Unit 8: Estimating with Confidence 8.3B What to do when we don’t know sigma

State & explain the three conditions (RIN) Objectives: State & explain the three conditions (RIN) Construct and interpret a Conf. Int for a pop. mean when sigma is unknown Determine critical values (𝒕 ∗ ) for a Conf. Int using a table Emphasize the appearance of “and confidence interval” b\c there is a huge difference between the two parts of this objective. Let students know that the conditions required to use a ConfInt should be familiar.

When the sampl. distr. of 𝑥 is approx. Normal When the sampl. distr. of 𝑥 is approx. Normal* we find probabilities by standardizing: What would have to happen if we did NOT know 𝜎?

This new statistic does not have a Normal distribution! Using an estimate for sigma: This new statistic does not have a Normal distribution!

Using 𝒔 𝒙 as an estimate for 𝝈 produces a statistic that has a t-distribution.

The t Distributions; Degrees of Freedom Draw an SRS of size n from a large population that has a Normal distribution with mean µ and standard deviation σ. This statistic has the t distribution with degrees of freedom df = n – 1. The statistic will have approx. a tn – 1 distribution as long as the sampl distrib of 𝑥 is close to Normal.

Quality control engineer for the Guinness Brewery in Dublin, Ireland William Sealy Gosset 1876-1937 Quality control engineer for the Guinness Brewery in Dublin, Ireland

Example 1 (BVD3e p. 530) To check adherence to the speed limit on a particular stretch of road the speed was recorded for a random sample of 37 vehicles. The average was 38.42 mph with a standard deviation of 5.01 mph.

Example 2 (BVD3e p. 558) How far does a pro drive the golf ball? To estimate this, a random sample of 63 distances was recorded with a mean of 288.6 yards and a standard deviation of 9.31 yards.

Using t Procedures Wisely The stated confidence level of a one- sample t interval for µ is exactly correct when the population distribution is exactly Normal. No population of real data is exactly Normal. The usefulness of the t procedures in practice therefore depends on how strongly they are affected by lack of Normality.

Definition: An inference procedure is called robust if the probability calculations involved in the procedure remain fairly accurate when a condition for using the procedures is violated.

Using One-Sample t Procedures: The Normal Condition • Large Samples: The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n ≥ 30 • Small samples: Use t procedures if the data appear close to Normal (roughly symmetric, single peak, no outliers). If the data are clearly skewed or if outliers are present, do not use t.

Example 3 (BVD3e p. 558) In 1998 Nabisco Foods advertised that each 18-oz bag of Chips Ahoy cookies contained at least 1000 chocolate chips. Air Force Academy statistics students purchased randomly selected bags of cookies and counted the chocolate chips.

Example 3 (BVD3e p. 558) 1219 1214 1087 1200 1419 1121 1325 1345 1244 1258 1356 1132 1191 1270 1295 1135

Example 4 (BVD3e p. 558) A researcher tests a maze on several rats, collecting time to complete in minutes. 38.4 46.2 62.5 38.0 62.8 33.9 50.4 35.0 52.8 60.1 55.1 57.6 55.5 49.5 40.9 44.3 93.8 47.9 69.2

The parameter doesn’t vary! What can go wrong? The parameter doesn’t vary!  There is a 95% chance that the true proportion is between 0.54 and 0.67 Confidence is not certainty!  The population mean is between 97.9 and 99.3 It’s about the parameter!  I am 95% confident that 𝒙 is between 5.4 and 6.8 inches Give examples of intervals and have students identify the pt. est. and the ME. The blank in bullet 3 is predetermined when we declare how confident we wish to be….