Estimating Population Means using Confidence Intervals - Prof. Murali S. Shanker, Study notes of Business Statistics

Download Estimating Population Means using Confidence Intervals - Prof. Murali S. Shanker and more Study notes Business Statistics in PDF only on Docsity! Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-1 Fundamentals of Business Statistics – Murali Shanker Chap 7-1 Business Statistics: A Decision-Making Approach 6th Edition Chapter 7 Estimating Population Values Fundamentals of Business Statistics – Murali Shanker Chap 7-2 Confidence Intervals Content of this chapter Confidence Intervals for the Population Mean, μ when Population Standard Deviation σ is Known when Population Standard Deviation σ is Unknown Determining the Required Sample Size Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-2 Fundamentals of Business Statistics – Murali Shanker Chap 7-3 Confidence Interval Estimation for μ Suppose you are interested in estimating the average amount of money a Kent State Student (population) carries. How would you find out? Fundamentals of Business Statistics – Murali Shanker Chap 7-4 Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-5 (1-α)x100% Confidence Interval for μ α−1 2 α 2 α Lower Limit Upper Limitμ Half Width H Half Width H X Fundamentals of Business Statistics – Murali Shanker Chap 7-10 CI Derivation Continued 1. Parameter = Statistic ± Error (Half Width) nZX nZH n H n XXZ XXH HX X / / // or σμ σ σσ μ σ μ μμ μ ×±= ×= = − = − = +−= ±= Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-6 Fundamentals of Business Statistics – Murali Shanker Chap 7-11 Confidence Interval for μ (σ Known) Assumptions Population standard deviation σ is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate ( ) n σzx α/2-1± (1-α)x100% CI α−1 2 α 2 α 0 μ ZZ(1-α/2)Z(α/2) X 99 95 0.9500.100.9090 Z(1-α/2)(1-α/2)α(1-α)Conf. Level Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-7 Fundamentals of Business Statistics – Murali Shanker Chap 7-13 μμx = Interpretation Confidence Intervals 100(1-α)% of intervals constructed contain μ; 100α% do not. Sampling Distribution of the Mean x x1 x2 /2α /2αα−1 Fundamentals of Business Statistics – Murali Shanker Chap 7-14 Factors Affecting Half Width Data variation, σ : H as σ Sample size, n : H as n Level of confidence, 1 - α : H if 1 - α n σ /2αzH = Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-10 Fundamentals of Business Statistics – Murali Shanker Chap 7-19 Student’s t Distribution The t is a family of distributions The t value depends on degrees of freedom (d.f.) Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1 Fundamentals of Business Statistics – Murali Shanker Chap 7-20 Student’s t Distribution t0 t (df = 5) t (df = 13) t-distributions are bell- shaped and symmetric, but have ‘fatter’ tails than the normal Standard Normal (t with df = ∞) Note: t z as n increases Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-11 Fundamentals of Business Statistics – Murali Shanker Chap 7-21 Student’s t Table Upper Tail Area df .25 .10 .05 1 1.000 3.078 6.314 2 0.817 1.886 2.920 3 0.765 1.638 2.353 t0 2.920 The body of the table contains t values, not probabilities Let: n = 3 df = n - 1 = 2 α = .10 α/2 =.05 α/2 = .05 Fundamentals of Business Statistics – Murali Shanker Chap 7-22 t distribution values With comparison to the z value Confidence t t t z Level (10 d.f.) (20 d.f.) (30 d.f.) ____ .80 1.372 1.325 1.310 1.28 .90 1.812 1.725 1.697 1.64 .95 2.228 2.086 2.042 1.96 .99 3.169 2.845 2.750 2.58 Note: t z as n increases Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-12 Fundamentals of Business Statistics – Murali Shanker Chap 7-23 Example A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ Fundamentals of Business Statistics – Murali Shanker Chap 7-24 Approximation for Large Samples Since t approaches z as the sample size increases, an approximation is sometimes used when n ≥ 30: ( ) ( ) n stX n 1 /21 − − ± α ( ) n szX /21 α−± Correct formula Approximation for large n Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-15 Fundamentals of Business Statistics – Murali Shanker Chap 7-29 Confidence Intervals 1. Standard Normal 2. T distribution ( ) ( ) ( ) n ZX n ZX n ZX σμ σμ σ α α α − − − −≥ +≤ ± 1 1 2/1 :Lower sided-One : Uppersided-One :sided-Two ( ) ( ) ( ) ( ) ( ) ( ) n stX n stX n stX n n n 1 1 1 1 2/1 :Lower sided-One : Uppersided-One :sided-Two 1 − − − − − −≥ +≤ ± − α α μ μ α Fundamentals of Business Statistics – Murali Shanker Chap 7-30 YDI 10.17 A beverage dispensing machine is calibrated so that the amount of beverage dispensed is approximately normally distributed with a population standard deviation of 0.15 deciliters (dL). Compute a 95% confidence interval for the mean amount of beverage dispensed by this machine based on a random sample of 36 drinks dispensing an average of 2.25 dL. Would a 90% confidence interval be wider or narrower than the interval above. How large of a sample would you need if you want the width of the 95% confidence interval to be 0.04? Fundamentals of Business Statistics – Murali Shanker Chapter 7 Student Lecture Notes 7-16 Fundamentals of Business Statistics – Murali Shanker Chap 7-31 YDI 10.18 A restaurant owner believed that customer spending was below the usual spending level. The owner takes a simple random sample of 26 receipts from the previous weeks receipts. The amount spent per customer served (in dollars) was recorded and some summary measures are provided: n = 26, X = 10. 44, s2 = 7. 968 Assuming that customer spending is approximately normally distributed, compute a 90% confidence interval for the mean amount of money spent per customer served. Interpret what the 90% confidence interval means.