Lecture PowerPoint Slides Basic Practice of Statistics 7 th Edition

Questions on Quiz 17/18? Questions on the reading? Let’s take a look at SA9

In Chapter 20, We Cover … Conditions for inference about a mean The t distributions The one-sample t confidence interval The one-sample t test Using technology Matched pairs t procedures Robustness of t procedures Resampling and standard errors*

So what really happens when we have small samples, non-Normal distribution, and don’t know σ?

The t Distributions

What do you notice about these distributions?

The t Distributions

When can we use a t-test? When we don’t know the population standard deviation σ and… Is the sample size “large” (i.e., above 40)? You can use a t-test Is the data roughly Normal? Is the data heavily skewed or are there strong outliers? Yes No Is the sample size at least 15? Yes Start HERE Don’t No Yes Don’t

Reading comments on spreadsheets Numerator of the z-score formula Questions over the reading New reading check-ins (trial runs)

So, how does it actually “work”?

Using Table C Suppose you want to construct a 95% confidence interval for the mean µ of a Normal population based on an SRS of size n = 12. What critical t* should you use? In Table C, we consult the row corresponding to df = n – 1 = 11. The desired critical value is t * = We move across that row to the entry that is directly above the 95% confidence level. Confidence Level C df90%95%96%98% z*

One-Sample t Confidence Interval

Does the expectation of good weather affect how generous someone is?

How would you test this?

Find and Interpret the 95% CI

Now conduct a t-test to see if this “fair weather” average is different from 21%

Could we have figured that out just by looking at the confidence interval? (Yes, at least for a 2-tailed test)

Hypothesis Test Comparisons p-value to alpha level (α) z statistic to z* t statistic to t* Measured in % of the Distribution Measured in Number of Standard Errors *

Using a confidence interval for a two-tailed test Lower limit of CI Upper limit of CI

Are there unsafe levels of feces in Ohio’s swimming areas?

How would you test this?

Matched Pairs t Procedures Comparative studies are more convincing than single-sample investigations. For that reason, one-sample inference is less common than comparative inference. Study designs that involve making two observations on the same individual, or one observation on each of two similar individuals, result in paired data. MATCHED PAIRS t PROCEDURES To compare the responses to the two treatments in a matched pairs design, find the difference between the responses within each pair. Then apply the one-sample t procedures to these differences.

Do women talk more than men in a relationship? *Voyagerix via Getty images

How would you test this?

Example Do women talk more than men in a relationship? Couple NumberManWoman 11,0081,354 21,5932,094 34,4922,592 43,5694,021 53,8025, ,052 72,8972,540 82,0052,634 93,4593,400

Now conduct a t-test (at 95% confidence) to see if women indeed talk more in a relationship than their partner.