BUAL 2650 FINAL EXAM QUESTIONS AND ANSWERS WITH COMPLETE SOLUTIONS VERIFIED

BUAL 2650 FINAL EXAM QUESTIONS AND ANSWERS WITH COMPLETE SOLUTIONS VERIFIED 5.0 (1 review) Terms in this set (125) Chapter 5: Sampling Distribution .. Parameter Numerical description measure of a population (almost always unknown) Sample statistic a numerical descriptive measure of a sample. It is calculated from the observations in the sample Sampling Distribution the distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic Point Estimator rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the population parameter If the sampling distribution of a sample statistic has a mean equal to the population parameter the statistic is intended to estimate, the statistic is said to be an of the parameter unbiased estimate If the mean of the sampling distribution is not equal to the parameter, the statistic is said to be a of the parameter biased estimate Will the sampling distribution of x-bar always be approximately normally distributed? Explain. No, because the Central Limit Theorem states that the sampling distribution of x-bar is approximately normally distributed only if the sample size is large enough. Theorem 5.1 If a random sample of n observations is selected from a population with a normal distribution, the sampling distribution of x-bar will be a normal distribution. x-bar is the minimum-variance unbiased estimator (MVUE) of µ p-hat is the MVUE of p Unbiased estimate if the sampling distribution of a sample statistic has a mean equal to the population parameter the statistic is intended to estimate, the statistic is said to be unbiased (mean = population parameter = unbiased) Biased estimate if the mean of the sampling distribution of a sample statistic is not equal to the population parameter, it's biased. (mean ≠ population parameter = biased) If a value uses a population standard deviation in the calculation then what is it standard deviation If a value uses a sample standard deviation in the calculation then what is it Standard Error If a random sample of n observations is selected from a population with a normal distribution then what? the sampling distribution of x-bar will be a normal distribution What's the general guideline for the normal distribution to be justified? n is greater than or equal to 30 Properties of the Sampling Distribution of x-bar 1. Mean of the sampling distribution equals mean of sampled population. μx-bar = E(x-bar)= μ 2. Standard deviation of sampling distribution equals standard deviation of sampled population / square root of sample size σx = σ/ √n. Theorem 5.2 (Central Limit Theorem) If you have a population with a mean and a standard deviation, and you take a sufficiently large sample from that population with replacement, then your sample distribution will be approximately normal. the larger n is, the better approximately that your distribution will be normal. Your sample needs to be sufficiently large, the sample can not be more than 10% of the population which you are taking the same from, the sample needs to be randomly selected, and if you are taking a proportion the samples must be independent from each other. This is all in efforts to validate your sample for an approximately normal distribution. When we use z and t scores of a sample, that is based off of the central limit theorem; we are validating the hypothesis being tested with these assumptions. The standard deviation of the sample distribution decreases as the sample size increases Central Limit Theorem offers an explanation for the fact that many relative frequency distributions of data possess mound-shaped distributions Sampling Distribution of p-hat 1. Mean of the sampling distribution is equal to the true binomial proportion, p; that is, E(p-hat) = p. Consequently, is an unbiased estimator of p. 2. Standard deviation of the sampling distribution is equal to that is, √p(1-p)/n 3. For large samples, the sampling distribution is approximately normal. (A sample is considered large if np-hat ≥ 15 and n(1-p) ≥ 15 Chapter 6: Inferences Based on a Single Sample ... Target Parameter the unknown population parameter that we are interested in estimating A point estimator a single number calculated from the sample that estimates a target population parameter Example: we'll use the sample mean, x-bar, to estimate he population mean μ An interval estimator (or confidence interval) a range of numbers that contain the target parameter with a high degree of confidence. Large sample sample being of sufficiently large size that we can apply the CLT and the normal (z) statistic to determine the form of the sampling distribution of x-bar. note: the large-sample interval estimator requires knowing the value of the population standard deviation, σ. (in most business cases however, σ is unknown) Confidence coefficient The probability that a randomly selected confidence interval encloses the population parameter Confidence Level the confidence coefficient expressed as a percentage. Example: if our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain μ and 5% will not The value Za is defined as the value of the standard normal random variable 'z' such that the area 'a' will lie to its right for a confidence coefficient of .90, we have (1-a) =.90, a=.10 and a/2= .05; z.05= 1.645 When σ is unknown and 'n' is large (n≥ 30), the confidence interval is approx. equal to where 's' os the sample standard deviation Conditions Required for a Valid Large- Sample Confidence Interval for μ 1. A random sample is selected from the target population. 2. The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of σ . T-stat vs Z-stat the primary difference between the sampling distributions of t and z is that the t- stat is more variable than the z, which follows intuitively when you realize that t contains two random quantities (x-bar and s) whereas z contains only one (x-bar) the sampling distribution of t depends on the sample size n, the t-stat has (n-1) degrees of freedom(df) Degrees of Freedom the actual amount of variability in the sampling distribution of 't' depends on the sample size 'n'. Conditions Required for a Valid Small- Sample Confidence Interval of μ 1. A random sample is selected from the target population. 2. The population has a relative frequency distribution that is approximately normal. Sampling Distribution of p-hat 1. The mean of the sampling distribution of p-hat is p; that is, is an unbiased estimator of p. 2. The standard deviation of the sampling distribution of p-hat is √pq/n where q=1- p 3. For large samples, the sampling distribution of is approximately normal. A sample size is considered large if both np-hat ≥ 15 and nq-hat ≥ 15.