Chapter 8 Estimating Single Population Parameters Business Statistics: A Decision-Making Approach 8th Edition Chapter 8 Estimating Single Population Parameters

Chapter Goals After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate Construct and interpret a confidence interval estimate for a single population mean using both the z and t distributions Determine the required sample size to estimate a single population mean or a proportion within a specified margin of error Form and interpret a confidence interval estimate for a single population proportion

Overview of the Chapter Builds upon the material from Chapter 1 and 7 Introduces using sample statistics to estimate population parameters Because gaining access to population parameters can be expensive, time consuming and sometimes not feasible Confidence Intervals for the Population Mean, μ when Population Standard Deviation σ is Known when Population Standard Deviation σ is Unknown Confidence Intervals for the Population Proportion, p Determining the Required Sample Size for means and proportions

Estimation Process Population Confidence Level Random Sample Mean (point estimate) I am 95% confident that μ is between 40 & 60. Population Mean x = 50 (mean, μ, is unknown) Sample confidence interval

Point Estimate Suppose a poll indicate that 62% (sample mean) of the people favor limiting property taxes to 1% of the market value of the property. The 62% is the point estimate of the true population of people who favor the property-tax limitation. EPA tested average Automobile Mileage (point estimate)

Confidence Interval The point estimate (sample mean) is not likely to exactly equal the population parameter because of sampling error. Probability of “sample mean = population mean” is zero Problem: how far the sample mean is from the population mean. To overcome this problem, “confidence interval” can be used as the most common procedure. An interval developed from sample values such that if all possible intervals of a given width were constructed, a percentage of these intervals, known as the confidence level, would include the true population parameter.

Point and Interval Estimates So, a confidence interval provides additional information about variability within a range. The interval incorporates the sampling error Lower Confidence Limit Upper Confidence Limit Point Estimate Width of confidence interval

Confidence Level Confidence level = 95% Meaning: In the long run, 95% of all the confidence intervals will contain the true parameter Describes how strongly we believe that a particular sampling method will produce a confidence interval that includes the true population parameter. A percentage (less than 100%) Most common: 90% (α = 0.1), 95% (α = 0.05), 99%

Common Levels of Confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Level Critical value, z 80% 90% 95% 98% 99% 99.8% 99.9% 1.28 1.645 1.96 2.33 2.58 3.08 3.27

Point Estimate  Margin of Error (z or t - Value) * (Standard Error) General Formula The general formula for the confidence interval is: Point Estimate  Margin of Error Margin of Error: Margin of Error (e): the amount added and subtracted to the point estimate to form the confidence interval (z or t - Value) * (Standard Error)

Point Estimate  (Critical Value)(Standard Error) General Formula According to our textbook Point Estimate  (Critical Value)(Standard Error) z or t -value based on the level of confidence desired Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

Conditions for finding the Critical Value The Central Limit Theorem states that the sampling distribution of a statistic will be normally (nearly normally) distributed if at least n > 30. The sampling distribution of the mean is normally distributed when the population distribution is normally distributed.

How to Find the z Value When one of these conditions is satisfied, the critical value can be expressed as a z score. To find the z value, see the next slide.

Finding the z Value -z = -1.96 z = 1.96 Consider a 95% confidence interval: 0.95/2 (because of lower and upper limit) = 0.475 find on z table = 1.96 -z = -1.96 z = 1.96 z units: Lower Confidence Limit Upper Confidence Limit x units: Point Estimate Point Estimate

How to Find the t value d.f. = n – 1 When the population distribution type is unknown or when the sample size is small (less than 30), the t value is preferred. 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 Only n-1 independent pieces of data information left in the sample because the sample mean has already been obtained

t Distribution: Try the t simulation one the website Note: t compared to z as n increases As n the estimate of s becomes better so t converges to z Standard Normal (t with df = ) t (df = 13) t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal t (df = 5) t

t Table 0.90 2 t /2 = 0.05 2.920 Confidence Level df 0.50 0.80 1 Let: n = 3 df = n - 1 = 2 confidence level: 90% 0.90 df 0.50 0.80 1 1.000 3.078 6.314 2 0.817 1.886 2.920 /2 = 0.05 3 0.765 1.638 2.353 The body of the table contains t values, not probabilities t 2.920

