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Eye shadow, mascara, and nickel allergies. 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.If you wish to estimate a population mean to within .2 with a 95% confidence interval and you know from previous sampling that 02 is approximately equal to 5.4, how many observations would you have to include in your sample?If nothing is known about p. .5 can be substituted for p in the sample-size formula for a population proportion. But when this is done, the resulting sample size may be larger than needed. Under what circumstances will using p = .5 in the sample-size formula yield a sample size larger than is needed to construct a confidence interval for p with a specified bound and a specified confidence level?Suppose you wish to estimate a population mean correct to within. 15 with a confidence level of 90. You do not know 02, but you know that the observations will range in value between 31 and 39. a. Find the approximate sample size that will produce the desired accuracy of the estimate. You wish to be conservative to ensure that the sample size will be ample for achieving the desired accuracy of the estimate. [Hint: Using your knowledge of data variation from Section 2.5, assume that the range of the observations will equal 40.] b. Calculate the approximate sample size, making the less conservative assumption that the range of the observations is equal to 60.In each case, find the approximate sample size required to construct a 95% confidence interval for p that has sampling error SE = .08. a. Assume that p is near .3. b. Assume that you have no prior knowledge about p. but you wish to be certain that your sample is large enough to achieve the specified accuracy for the estimate.The following is a 90%% confidence interval for p: (.28, .54). How large was the sample used to construct this interval?It costs you $10 to draw a sample of size n = 1 and measure the attribute of interest. You have a budget of $1 200. 3. Do you have sufficient funds to estimate the population mean for the attribute of interest with a 95% confidenceinterval 4 units in width? Assume that o = 12. b. If a 90% confidence level were used, would your answer to part a change? Explain.Giraffes have excellent vision. Refer to the African Zoology (Oct. 2013) study of a giraffe's eyesight, Exercise 7.38 (p. 334). Recall that the researchers measured the eye mass for a sample of 27 giraffes native to Zimbabwe, Africa, and found y = 53.4 grams and s = 8.0 grams. Suppose the objective is to sample enough giraffes in order to obtain an estimate of the mean eye mass to within 3 grams of its true value with a 99% confidence interval. a. Identify the confidence coefficient for this study. b. Identify the desired sampling error for this study. c. Find the sample size required to obtain the desired estimate of the true mean. (Reference Exercise 7.39) Giraffes have excellent vision. Due to habitat, giraffes travel in small groups. Hence, they require excellent vision in order to detect predators. The eyesight of giraffes was studied in African Zoology (Oct. 2013). The researchers measured a variety of eye characteristics for 27 giraffes native to Zimbabwe, Africa. One variable measured was eye mass ([in grams). The study reported * = 53.4 g and s = 8.0 g. a. Use this information to find a 90% confidence interval for the true mean eye mass of giraffes native to Zimbabwe. b. Suppose it is known that the mean eye mass of an African water buffalo is A . = 31 grams. Is it likely that the true mean eye mass of a giraffe is larger or smaller than this mean? Explain. c. Suppose it is known that the mean eye mass of an African elephant is A . = 58 grams. Is it likely that the true mean eye mass of a giraffe is larger or smaller than this mean? Explain.Risk of home burglary in cul-de-sacs. Research published in the Journal of Quantitative Criminology (March 2010) revealed that the risk of burglaries in homes located on cul-de-sacs is lower than for homes on major roads. Suppose you want to estimate the true percentage of culdesac homes in your home city that were burglarized in the past year. Devise a sampling plan so that your estimate will be accurate to within 2%% of the true value using a confidence coefficient of 95%%. How many cul-de-sac homes need to be sampled and what information do you need to collect for each sampled home
