Collaborative Statistics by Barbara Illowsky, Ph.D. and Susan Dean - HTML preview

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Introduction

____ State the bivariate data your group is going to study.

EXAMPLES: Here are two examples, but you may NOT use them: height vs. weight and age

vs. running distance.

____ Describe how your group is going to collect the data (for instance, collect data from the web, survey

students on campus).

____ Describe your sampling technique in detail. Use cluster, stratified, systematic, or simple random

sampling (using a random number generator) sampling. Convenience sampling is NOT acceptable.

____ Conduct your survey. Your number of pairs must be at least 30.

____ Print out a copy of your data.

Analysis

____ On a separate sheet of paper construct a scatter plot of the data. Label and scale both axes.

____ State the least squares line and the correlation coefficient.

____ On your scatter plot, in a different color, construct the least squares line.

____ Is the correlation coefficient significant? Explain and show how you determined this.

____ Interpret the slope of the linear regression line in the context of the data in your project. Relate the

explanation to your data, and quantify what the slope tells you.

____ Does the regression line seem to fit the data? Why or why not? If the data does not seem to be linear,

explain if any other model seems to fit the data better.

____ Are there any outliers? If so, what are they? Show your work in how you used the potential outlier

formula in the Linear Regression and Correlation chapter (since you have bivariate data) to determine

whether or not any pairs might be outliers.

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14.4.5.4 Part II: Univariate Data

In this section, you will use the data for ONE variable only. Pick the variable that is more interesting to

analyze. For example: if your independent variable is sequential data such as year with 30 years and one

piece of data per year, your x-values might be 1971, 1972, 1973, 1974, . . ., 2000. This would not be interesting

to analyze. In that case, choose to use the dependent variable to analyze for this part of the project.

_____ Summarize your data in a chart with columns showing data value, frequency, relative frequency,

and cumulative relative frequency.

_____ Answer the following, rounded to 2 decimal places:

1. Sample mean =

2. Sample standard deviation =

3. First quartile =

4. Third quartile =

5. Median =

6. 70th percentile =

7. Value that is 2 standard deviations above the mean =

8. Value that is 1.5 standard deviations below the mean =

_____ Construct a histogram displaying your data. Group your data into 6 – 10 intervals of equal width.

Pick regularly spaced intervals that make sense in relation to your data. For example, do NOT group

data by age as 20-26,27-33,34-40,41-47,48-54,55-61 . . . Instead, maybe use age groups 19.5-24.5, 24.5-

29.5, . . . or 19.5-29.5, 29.5-39.5, 39.5-49.5, . . .

_____ In complete sentences, describe the shape of your histogram.

_____ Are there any potential outliers? Which values are they? Show your work and calculations as to

how you used the potential outlier formula in chapter 2 (since you are now using univariate data) to

determine which values might be outliers.

_____ Construct a box plot of your data.

_____ Does the middle 50% of your data appear to be concentrated together or spread out? Explain how

you determined this.

_____ Looking at both the histogram AND the box plot, discuss the distribution of your data. For example:

how does the spread of the middle 50% of your data compare to the spread of the rest of the data rep-

resented in the box plot; how does this correspond to your description of the shape of the histogram;

how does the graphical display show any outliers you may have found; does the histogram show any

gaps in the data that are not visible in the box plot; are there any interesting features of your data that

you should point out.

14.4.5.5 Due Dates

• Part I, Intro: __________ (keep a copy for your records)

• Part I, Analysis: __________ (keep a copy for your records)

• Entire Project, typed and stapled: __________

____ Cover sheet: names, class time, and name of your study.

____ Part I: label the sections “Intro” and “Analysis.”

____ Part II:

____ Summary page containing several paragraphs written in complete sentences describing the ex-

periment, including what you studied and how you collected your data. The summary page

should also include answers to ALL the questions asked above.

____ All graphs requested in the project.

____ All calculations requested to support questions in data.

____ Description: what you learned by doing this project, what challenges you had, how you over-

came the challenges.

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645

NOTE:

Include answers to ALL questions asked, even if not explicitly repeated in the items

above.

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index-656_1.png

index-656_2.png

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APPENDIX

14.5 Solution Sheets

14.5.1 Solution Sheet: Hypothesis Testing for Single Mean and Single Proportion9

Class Time:

Name:

a. Ho:

b. Ha:

c. In words, CLEARLY state what your random variable X or P’ represents.

d. State the distribution to use for the test.

e. What is the test statistic?

f. What is the p-value? In 1 – 2 complete sentences, explain what the p-value means for this problem.

g. Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizon-

tal axis and shade the region(s) corresponding to the p-value.

Figure 14.1

h. Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write

an appropriate conclusion, using complete sentences.

i. Alpha:

ii. Decision:

iii. Reason for decision:

iv. Conclusion:

i. Construct a 95% Confidence Interval for the true mean or proportion. Include a sketch of the graph of

the situation. Label the point estimate and the lower and upper bounds of the Confidence Interval.