With comparison to the z value t Distribution Values With comparison to the z value Note: t compared to z as n increases Confidence t t t z Level (10 d.f.) (20 d.f.) (30 d.f.) ____ 0.80 1.372 1.325 1.310 1.28 0.90 1.812 1.725 1.697 1.64 0.95 2.228 2.086 2.042 1.96 0.99 3.169 2.845 2.750 2.58

Example A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ d.f. = n – 1 = 24, so The confidence interval is 46.698 53.302

Summary of When to use z or t Use z value (or score) Population is normally distributed (very rare and even though sample size is small) Sample size is n > 30 Use t value (or score) Do not know population distribution type (not normal) Do not know Pop. Mean and Std Many real world situations Sample size is less than 30

Using Analysis ToolPak Download “Confidence Interval Example” Excel file

Using Analysis ToolPak small sample: apply “t” distribution automatically) Confidence Interval 119.9 (mean) ± 2.59 117.31 ------ 122.49 Don’t even worry about p* = 1 - α/2 Margin of Error

Using Analysis ToolPak (large sample: use normal (z) distribution automatically )

Determining Required Sample Size Wishful situation High confidence level, low margin of error, and right sample size In reality, conflict among three…. For a given sample size, a high confidence level will tend to generate a large margin of error For a given confidence level, a small sample size will result in an increased margin of error Reducing of margin of error requires either reducing the confidence level or increasing the sample size, or both

Determining Required Sample Size How large a sample size do I really need? Sampling budget constraint Cost of selecting each item in the sample

Determining Required Sample Size When σ is known The required sample size can be found to reach a desired margin of error (e) and level of confidence (1 - ) Required sample size, σ known:

Required Sample Size Example If  = 45 (known), what sample size is needed to be 90% confident of being correct within ± 5? So the required sample size is n = 220 (Always round up)

If σ is unknown (most real situation) If σ is unknown, three possible approaches Use a value for σ that is expected to be at least as large as the true σ Select a pilot sample (smaller than anticipated sample size) and then estimate σ with the pilot sample standard deviation, s Use the range of the population to estimate the population’s Std Dev. As we know, µ ± 3σ contains virtually all of the data. Range = max – min. Thus, R = (µ + 3σ) – (µ - 3σ) = 6σ. Therefore, σ = R/6 (or R/4 for a more conservative estimate, producing a larger sample size)

Example when σ is unknown Example 8-5: Jackson’s Convenience Stores Using a pilot sample approach Available on the class website Make sure to review all the examples from page 306 to 327.

Confidence Intervals for the Population Proportion, π Try by yourself! An interval estimate for the population proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p ). For example, estimation of the proportion of customers who are satisfied with the service provided by your company Sample proportion, p = x/n

Confidence Intervals for the Population Proportion, π (continued) Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation We will estimate this with sample data: See Chpt. 7!!

Confidence Interval Endpoints Upper and lower confidence limits for the population proportion are calculated with the formula where z is the standard normal value for the level of confidence desired p is the sample proportion n is the sample size

Example A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers

Example (continued) A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. 1. 2. 3.

Interpretation We are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49% Although this range may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.

Changing the sample size Increases in the sample size reduce the width of the confidence interval. Example: If the sample size in the above example is doubled to 200, and if 50 are left-handed in the sample, then the interval is still centered at 0.25, but the width shrinks to 0.19 0.31

Finding the Required Sample Size for Proportion Problems Define the margin of error: Solve for n: Will be in % units π can be estimated with a pilot sample, if necessary (or conservatively use π = 0.50 – worst possible variation thus the largest sample size)

What sample size...? How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (Assume a pilot sample yields p = 0.12)

What sample size...? Solution: For 95% confidence, use Z = 1.96 (continued) Solution: For 95% confidence, use Z = 1.96 e = 0.03 p = 0.12, so use this to estimate π So use n = 451

Using PHStat PHStat can be used for confidence intervals for the mean or proportion Two options for the mean: known and unknown population standard deviation Required sample size can also be found Download from the textbook website The link available on the class website

PHStat Interval Options

PHStat Sample Size Options

Using PHStat (for μ, σ unknown) A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for μ

Using PHStat (sample size for proportion) How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (Assume a pilot sample yields p = 0.12)

Chapter Summary Discussed point estimates Introduced interval estimates Discussed confidence interval estimation for the mean [σ known] Discussed confidence interval estimation for the mean [σ unknown] Addressed determining sample size for mean and proportion problems Discussed confidence interval estimation for the proportion

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