The Central Limit Theorem (Summarized) For a sampling distribution with sample size n that is drawn from any population with mean u and standard deviation 6 The distribution of sample means has the following properties: - The distribution is normal 0 The mean is: p1,; = p - The standard deviation is: an; = \1. Some sportsmen fish for snapping turtles, as they are good to eat and fun to catch (at least according to some sportsmen). The average length of a snapping turtle is 32.5 inches, and the standard deviation is approximately 6.3 inches. Assume the distribution is normal. a. In this state, a snapping turtle must be 27 inches in length or longer in order to keep it. What percent of snapping turtles are legal to keep? b. The department of natural resources wishes to restrict fishing for snapping turtles in a certain area where their population is declining. What minimum length should be set so that fishermen are only allowed to keep snapping turtles in the 80th percentile of size? Round to the nearest tenth. c. A sample of 36 snapping turtles is taken from the population. What is the probability that the mean of the sample will be less than 31.5 inches? d. A sample of 36 snapping turtles is taken from the population. What is the probability that the mean of the sample will be greater than 35 inches? Is this unusual? e. If you found a sample of 36 snapping turtles where the mean was greater than 35 inches, what might you suspect is really true about the mean size of snapping turtles in this area? Explain.2. The FDA maintains quality standards on food products in the United States. As an example, a 100-gram sample of apple butter is permitted to contain an average of no more than 4 rodent hairs per 100-gram sample. If contaminants are not within this limit, the product must be recalled. Assume the standard deviation 0.38. U.S. Food and Drug Administration (2018, Sept 7). Food Defect Levels Handbook. FDA. https://www.fda gow/food/ingredients-additives-gras-packaging- guidance-documents-regulatory-information/food-defect-levels-handbook#commodities a. A quality assurance expert at the FDA takes a sample of 50 such jars of apple butter and tests for contaminants. He finds a sample mean of 4.21 rodent hairs. Find the z-score for this sample mean. Assume the population is normally distributed with mean 4 hairs with standard deviation 0.38. Round to the nearest hundredth. b. Find the probability that the mean level of rodent hair contaminant in the sample of 50 jars would be 4.21 or greater. c. If an inspector found that the mean level of rodent hair contaminant was x = 4.21 in this sample of 50, what would the quality assurance expert conclude about the population of jars from this brand of apple butter?3. The mean exam score for a certain exam in a Psychology course is 71.3 points with standard deviation 8.7 points. The population distribution is graphed below: True mean = 71.3 Frequency TO ! Individual X value a. Assume you were asked to find the probability that an individual student in this population would have a score on the exam of 74 points or lower. Could you use the normal distribution to find this probability? If so, find the probability. If not, explain why it is not possible. b. A sample of 49 students from this population was taken. Assume you were asked to find the probability that the sample mean would be 74 points or lower. Would it be possible to use a normal distribution to find this probability? Explain why or why not. If it is possible, find the probability. c. A sample of 5 students from this population was taken. Assume you were asked to find the probability that the sample mean would be 74 points or lower. Would it be possible to use a normal distribution to find this probability? Explain why or why not. If it is possible, find the probability. d. What would be more unusual to find: an individual student from this population who scored 65 points, or a sample of 49 students with a mean of 65 points? Why is this more unusual? Give an intuitive explanation for the reason why these are different; do not just name or repeat probabilities or z-scores