FREE-RESPONSE QUESTIONS. STUDENTS MUST SHOW WORK FOR FULL CREDIT!!!16. A sample of size 36 from a normally distributed population variable with population standarddeviation 20 had a sample mean of 88. Find a 90% confidence interval for the population mean.17. A sample of size 90 from a population variable had a sample mean of 4.74 and a samplestandard deviation of 0.71. Find a 95% confidence interval for the population mean.18. We wish to estimate the population mean of a variable that has standard deviation 70.5. Wewant to estimate it with an error no greater than 5 units with probability 0.99. How big a sampleshould we take from the population? What happens if the standard deviation and the margin oferror are both doubled?19. Here is a sample of ACT scores (average of the Math, English, Social Science, and NaturalScience scores) for students taking college freshman calculus:24.00 28.00 27.75 27.00 24.25 23.50 26.2524.00 25.00 30.00 23.25 26.25 21.50 26.0028.00 24.50 22.50 28.25 21.25 19.75a. Using an appropriate graph, see if it is plausible that the observationswere selected from a normal distribution.b. Calculate a 95% confidence interval for the population mean.c. The university ACT average for entire freshmen that year was about 21.Are the calculus students better than the average as measured by theACT?20. For each of the following scenarios state whether H0 should be rejected or not. State anyassumptions that you make beyond the information that is given.(a) H0 : Âµ = 4, H1 : Âµ 6= 4, n = 15, X = 3.4, S = 1.5, ? = .05.(b) H0 : Âµ = 21, H1 : Âµ < 21, n = 75, X = 20.12, S = 2.1, ? = .10.(c) H0 : Âµ = 10, H1 : Âµ 6= 10, n = 36, p-value = 0.061.21.*Water samples are taken from water used for cooling as it is being discharged from apower plant into a river. It has been determined that as long as the mean temperature of thedischarged water is at most 150? F, there will be no negative effects on the riverâs ecosystem. Toinvestigate whether the plant is in compliance with regulations that prohibit a mean dischargewater temperature above 150? , 50 water sample will be tank at randomly selected times, andthe temperature of each sample recorded.a. Determine an appropriate null and alternative hypothesis for this test.b. In the context of this situation, describe type I and type II errors.c. Which type of error would you consider more serious? Explain. 22. A sample of 12 radon detectors of a certain type was selected, and each was exposed to100 pCI/L of radon. The resulting readings were as follows:105.6 90.9 91.2 96.9 96.5 91.3 100.1 105.0 99.6 107.7 103.3 92.4 Does this data suggest that the population mean reading under these conditions differs from100? Set up an appropriate hypothesis test to answer this question. 23. In each of the following situations, for parts a â e write a short explanation of what is wrongand why.a. A researcher wants to test H0: ? = 30 versus the two sided alternative Ha: ? ? 30.b. A two-sample t statistic gave a P-value of 0.97. From this we can reject the nullhypothesis with 95% confidence.c. A manager wants to test the null hypothesis that average weekly demand is not equal to100 units.d. The z statistic had a value of 0.014, and the null hypothesis was rejected at the 5% levelbecause 0.014 < 0.05.e. Suppose the P-value is 0.03. Explain what is wrong with stating, âThe probability that thenull hypothesis is true is 0.03.âf. In terms of probability language, a P-value is a conditional probability. Define the eventD as âobserving a test statistic as extreme or more extreme than actually observed.â Usingthe definition of a P-value, which of the two conditional probabilities represents a P-value:i.P(H0 is true | D)ii.P(D | H0 is true)
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