Question

I want red! A candy maker offers Child and Adult bags of jelly beans with different color mixes. The company claims that the Child mix has 30% red jelly beans, while the Adult mix contains 15% red jelly beans. Assume that the candy maker's claim is true. Suppose we take a random sample of 50 jelly beans from the Child mix and a separate random sample of 100 jelly beans from the Adult mix. Let and be the sample proportions of red jelly beans from the Child and Adult mixes, respectively. (a) What is the shape of the sampling distribution of? Why? (b) Find the mean of the sampling distribution. Show your work. (e) Calculate and interpret the standard deviation of the sampling distribution. Show your work. (d) Find the probability that the proportion of jelly beans in the Child sample is less than or equal to the proportion of red jelly beans in the Adult sample, assuming that the company's claim is true. (e) Based on your result on part (d), would this give you reason to doubt the company's claim? Explain your reasoning. 6. Literacy A researcher reports that 80% of high school graduates, but only 40% of high school dropouts, would pass a basic literacy test.5 Assume that the researcher's claim is true. Suppose we give a basic literacy test to a random sample of 60 high school graduates and a separate random sample of 75 high school dropouts. Let and be the sample proportions of graduates and dropouts, respectively, who pass the test. (a) What is the shape of the sampling distribution of? Why? (b) Find the mean of the sampling distribution. Show your work. (c) Calculate and interpret the standard deviation of the sampling distribution. Show your work. (d) Find the probability that the proportion of graduates who pass the test is at most 0.20 higher than the proportion of dropouts who pass, assuming that the researcher's report is correct. (e) Based on your result on part (d), would this give you reason to doubt the researcher's claim? Explain your reasoning.

Answer 1

Touches on statistics topics such as probability, sampling distributions, hypothesis testing, and interpretation of results related to mean and standard deviation calculations in different scenarios.

Step-by-step explanation:

Probability, sampling distributions, and hypothesis testing in the field of statistics. Specific topics include the sampling distribution of proportions, calculating means and standard deviations, constructing confidence intervals, and conducting hypothesis tests for population means and proportions.

For example, to calculate the mean of the sampling distribution of red jelly beans in the candy maker's scenario, we use the given population proportion and the sample size. Here, the mean for the children's mix would be 0.30, since 30% of the jelly beans are red according to the company's claim. For the adult mix with 100 jelly beans, the mean would be 0.15. Similarly, the standard deviation of the sampling distribution would be calculated using the formula √[p(1-p)/n] where p is the proportion of red jelly beans and n is the sample size.

Hypothesis testing, like the ones described involving vaccination rates and life expectancies, generally involves setting up null and alternative hypotheses, selecting the appropriate test statistic, and making decisions based on p-values and significance levels (α).