Multiple Choice Questions Sampling Distributions

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1 Multiple Choice Questions Sampling Distributions 1. The Gallup Poll has decided to increase the size of its random sample of Canadian voters from about 1500 people to about 4000 people. The effect of this increase is to: (a) reduce the bias of the estimate. (b) increase the standard error of the estimate. (c) reduce the variability of the estimate. (d) increase the confidence interval width for the parameter. (e) have no effect because the population size is the same. 2. An airplane is only allowed a gross passenger weight of 8000 kg. If the weights of passengers traveling by air between Toronto and Vancouver have a mean of 78 kg and a standard deviation of 7 kg, the approximate probability that the combined weight of 100 passengers will exceed 8,000 kg is: (a) (b) (c) (d) (e) Government regulations indicate that the total weight of cargo in a certain kind of airplane cannot exceed 330 kg. On a particular day a plane is loaded with 100 boxes of goods. If the weight distribution for individual boxes is normal with mean 3.2 kg and standard deviation 7 kg, what is the probability that the regulations will NOT be met: (a) 1.5% (b) 92% (c) 8% (d) 15% 1

2 (e) 85% 4. The time required to assemble an electronic component is normally distributed with a mean of 12 minutes and a standard deviation of 1.5 min. Find the probability that the time required to assemble all nine components (i.e. the total assembly time) is greater than 117 minutes. (a) 2514 (b).2486 (c).4772 (d).0228 (e) A wholesale distributor has found that the amount of a customer s order is a normal random variable with a mean of $200 and a standard deviation of $50. What is the probability that the total amount in a random sample of 20 orders is greater than $4500? (a).1915 (b).0125 (c).3085 (d).0228 (e) A random sample of 100 observations is to be drawn from a population with a mean of 40 and a standard deviation of 25. The probability that the mean of the sample will exceed 45 is: (a) (b) (c) (d) (e) not possible to compute, based on the information provided. 7. Which of the following statements is INCORRECT about the sampling distribution of the sample mean: (a) The standard error of the sample mean will decrease as the sample size increases. (b) The standard error of the sample mean is a measure of the variability of the sample mean among repeated samples. (c) The sample mean is unbiased for the true (unknown) population mean. c 2006 Carl James Schwarz 2

3 (d) The sampling distribution shows how the sample mean will vary among repeated samples. (e) The sampling distribution shows how the sample was distributed around the sample mean. 8. The sample mean is an unbiased estimator for the population mean. This means: (a) The sample mean always equals the population mean. (b) The average sample mean, over all possible samples, equals the population mean. (c) The sample mean is always very close to the population mean. (d) The sample mean will only vary a little from the population mean. (e) The sample mean has a normal distribution. 9. Which of the following statements is NOT CORRECT? (a) In a proper random sampling, every element of the population has a known (and often equal) chance of being selected. (b) The precision of a sample mean or sample proportion depends only upon the sample size (and not the population size) in a proper random sample. (c) Convenience sampling often leads to biases in estimates because the sample is often not representative of the population. (d) If a sample of 1,000,000 families is randomly selected from all of Canada (with about 8,000,000 families) and the average family income is computed, then the true value of the family income for all families in Canada is known. (e) The sampling distribution of the sample mean describes how the sample mean will vary among repeated samples. 10. The sampling distribution of refers to: (a) the distribution of the various sample sizes which might be used in a given study (b) the distribution of the different possible values of the sample mean together with their respective probabilities of occurrence (c) the distribution of the values of the items in the population (d) the distribution of the values of the items actually selected in a given sample (e) none of the above c 2006 Carl James Schwarz 3

