Essentials of Statistics for Business and Economics, Part 1

University:
California State University, Northridge
Course:
SOM 120 | Basic Business Statistics
Academic year:

2024
Views:

440

Pages:

23
Author:

Jerome Sherman

Essentials of Statistics for Business and Economics Essentials of Statistics for Business and Economics Part 1 Chapter 8 Interval Estimation • Population Mean: σ Known • Population Mean: σ Unknown • Determining the Sample Size • Population Proportion 2 Margin of error and the Interval Estimate • A point estimator cannot be expected to provide the exact value of the population parameter. • An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. • Point estimate ± Margin of error The purpose of an interval estimate is to provide information about how close the point estimate, is to the value of the parameter. 3 Margin of Error and the Interval Estimate • The general form of an interval estimate of a population mean is 𝑥𝑥̅ + Margin of error 4 Interval Estimate of a Population Mean: σ Known • In order to develop an interval estimate of a population mean, the margin of error must be computed using either: • the population standard deviation σ , or • the sample standard deviation s • σ is rarely known exactly. But often a good estimate can be obtained based on historical data or other information. • We refer to such cases as the σ known case. 5 Interval Estimate of a Population Mean: σ Known There is a 1 - α probability that the value of a sample mean will provide a margin of error of 𝑧𝑧𝛼𝛼/2 𝜎𝜎𝑥𝑥̅ or less. α/2 1 - α of all 𝑥𝑥̅ values µ Sampling distribution of 𝑥𝑥̅ ← 𝑧𝑧𝛼𝛼/2 𝜎𝜎𝑥𝑥̅ →← 𝑧𝑧𝛼𝛼/2 𝜎𝜎𝑥𝑥̅ → α/2 𝑥𝑥̅ 6 Interval Estimate of a Population Mean: σ Known 1 - α of all 𝑥𝑥̅ values α/2 Sampling distribution of 𝑥𝑥̅ α/2 µ interval does not include µ ← 𝑧𝑧𝛼𝛼/2 𝜎𝜎𝑥𝑥̅ → ← 𝑧𝑧𝛼𝛼/2 𝜎𝜎𝑥𝑥̅ → −−−− −𝑥𝑥̅ −−−− − −−−− −𝑥𝑥̅ −−−− − 𝑥𝑥̅ interval includes µ 7 Interval Estimate of a Population Mean: σ Known • Interval Estimate of µ where: 𝑥𝑥̅ ± 𝑧𝑧𝛼𝛼/2 𝜎𝜎 𝑛𝑛 𝑥𝑥̅ is the sample mean 1 - α is the confidence coefficient zα/2 is the z value providing an area of α/2 in the upper tail of the standard normal probability distribution σ is the population standard deviation n is the sample size 8 Interval Estimate of a Population Mean: σ Known • Values of zα/2 for the Most Commonly Used Confidence Levels Confidence Level α 90% 95% 99% .10 .05 .01 α/2 Table Look-up Area zα/2 .05 .025 .005 .9500 .9750 .9950 1.645 1.960 2.576 9 Meaning of Confidence • Because 90% of all the intervals constructed using 𝑥𝑥� + 1.645𝜎𝜎𝑥𝑥̅ will contain the population mean, we say we are 90% confident that the interval 𝑥𝑥� + 1.645𝜎𝜎𝑥𝑥̅ includes the population mean µ. • We say that this interval has been established at the 90% confidence level. • The value .90 is referred to as the confidence coefficient. 10 Interval Estimate of a Population Mean: σ Known • Example: Discount Sounds Discount Sounds has 260 retail outlets throughout the United States. The firm is evaluating a potential location for a new outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location. A sample of size n = 36 was taken; the sample mean income is $41,100. The population is not believed to be highly skewed. The population standard deviation is estimated to be $4,500, and the confidence coefficient to be used in the interval estimate is .95. 11 Interval Estimate of a Population Mean: σ Known • Example: Discount Sounds 95% of the sample means that can be observed are within + 1.96 𝜎𝜎𝑥𝑥̅ of the population mean µ. The margin of error is: 𝑧𝑧𝛼𝛼/2 𝜎𝜎 4,500 = 1,470 = 1.96 𝑛𝑛 36 Thus at 95% confidence, the margin of error is $1,470. 12 Interval Estimate of a Population Mean: σ Known • Example: Discount Sounds Interval estimate of µ is: $41,100 + $1,470 or $39,630 to $42,570 We are 95% confident that the interval contains the population mean. 13 Interval Estimate of a Population Mean: σ Known • Example: Discount Sounds Confidence level Margin of error Interval estimate 90% 95% 99% 1,234 1,470 1,932 39,866 to 42,334 39,630 to 42,570 39,168 to 43,032 In order to have a higher degree of confidence, the margin of error and thus the width of the confidence interval must be larger. 14 Interval Estimate of a Population Mean: σ Known • Adequate Sample Size • In most applications, a sample size of n≥ 30 is adequate. • If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended. 15 Interval Estimate of a Population Mean: σ Known • Adequate Sample Size (continued) • If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 will suffice. • If the population is believed to be at least approximately normal, a sample size of less than 15 can be used. 16 Interval Estimate of a Population Mean: σ Unknown • If an estimate of the population standard deviation σ cannot be developed prior to sampling, we use the sample standard deviation s to estimate σ . • This is the σ unknown case. • In this case, the interval estimate for µ is based on the t distribution. • (We’ll assume for now that the population is normally distributed.) 17 t Distribution • William Gosset, writing under the name “Student”, is the founder of the t distribution. • Gosset was an Oxford graduate in mathematics and worked for the Guinness Brewery in Dublin. • He developed the t distribution while working on small-scale materials and temperature experiments. 18 t Distribution • The t distribution is a family of similar probability distributions. • A specific t distribution depends on a parameter known as the degrees of freedom. • Degrees of freedom refer to the number of independent pieces of information that go into the computation of s. 19 t Distribution • A t distribution with more degrees of freedom has less dispersion. • As the degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller. 20 t Distribution t distribution (20 degrees of freedom) Standard normal distribution t distribution (10 degrees of freedom) z, t 0 21 t Distribution • For more than 100 degrees of freedom, the standard normal z value provides a good approximation to the t value. • The standard normal z values can be found in the infinite degrees (∞ ) row of the t distribution table. 22 t Distribution Degrees of Freedom . Area in Upper Tail .20 .10 .05 . . . .025 . .01 . .005 . 50 .849 1.299 1.676 2.009 2.403 2.678 60 .848 1.296 1.671 2.000 2.390 2.660 80 .846 1.292 1.664 1.990 2.374 2.639 100 .845 1.290 1.660 1.984 2.364 2.626 ∞ .842 1.282 1.645 1.960 2.326 2.576 (bottom row is standard normal z values) 23

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