Early stages little is known about population parameters Less precision is

Early stages little is known about population

This preview shows page 86 - 89 out of 94 pages.

Early stages – little is known about population parameters - Less precision is needed 95% confidence interval - Common and “workhorse” 99% confidence interval - Medical and scientific research - Where making wrong conclusion is costly Putting Confidence Intervals to Work in Business Test Claims Example: claims average of 338 minutes – sample average of 35 tests is 324.6 minutes with a SD of 32 minutes 95% confidence interval around mean - Where z-score always equals 1.96 Calculate standard error of the mean σx = σ / Ön - 32 ¸ Ö 35 = 5.41 Calculate upper and lower limits Upper: x + (z-score) • (standard error) - 324.6 + (1.96 • 5.41) = 335.2 Lower: x – (z-score) • (standard error) - 324.6 – (1.96 • 5.41) = 314.0 à 95% confident that the true population mean is w/in the interval of (314, 335.2). - Sample DOES NOT validate claim - b/c claimed 338 does not fall w/in interval Using Excel to Determine Confidence Intervals for the Mean (Sigma – Standard Deviation – Known) The Function =CONFIDENCE (alpha, SD, size)
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- Alpha = significance level (opposite of confidence level) - SD = standard deviation of the population - Size = sample size Resulting value = margin of error in confidence interval - Slightly different than manual calculation b/c we typically round numbers (excel does not) Calculating Confidence Intervals for the Mean with Small Samples When the SD of a Population is Known When sample size is less than 30 Can no longer depend on Central Limit Theorem Population MUST be normally distributed to construct a confidence interval When it’s the same as previously shown - Know the population is normally distributed - Sample size smaller than 30 - SD is known Example: reported 21.87 second average What we know - Sample size = 15 - Population SD = 2.5 seconds - Population follows a normal probability distribution Construct 99% confidence interval to estimate average time - Where z-score is always equal to 2.575 Calculating the standard error of the mean SD (σ) / Ön - 2.5 / Ö15 = 0.646 Calculating the upper and lower limits Upper = x + (z-score • standard-error) Lower = x – (z-score • standard-error) - UCL = 21.87 + (2.575)(0.646) = 23.53 seconds - LCL = 21.87 – (2.575)(0.646) = 20.21 seconds CONCLUSION - 99% confident that the true average time is b/w 20.21 and 23.53 - Claim is validated Ø 8.3 – CALCULATING CONFIDENCE INTERVALS FOR THE MEAN WHEN THE SD IS UNKNOWN o What to do
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§ Substitute with sample standard deviation - Have to calculate it the old fashion way - =STDEV (chapter 3 explained) § Can’t rely on normal distribution to provide critical z-score for confidence interval - Must use the Student’s T-Distribution Using the Student’s t-distribution Student’s t-distribution – used in place of the normal probability distribution when the sample SD is used in place of the population SD Continuous probability distribution that is bell-shaped and symmetrical around the mean Shape of curve depends on degrees of freedom - Degrees of freedom –t he number of values that are free to vary given that certain info is known - When dealing with sample mean, DF = (n – 1) Area under the curve is equal to 1.0
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