Stat 512Day 4: Quantitative Data

Last Time

p-values and statistical significance What p-values tell us (and do not tell us)

For now, approximating the p-value through simulating the randomization process

How small p-values provide evidence that the difference we observed did not occur “just by chance” (randomization) Assume there is no treatment effect…

If a randomized experiment, then can also draw cause and effect conclusions

Practice Problem

In (a), “controlling variables” Specify the explanatory variable

In (d), if no association… If no relationship, same “success proportion” in each group Not 1/2 since not equal group sizes

(“significant”) No inference here

Role of randomization test Don’t have to have equal sample sizes

In (f), Causal vs. relationship Don’t panic, sorry for my biased comments

Statistical Methods

Design: Planning and carrying out research studies Observational units, Number and types of

variables Descriptive: Summarizing and exploration

data Inference: Making predictions or generalizing

about phenomena represented by data What conclusions can we draw based on

each of these three steps?

Repeat the Process – Quantitative Data Consider data collection issues Consider appropriate numerical and

graphical summaries Several measures, what does each tell you? How do we get Minitab to do all the work?

Simulation of p-values to determine statistical significance Interpretation of p-values

Example 1: Cloud Seeding

“A Bayesian analysis of a multiplicative treatment effect in weather modification” Simpson, Alsen, Eden Technometrics, 17, 161-166 (1975)

Example 1

(a) Type of study, observational units?Experiment since randomly assigned the clouds

(b) EV and RVseeding

Clouds compare rainfall

no seeding

randomized

Example 1

With a quantitative response variable, can compare the groups through parallel dotplots

What to look for

Center

Spread

Shape

Unusual observations

Numerical Summaries

Five number summaryVariable treatment Minimum Q1 Median Q3 Maximum

rainfall seeded 4 79 222 445 2746

unseeded 1.0 23.7 44.2 183.3 1202.6

Numerical Summaries

Five number summary

Min, Q1, median, Q3, outliers

Mean vs. Median

Properties

The University of North Carolina took a survey of the students who had graduated as geology majors. In 1998, the average annual salary of geology majors who graduated from UNC was more than $500,000.

The next year it was less than $100,000.

Summary

Comparing the distribution of a quantitative variable between two or more groups Graphical summaries: (parallel) dotplots, boxplots,

side by side stemplots Center, spread, shape (skewed?), outliers

Numerical summaries Center: mean, median (five-number summary)

Mean = average of all values (not “resistant”) Median = “typical” value

Outliers: 1.5IQR criterion

x

Old Faithful

Histograms

Histograms

Geyser Eruptions

1978 Range = 95-42 = 53 minutes 2003 Range = 110-56 = 54 minutes

Without outliers: 110-70 = 40 minutes

9542

11056 70

Geyser Eruptions

1978 IQR = 81-58 = 23 minutes 2003 IQR = 98-87 = 11 minutes

Without outliers IQR = 98-87 = 11 minutes

Standard Deviation

Want to compare the distance of the observations from the mean Deviation from mean: yi- Absolute deviations Squared deviations

1

)(1

2

2

n

xxs

n

ii

y

1

)(1

2

n

yys

n

ii

Old Faithful

1978 SD = 13 minutes 2003 SD = 8.5 minutes

Without outliers SD=6.9 (SD is not resistant!)

Example 3

What do we mean by variability?

Notes on histograms

Left-hand endpoint rule Choice of interval widths

Also watch use of “even” in describing shape (flat vs. symmetric)

Notes on Using Minitab

Worksheets vs. Projects Saving graph windows Stacked vs. unstacked data

To Do

For Tuesday: PP 4 For Thursday: PP 5 and reading HW 3 by Friday

Heavy Minitab component Favor

Upcoming: Project proposal