Inference for Distributions Chapter 11 Inference for Distributions

The one-sample t statistic Draw an SRS of size n from a population that has the normal distribution with mean µ and standard deviation σ. The one-sample t statistic has the t distribution with n-1 degrees of freedom.

Key features of t distributions Density curves are similar in shape to the standard normal curve Spread is a bit greater than that of the standard normal distribution. As the degrees of freedom increase, the density curve approaches the N(o,1) curve more closely.

Using the t-table What critical value from Table C would you use for a t-distribution with 18 degrees of freedom having probability 0.90 to the left of t? When using a t-table the probability given is to the right – we need 1-0.90 = 0.10 with df of 18. t* = 1.330

More on Using t-table Suppose that we want to construct a 95% confidence interval for the mean µ of a population based on an SRS of size n=12. What critical value t should you use? t* = 2.201

Constructing a confidence interval Identify the population of interest and the parameter you want to draw conclusions about. Choose the appropriate inference procedure. Verify the conditions for using the selected procedure Carry out the inference procedure. Interpret your results in the context of the problem.

Example 11.2 Auto Pollution: the major pollutants in auto exhaust are hydrocarbons, monoxide, and nitrogen oxide (NOX). Given the NOX levels for a sample of light-duty engines of the same type, construct a 95% confidence interval for the mean amount of NOX emitted.

Table of values

Step 1 Identify the population of interest and the parameter you want to draw conclusions about. The population of interest is all light-duty engines of this type. We want to estimate µ, the mean amount of pollutant NOX emitted

Step 2 Choose the appropriate inference procedure Verify the conditions for using the selected procedure Since we do not know σ, we should use a one-sample t procedure. The data come from 46 engines, therefore CLT tells us that the distribution of sample means will be app. Normal. We do not know if the sample is an SRS.

Step 3 If the conditions are met, carry out the inference procedure Confidence interval formula: df = 46 – 1 = 45

Calculations 1.329 + 2.021 (0.484/√46) 1.329 + 0.144 (1.185, 1.473)

Step 4 Interpret your results in the context of the problem We are 95% confident that the true mean level of nitrogen oxides emitted by this type of light-duty engine is between 1.185 grams/mile and 1.473 grams/mile.

Using Technology Look at test under STAT and run a Tinterval for the previous problem.

Significance test for µ when σ is unknown Cola makers test new recipes for loss of sweetness during storage. Following are the sweetness losses found by 10 tasters for one new cola recipe: 2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3 Are these good evidence that the cola lost sweetness?

Step 1 Identify the population of interest and the parameter you want to draw conclusions about. State the null and alternative hypothesis in words and symbols Tasters vary in their perception of sweetness loss H0: µDIFF = 0 Ha: µDIFF > 0

H0: µDIFF = 0 The mean sweetness loss for the population of tasters is 0 Ha: µDIFF > 0 The mean sweetness loss for the population of tasters is positive.

Step 2 Choose the appropriate inference procedure. Verify the conditions for using the selected procedure. Since we do not know the standard deviation of sweetness loss in the population of tasters, we must use a one-sample t test

Verify conditions We must treat the 10 tasters as a SRS from the population of tasters. We only have 10 observations, therefore the assumption that the population distribution is normal cannot be effectively checked. So we proceed with caution.

Step 3 If the conditions are met, carry out the inference procedure Calculate the test statistic. X = 1.02 S = 1.196 = (1.02 – 0) /(1.196/√10) = 2.70

Step 3 continued Find the P-value The P-value for t=2.70 and df = 10-1 = 9 It is somewhere between 0.01 and 0.02.

Step 4 Interpret your results in the context of the problem A P-value this low gives quite strong evident against the null hypothesis. We reject H0 and conclude that the cola has lost sweetness during storage.

Exercises 11.7 and 11.9

Matched pairs t procedure In matched pairs design, subjects are matched in pairs and each treatment is given to one subject in each pair.

Robustness of t procedures A confidence interval or significance test is called robust if the confidence level or P-value does not change very much when the assumptions of the procedure are violated Since mean and sample standard deviation are strongly influenced by outliers – the t-test will be also.

Can we use t procedures for these data Percent of residents aged 65 and over in the states NO : this is the entire population, not a sample

Can we use t procedures for these data Times of first lightning strikes each day at a site in Colorado. Yes: there are over 70 observations with a symmetric distribution.

Can we use t procedures for these data Word lengths in Shakespeare’s plays. Yes, if the sample is large enough to overcome the right skewness.

Exercises to practice 11.17 and 11.20