Stat 13, UCLA – Chapter 7 1 UCLA STAT 13 Statistical Methods Final Exam Review Chapter 7 – Sampling Distributions of Estimates 1. A random sample of size n is drawn from a population with mean, µ, and standard deviation, σ. Let X be the sample mean. (a) What is the: (i) mean of X ? (ii) standard deviation of X ? (b) If we are sampling from a Normal distribution then X is exactly / approximately (circle one) Normally distributed. (c) (i) If we are sampling from a non-Normal distribution then for large samples (ie, n is large) X is exactly / approximately (circle one) Normally distributed. (ii) The result in (i) is called the 2. A random sample of size n is drawn from a population in which a proportion p has a characteristic of interest. Let Pˆ be the sample proportion. (a) What is the: (i) mean of Pˆ? (ii) standard deviation of Pˆ? (b) For large samples Pˆ is exactly / approximately (circle one) Normally distributed. 3. A __________________________________________ is a numerical characteristic of a population. 4. An ______________________________________ is a quantity calculated from data in order to estimate an unknown . 5. Suppose that X1, X2, . . . , X16 is a random sample from a Normal distribution with mean of 50 and a standard deviation of 10. Then the distribution of the sample mean 161621XXXX+++=K has mean, Xµ, and standard deviation, Xσ, given by: (1) Xµ = 50, Xσ = 6.25 (2) Xµ = 50, Xσ = 0.625 (3) Xµ = 800, Xσ = 10 (4) Xµ = 50, Xσ = 2.5 (5) cannot be determined because n = 16 is too small for the central limit effect to take effect. Stat 13, UCLA Final Exam – Chapter 7 2 6. The distribution of all long-distance telephone calls is approximately Normally distributed with a mean of 280 seconds and a standard deviation of 60 seconds. A random sample of sixteen calls is chosen from telephone company records. Let X be the sample mean of sixteen such calls. (a) Describe the distribution of X . (b) Calculate the probability that the sample mean exceeds 240 seconds. Use the output below to help you. Cumulative Distribution Function Normal with mean = 280.000 and standard deviation = 60.0000 x P( X <= x) 240.0000 0.2525 Cumulative Distribution Function Normal with mean = 280.000 and standard deviation = 15.0000 x P( X <= x) 240.0000 0.0038 Cumulative Distribution Function Normal with mean = 280.000 and standard deviation = 3.75000 x P( X <= x) 240.0000 0.0000 7. The fuel consumption, in litres per 100 kilometres, of all cars of a particular model has mean of 7.15 and a standard deviation of 1.2. A random sample of these cars is taken. (a) Calculate the mean and standard deviation of the sample if: (i) one observation is taken. (ii) four observations are taken. (iii) sixteen observations are taken.Stat 13, UCLA Final Exam – Chapter 7 3 (b) In what way do your answers in (a) differ? Why? 8. About 65% of all university students belong to the student loan scheme. Consider a random sample of 50 students. Let Pˆ be the proportion of these 50 students who belong to the student loan scheme. (a) In words, describe p. (b) State the distribution of Pˆ. (c) What is the probability that the sample proportion is more than 70%? Use the output below to help you. (d) What is the probability that the sample proportion is between 0.45 and 0.55? Use the output below to help you. Cumulative Distribution Function Normal with mean = 0.650000 and standard deviation = 0.0674537 x P( X <= x) 0.4000 0.0001 0.4500 0.0015 0.5000 0.0131 0.5500 0.0691 0.6000 0.2293 0.6500 0.5000 0.7000 0.7707 0.7500 0.9309 Stat 13, UCLA Final Exam – Chapter 7 4 9. The owner of a large fleet of courier vans is trying to estimate her costs for next year’s operations. Fuel purchases are a major cost. A random sample of 8 vans yields the following fuel consumption data (in km/L): 10.3 9.7 10.8 12.0 13.4 7.5 8.2 9.1 Assume that the distribution of fuel consumption of the vans is approximately Normal. (a) Calculate the sample mean and the sample standard deviation. (b) Construct a two-standard-error interval for the mean fuel consumption of all of her vans. (c) Without doing any calculations specify what happens to the width of the two-standard error interval in the following cases: (i) the sample standard deviation increases. (ii) the sample mean decreases. (iii) the sample size increases. 10. A large department store wants to estimate the proportion of their customers who have a charge card for the store. They take a random sample of 120 shoppers. They find that 36 of these shoppers have a charge card for the store. Construct a two-standard-error interval for the proportion of all of their customers who have a charge card for the store. 11. Which one of the following statements is true? (1) In a poll, all estimates of population proportions, including estimates for subgroups of the population, will have the same standard error. (2) If X is Normal, then the Student’s t-distribution is used instead of the standard Normal distribution for the distribution of nXXσµ− when the population standard deviation is replaced by the sample standard deviation. (3) The Central Limit Effect can only be detected for sample sizes that are greater than 30. (4) When sampling, taking a large sample guarantees an accurate estimate of the parameter of interest. (5) For small samples, the shape of the distribution of the sample mean, X , is always Normal regardless of the shape of the distribution of the random variable X.Stat 13 Review Chapter 8 1 UCLA Stat 13 Review Final Exam Chapter 8 – Confidence Intervals Section A: Confidence intervals for a mean, proportion and difference between means 1. An earlier Stat 13 exam had a possible total of 64 points. A random sample of 30 was selected from all of exam scores. The data was collected and summary statistics follow. 46 32 24 20 51 33 35 43 26 29 59 41 30 35 49 53 32 50 52 23 25 53 51 34 26 29 40 38 45 42 sample size sample mean sample standard deviation 30 38.20 10.85 You will use this sample to construct a 95% confidence interval for the mean. (a) State the parameter θ (using a symbol and in
View Full Document
Unlocking...