Questions and Answers > CHAPTER FIVE SAMPLING DISTRIBUTIONS

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CHAPTER FIVE SAMPLING DISTRIBUTIONS After completing these tutorials, students should be able to:  Construct a frequency distribution  Construct a histogram  Construct a frequency... polygon.  Construct a relative frequency polygon.  Construct an ogive.  Determine the mean, variance and standard deviation for the given data set.  Determine the mode, midrange and range for the given data set. Review Exercises - Question 1 The heights of UTeM students are approximately normally distributed with a mean of 174.5 centimetres and a standard deviation of 6.9 centimetres. A random sample of size 36 is drawn from this population. Determine: (a) the mean and the standard deviation of the sampling distribution of x .  1.15 (b) the probability that the average height is between 172.5 and 176.5 centimetres, inclusive. (c) the probability that the average height is below 170.5.  Review Exercises - Question 2 The amount of time that a telemarketer spends on a customer is a random variable with a mean, μ = 3.2 minutes and a standard deviation, σ = 1.6 minutes. If a random sample of 64 customers is observed from a normally distributed population, find the probability that the mean time of this sample is: (a) more than 3.5 minutes. (b) at least 3.2 minutes but less than 3.4 minutes. Review Exercises - Question 3 The average life of a bread-making machine is 7 years, with a standard deviation of 1 year. Assuming that the lives of these machines follow approximately a normal distribution, find: (a) the probability that the mean life of a random sample of 9 such machines falls between 6.4 and 7.5 years. Solution: (b) the probability that the mean life is more than the population mean by at least 2 standard deviations (n = 9). Solution: (c) the value of x to the right of which 15% of the means computed from random samples of size 9 would fall. Solution: Review Exercises - Question 4 Suppose that a printer needs an average of Y minutes to print several designs. Given that the standard deviation to print all design is 1.4 minutes. If a random sample of 49 designs is selected, (a) what is the value of Y so that there is 99% chance that the sample mean will be at least 13.634 minutes? Solution: (b) find the probability that the sample mean will be below 13.5 or above 14.8 minutes. Review Exercises - Question 5 Researchers are experimenting with a new compound used to bond A to steel. The drying time that the compound requires is being monitored and it is known that it is approximately normally distributed, with an average drying time of 4.70 minutes and a standard deviation of 0.40 minutes. Suppose that a sample of 25 drying times is selected, (a) what is the probability that the sample mean will be at least 4.57 minutes? (b) there is an 85 percent chance that the sample average will fall between two values symmetrically distributed around the population mean. What are those two values? Solution: Review Exercises - Question 6 The test scores for 300 UTeM students were entered into a computer, analyzed, and stored in a file. Knowing that 30% of the mean scores were below 65 and 15% of the mean scores were above 90, find their mean and standard deviation (assuming the scores are normally distributed). Solution: Review Exercises - Question 7 For a population with p = 76% and  ^  0.2 , find the probability that p (a) pˆ > 80%. Review Exercises - Question 8 A survey conducted by UTeM shows that 77% of the Information Technology students have laptops at home. If a random sample of 120 Information Technology students is selected, find the probability that the value of pˆ is Review Exercises - Question 9 A company that manufactures soda drink claims that 85% of their soda drinks are good for 4 years or longer. Assume that the claim is true. Let pˆ be the proportion in a sample of 100 soda drinks that are good for 4 years or longer. (a) What is the probability that this sample proportion is within 0.05 of the population proportion? Solution: (b) What is the probability that this sample proportion is less than the population proportion by 0.06 or more? Solution: (c) What is the probability that this sample proportion is greater than the population proportion by 0.07 or more? Solution: Review Exercises - Question 10 The proportion of females in an organization is 85%. We have a random sample of n = 500 individuals. (a) What are the mean and standard deviation of pˆ , the sample proportion of females in the organization? Solution:  0.016 (b) Is the distribution of pˆ approximately normal? Justify your answer. Solution: np  500  0.85  425  5 nq  500  0.15  75  5 Sample size is sufficiently large,  approximately normal (c) What is the probability that the sample proportion exceeds 82%? Solution: (d) What is the probability that the sample proportion lies between 83% and 88%? Solution: (e) 99% of the time, the sample proportion would lie between what two symmetrical limits? Solution: Review Exercises - Question 11 A city is planning to build a shopping complex building. A local newspaper found that 55% of the voters in this city favour the construction of this building. Assume that this result holds true for the population of all voters in this city. (a) What is the probability that more than 50% of the voters in a random sample of 150 voters selected from this city will favour the construction of this plant? Solution: (b) A politician would like to take a random sample of voters in which over 50% would favour the plant construction. How large a sample should be selected so that the politician is 95.5% sure of this outcome? Solution: Review Exercises - Question 12 A survey conducted by UTeM Exam Unit shows that 88% of UTeM students passed all subjects and completed their studies in 4 years’ time. If a random sample of 132 students is selected, find the probability that the sample proportion is within 0.10 of the population mean. Next, find the number of samples that are selected, if the probability is 0.38 that at most 81% of UTeM students passed all subjects and completed their studies in 4 years’ time. Solution: [Show More]

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