The image displays the results of a one-way ANOVA test, which is used to compare the means of different groups to determine if there are any statistically significant differences among them.**Analysis of Variance Table:**- **Source:** - **Group:** - Degrees of Freedom (DF): 3 - Sum of Squares: 4.658577 - Mean Square: 1.55286 - F Ratio: 3.5813 - Prob > F: 0.0320* - **Error:** - DF: 20 - Sum of Squares: 8.672101 - Mean Square: 0.43361 - **C. Total:** - DF: 23 - Sum of Squares: 13.330678The probability value (Prob > F) is 0.0320, indicating that there is a statistically significant difference among the group means at the 5% significance level.**Means and Standard Deviations:**- **Level:** - **Cream:** - Number: 6 - Mean: -0.915479 - Standard Deviation (Std Dev): 0.6297893 - Standard Error of Mean (Std Err Mean): 0.2571104 - Lower 95% Confidence Interval: -1.576402 - Upper 95% Confidence Interval: -0.254556 - **Lotion:** - Number: 6 - Mean: 0.2225703 - Std Dev: 0.7306505 - Std Err Mean: 0.2982868 - Lower 95% CI: -0.5442 - Upper 95% CI: 0.989341 - **Spray 1:** - Number: 6 - Mean: 0.0804044 - Std Dev: 0.6380284 - Std Err Mean: 0.260474 - Lower 95% CI: -0.589165 - Upper 95% CI: 0.749741 - **Spray 2:** - Number: 6 - Mean: -0.30336

Anova is used to measure the significance between the mean groups. If one categorical variable means…

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. Researchers recruited 24 healthy volunteers to investigate avobenzone absorption for four different commercially available sunscreens: two sprays, a lotion, and a cream. Do the types of sunscreens (spray, lotion, and cream) differ in terms of how much avobenzone is absorbed into the blood? Subjects were randomly assigned to one of the four sunscreens, with six subjects for each. Subjects had 2 milligrams of sunscreen per 1 cm? applied to 75% of their body surface area (area outside of normal swimwear) four times per day for four days. The amount of avobenzone absorbed into the bloodstream (in nanograms per milliliter ap-plied, ng/mL) after 150 hours was then measured for each subject. For the purpose of analyzing the data us ing ANOVA, the logarithms of the measurements were used. Thus, for example, a value of 0 corresponds to an absorption of 1 ng/ mL and negative values correspond to absorptions less than 1 ng/ mL. Larger values of the logarithms correspond to larger levels of absorption. Figure 27.13 shows JMP ANOVA output for these data. Show the work on how the values are acquired. a. Do these data satisfy the conditions for ANOVA? b. State Ho and H. for the ANOVA F test and explain in words what ANOVA tests in this setting. c. Report your overall conclusions about whether the types of sunscreen differ in the mean amount of avobenzone absorbed into the bloodstream after 150 hours.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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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. Researchers recruited 24 healthy volunteers to investigate avobenzone absorption for four different commercially available sunscreens: two sprays, a lotion, and a cream. Do the types of sunscreens (spray, lotion, and cream) differ in terms of how much avobenzone is absorbed into the blood? Subjects were randomly assigned to one of the four sunscreens, with six subjects for each. Subjects had 2 milligrams of sunscreen per 1 cm? applied to 75% of their body surface area (area outside of normal swimwear) four times per day for four days. The amount of avobenzone absorbed into the bloodstream (in nanograms per milliliter ap-plied, ng/mL) after 150 hours was then measured for each subject. For the purpose of analyzing the data us ing ANOVA, the logarithms of the measurements were used. Thus, for example, a value of 0 corresponds to an absorption of 1 ng/ mL and negative values correspond to absorptions less than 1 ng/ mL. Larger values of the logarithms correspond to larger levels of absorption.

Figure 27.13 shows JMP ANOVA output for these data. Show the work on how the values are acquired.

a. Do these data satisfy the conditions for ANOVA?

b. State Ho and H. for the ANOVA F test and explain in words what ANOVA tests in this setting.

c. Report your overall conclusions about whether the types of sunscreen differ in the mean amount of avobenzone absorbed into the bloodstream after 150 hours.

The table displays comparative data for different topical treatments over several entries. Each treatment option—Spray 1, Spray 2, Lotion, and Cream—is associated with a set of numerical values that likely represent measurements or evaluations under various conditions.

### Table: Comparative Analysis of Topical Treatments

| Treatment | Entry 1 | Entry 2 | Entry 3 | Entry 4 | Entry 5 | Entry 6 |
|-----------|---------|---------|---------|---------|---------|---------|
| Spray 1 | 0.693 | 0.405 | 0.405 | -0.511 | -0.916 |
| Spray 2 | 0.588 | 0.000 | 0.000 | -0.511 | -0.693 | -1.204 |
| Lotion | 1.386 | 0.693 | 0.182 | 0.095 | -0.511 | 0.511 |
| Cream | -0.357 | -0.357 | -0.357 | -1.204 | -1.609 | -1.609 |

### Explanation of Data:
- **Spray 1:** Positive initial values, decreasing in later entries.
- **Spray 2:** Starts with positive and zero values, gradually decreases into negative territory.
- **Lotion:** Begins with a high positive value, fluctuates with both positive and negative entries.
- **Cream:** Consistently negative, with increasingly negative values in later entries.

This table could be used for educational purposes to illustrate statistical evaluation or comparative studies in dermatological research or product analysis.

**Oneway Anova Analysis**

The image displays the results of a one-way ANOVA test, which is used to compare the means of different groups to determine if there are any statistically significant differences among them.

**Analysis of Variance Table:**

- **Source:**
- **Group:**
- Degrees of Freedom (DF): 3
- Sum of Squares: 4.658577
- Mean Square: 1.55286
- F Ratio: 3.5813
- Prob > F: 0.0320*
- **Error:**
- DF: 20
- Sum of Squares: 8.672101
- Mean Square: 0.43361
- **C. Total:**
- DF: 23
- Sum of Squares: 13.330678

The probability value (Prob > F) is 0.0320, indicating that there is a statistically significant difference among the group means at the 5% significance level.

**Means and Standard Deviations:**

- **Level:**
- **Cream:**
- Number: 6
- Mean: -0.915479
- Standard Deviation (Std Dev): 0.6297893
- Standard Error of Mean (Std Err Mean): 0.2571104
- Lower 95% Confidence Interval: -1.576402
- Upper 95% Confidence Interval: -0.254556

- **Lotion:**
- Number: 6
- Mean: 0.2225703
- Std Dev: 0.7306505
- Std Err Mean: 0.2982868
- Lower 95% CI: -0.5442
- Upper 95% CI: 0.989341

- **Spray 1:**
- Number: 6
- Mean: 0.0804044
- Std Dev: 0.6380284
- Std Err Mean: 0.260474
- Lower 95% CI: -0.589165
- Upper 95% CI: 0.749741

- **Spray 2:**
- Number: 6
- Mean: -0.30336

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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