4 11. The average monthly mortgage payment for recent home buyers in Winnipeg is µ = $732, with standard deviation of σ = $421 A random sample of 125 recent home buyers is selected. The approximate probability that their average monthly mortgage payment will be more than $782 is: (a) (b) (c) (d) (e) Can of salmon have a nominal net weight of 250 g. However, due to variation in the canning process, the actual net weight has an approximate normal distribution with a mean of 255 g and a standard deviation of 10 g. According to Consumer Affairs, a sample of 16 tins should have less than a 5% chance that the mean weight is less than 250 g. What is the actual probability that a sample of 16 tins will have a mean weight less than 250 g? (a).1915 (b).3085 (c).0228 (d).4772 (e) The Central Limit Theorem states that: (a) if n is large then the distribution of the sample can be approximated closely by a normal curve (b) if n is large, and if the population is normal, then the variance of the sample mean must be small. (c) if n is large, then the sampling distribution of the sample mean can be approximated closely by a normal curve (d) if n is large, and if the population is normal, then the sampling distribution of the sample mean can be approximated closely by a normal curve (e) if n is large, then the variance of the sample must be small. 14. A random sample of size n = 30 is taken from a population of size N = 300. Which statement is generally correct? (a) µ is an estimate of X; σ is an estimate of s. (b) X is an estimate of µ; s is an estimate of σ. c 2006 Carl James Schwarz 4

5 (c) µ is an estimate of X; s is an estimate of the standard deviation of the sample mean. (d) X is an estimate of µ; s is an estimate of the standard deviation of the sample mean. (e) X is an estimate of µ; s is the standard error of the sample mean. 15. The central limit theorem tells us that the sampling distribution of is approximately normal. Which of the following conditions are necessary for the theorem to be valid: (a) The sample size has to be large. (b) We have to be sampling from a normal population. (c) The population has to be symmetric. (d) Population variance has to be small (e) Both A and C. 16. The Central Limit Theorem is important in Statistics because it allows us to use the normal distribution to make inferences concerning the population mean: (a) provided that the population is normally distributed and the sample size is reasonably large. (b) provided that the population is normally distributed (for any sample size). (c) provided that the sample size is reasonably large (for any population). (d) provided that the population is normally distributed and the population variance is known (for any sample size). (e) provided that the population size is reasonably large (whether the population distribution is known or not). 17. The Central Limit Theorem is important in Statistics because: (a) it tells us that large samples do not need to be selected. (b) it guarantees that, when it applies, the samples that are drawn are always randomly selected. (c) it enables reasonably accurate probabilities to be determined for events involving the sample average when the sample size is large regardless of the distribution of the variable (d) it tells us that if several samples have produced sample averages which seem to be different than expected, the next sample average will likely be close to its expected value. (e) it is the basis for much of the theory that has been developed in the area of discrete random variables and their probability distributions. c 2006 Carl James Schwarz 5

6 18. One class decided to estimate the proportion of cars that are red in a parking lot. They took a random sample of the cars in the closest parking lot to the class. Which of the following is NOT correct? (a) Even though the sample was random sample of cars in the parking lot, the sample may not be representative of the population of cars driven by SFU students because the decision to park in B-lot is a self-selected sample. (b) If another sample of cars was taken, it is likely that a different proportion for Japanese made cars would be found. The set of all possible values for the proportion is known as the sampling distribution. (c) The confidence interval computed refers to the proportion of cars in the sample that were red. (d) The sample was a simple random sample from cars parked. This means that every car in the lot had an equal chance of being selected. (e) A convenience sample could be chosen by selecting the first 25 cars in the parking lot that are closest to the Applied Science Building. 19. Recall in one assignment you surveyed cars in a parking lot to estimate the proportion that were red or the proportion that were from a Japanese manufacturer. Which of the following is NOT CORRECT? (a) A convenience sample of the cars closest to the Applied Science building may give a biased estimate of the proportion of cars which are from a Japanese manufacturer. (b) Different students may get different answers for the proportion of cars that are red. (c) The sample proportion of cars that are red is an unbiased estimate of the population proportion if the sampling is a simple random sample. (d) A sample of 100 cars in a convenience sample is always better than a sample of 20 cars from a proper random sample. (e) A sample of 100 cars from a proper random sample will give more precise estimates of the proportion of cars that are red than a sample of 20 cars from a proper random sample. 20. Which statement is NOT CORRECT? (a) The sample standard deviation measures variability of our sample values. (b) A larger sample will give answers that vary less from the true value than smaller samples (assuming both are properly chosen). (c) The sampling distribution describes how our estimate (answer) will vary if a new sample is taken. c 2006 Carl James Schwarz 6

7 (d) The standard error measures how much our estimate (answer) may vary if a new sample of the same size is chosen using the same sampling method. (e) A large sample size always gives unbiased estimators regardless of how the sample is chosen. c 2006 Carl James Schwarz 7