## Related questions with answers

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.

Solution

Verified$\textbf{Eye Shadow}$

$\begin{align*} n&=\text{Sample size}=131 \\ x&=\text{Number of successes}=12 \\ c&=\text{Correlation coefficient}=95\%=0.95 &\color{#4257b2}(\text{Assumption}) \end{align*}$

Since the number of successes is not at least 15, the large sample conditions is not satisfied and thus we need to use the $\textit{adjusted}$ confidence interval for a population proportion.

The sample proportion is the number of successes divided by the sample size. Since we use the adjusted confidence interval, we add 2 failures and 2 successes to the sample.

$\tilde{p}=\dfrac{x+2}{n+4}=\dfrac{12+2}{131+4}=\frac{14}{135}\approx 0.1037$

For confidence level $1-\alpha=0.95$, determine $z_{\alpha/2}=z_{0.025}$ using the normal probability table in the appendix, which is the z-score corresponding to a probability of $0.5-\alpha/2=0.475$:

$z_{\alpha/2}=1.96$

The margin of error is then:

$E=z_{\alpha/2}\cdot \sqrt{\dfrac{\tilde{p}(1-\tilde{p})}{n+4}}=1.96\times \sqrt{\dfrac{0.1037(1-0.1037)}{135}}\approx 0.0514$

The boundaries of the confidence interval are then:

$\tilde{p}-E=0.1037-0.0514=0.0523$

$\tilde{p}+E=0.1037+0.0514=0.1551$

We are 95% confident that the true proportion of women diagnosed with a nickel allergy when using the eye shadow is between 0.0523 and 0.1551.

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