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Chapter 7: Problem 51

While the majority of people who are color blind are male, the National Eye Institute reports that \(0.5 \%\) of women of with Northern European ancestry have the common form of red-green color blindness. Suppose a random sample of 100 women with Northern European ancestry is selected. Can we find the probability that less than \(0.3 \%\) of the sample is color blind? If so, find the probability. If not, explain why this probability cannot be calculated.

Short Answer

Expert verified
No, the probability of less than 0.3% of these 100 women with Northern European ancestry being colorblind cannot be calculated using a binomial distribution because it involves a non-integer number of successes, which is impossible in this discrete model.

Step by step solution

01

Understanding the problem and identifying the given parameters

Here, the given parameters include the number of trials \(n\), which is the number of women sampled (100), and the probability of success \(p\), which is the likelihood of a woman having color blindness according to the National Eye Institute (0.005). However, the problem is asking for the probability of less than 0.3 percent of the women sampled being colorblind, which would be less than 1 woman since \(0.003 \times 100 = 0.3\) women. Since we can't have a fraction of a woman, this is not a value that we can use in the standard binomial formula.
02

Attempting to find the probability

While it would be possible to calculate the probability of exactly 0, 1, 2, etc. women being colorblind in the sample using the binomial formula, we cannot use it to find the probability of a fraction of a woman being colorblind. Therefore, we can't find this probability using a binomial distribution.
03

Final conclusion

The probability of less than 0.3% of these 100 women with Northern European ancestry being colorblind cannot be found using a binomial distribution because it would involve finding the probability of a non-integer, which is not possible with the discreet nature of this distribution model.

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Most popular questions from this chapter

Suppose it is known that \(60 \%\) of employees at a company use a Flexible Spending Account (FSA) benefit. a. If a random sample of 200 employees is selected, do we expect that exactly \(60 \%\) of the sample uses an FSA? Why or why not? b. Find the standard error for samples of size 200 drawn from this population. What adjustments could be made to the sampling method to produce a sample proportion that is more precise?

A 2016 Pew Research poll found that \(61 \%\) of U.S. adults believe that organic produce is better for health than conventionally grown varieties. Assume the sample size was 1000 and that the conditions for using the CLT are met. a. Find and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults to believe organic produce is better for health. b. Find and interpret an \(80 \%\) confidence interval for this population parameter. c. Which interval is wider? d. What happens to the width of a confidence interval as the confidence level decrease?

A Harris poll asked Americans in 2016 and 2017 if they were happy. In \(2016,31 \%\) reported being happy and in 2017 , \(33 \%\) reported being happy. Assume the sample size for each poll was 1000 . A \(95 \%\) confidence interval for the difference in these proportions \(p_{1}-p_{2}\) (where proportion 1 is proportion happy in 2016 and proportion 2 is the proportion happy in 2017) is \((-0.06,0.02)\). Interpret this confidence interval. Does the interval contain \(0 ?\) What does this tell us about happiness among American in 2016 and \(2017 ?\)

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Just the Boys Refer to Exercise \(7.77\) for information. This data set records results just for the boys. $$\begin{array}{|lcc|}\hline & \text { Preschool } & \text { No Preschool } \\\\\hline \text { Grad HS } & 16 & 21 \\\\\hline \text { No Grad HS } & 16 & 18 \\\\\hline\end{array}$$ a. Find and compare the percentages that graduated for each group, descriptively. Does this suggest that preschool was linked with a higher graduation rate? b. Verify that the conditions for a two-proportion confidence interval are satisfied. c. Indicate which one of the following statements is correct. i. The interval does not capture 0 , suggesting that it is plausible that the proportions are the same. ii. The interval does not capture 0 , suggesting that it is not plausible that the proportions are the same. iii. The interval captures 0 , suggesting that it is plausible that the population proportions are the same. iv. The interval captures 0 , suggesting that it is not plausible that the population proportions are the same. d. Would a \(99 \%\) confidence interval be wider or narrower?
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