Figure 14.2

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14.5.2 Solution Sheet: Hypothesis Testing for Two Means, Paired Data, and Two

Proportions10

Class Time:

Name:

a. Ho: _______

b. Ha: _______

c. In words, clearly state what your random variable X1 − X2, P1’ − P2’- or Xd represents.

d. State the distribution to use for the test.

e. What is the test statistic?

f. What is the p-value? In 1 – 2 complete sentences, explain what the p-value means for this problem.

g. Use the previous information to sketch a picture of this situation. CLEARLY label and scale the horizon-

tal axis and shade the region(s) corresponding to the p-value.

Figure 14.3

h. Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write

an appropriate conclusion, using complete sentences.

i. Alpha:

ii. Decision:

iii. Reason for decision:

iv. Conclusion:

i. In complete sentences, explain how you determined which distribution to use.

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APPENDIX

14.5.3 Solution Sheet: The Chi-Square Distribution11

Class Time:

Name:

a. Ho: _______

b. Ha: _______

c. What are the degrees of freedom?

d. State the distribution to use for the test.

e. What is the test statistic?

f. What is the p-value? In 1 – 2 complete sentences, explain what the p-value means for this problem.

g. Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal

axis and shade the region(s) corresponding to the p-value.

Figure 14.4

h. Indicate the correct decision (“reject” or “do not reject” the null hypothesis) and write appropriate con-

clusions, using complete sentences.

i. Alpha:

ii. Decision:

iii. Reason for decision:

iv. Conclusion:

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649

14.5.4 Solution Sheet: F Distribution and One-Way ANOVA12

Class Time:

Name:

a. Ho:

b. Ha:

c. df (n) = ______ df (d) = _______

d. State the distribution to use for the test.

e. What is the test statistic?

f. What is the p-value?

g. Use the previous information to sketch a picture of this situation. Clearly label and scale the horizontal

axis and shade the region(s) corresponding to the p-value.

Figure 14.5

h. Indicate the correct decision (“reject” or “do not reject” the null hypothesis) and write appropriate con-

clusions, using complete sentences.

i. Alpha:

ii. Decision:

iii. Reason for decision:

iv. Conclusion:

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14.6 English Phrases Written Mathematically13

14.6.1 English Phrases Written Mathematically

When the English says:

Interpret this as:

Xis at least 4.

X ≥ 4

The minimum of X is 4.

X ≥ 4

X is no less than 4.

X ≥ 4

X is greater than or equal to 4.

X ≥ 4

X is at most 4.

X ≤ 4

The maximum of X is 4.

X ≤ 4

Xis no more than 4.

X ≤ 4

X is less than or equal to 4.

X ≤ 4

Xdoes not exceed 4.

X ≤ 4

Xis greater than 4.

X > 4

X is more than 4.

X > 4

Xexceeds 4.

X > 4

Xis less than 4.

X < 4

There are fewer X than 4.

X < 4

Xis 4.

X = 4

Xis equal to 4.

X = 4

Xis the same as 4.

X = 4

Xis not 4.

X = 4

Xis not equal to 4.

X = 4

Xis not the same as 4.

X = 4

Xis different than 4.

X = 4

Table 14.16

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14.7 Symbols and their Meanings14

Symbols and their Meanings

Chapter (1st used)

Symbol

Spoken

Meaning

Sampling and Data

The square root of

same

Sampling and Data

π

Pi

3.14159. . . (a specific

number)

Descriptive Statistics

Q1

Quartile one

the first quartile

Descriptive Statistics

Q2

Quartile two

the second quartile

Descriptive Statistics

Q3

Quartile three

the third quartile

Descriptive Statistics

IQR

inter-quartile range

Q3-Q1=IQR

Descriptive Statistics

x

x-bar

sample mean

Descriptive Statistics

µ

mu

population mean

Descriptive Statistics

s sx sx

s

sample standard devia-

tion

Descriptive Statistics

s2 s2x

s-squared

sample variance

Descriptive Statistics

σ σ x σ x

sigma

population

standard

deviation

Descriptive Statistics

2

2

σ σ x

sigma-squared

population variance

Descriptive Statistics

Σ

capital sigma

sum

Probability Topics

{}

brackets

set notation

Probability Topics

S

S

sample space

Probability Topics

A

Event A

event A

Probability Topics

P (A)

probability of A

probability of A occur-

ring

Probability Topics

P (A | B)

probability of A given B

prob.

of A occurring

given B has occurred

Probability Topics

P (AorB)

prob. of A or B

prob. of A or B or both

occurring

continued on next page

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APPENDIX

Probability Topics

P (AandB)

prob. of A and B

prob. of both A and B

occurring (same time)

Probability Topics

A’

A-prime, complement

complement of A, not A

of A

Probability Topics

P (A’)

prob. of complement of

same

A

Probability Topics

G1

green on first pick

same

Probability Topics

P (G1)

prob. of green on first

same

pick

Discrete Random Vari-

PDF

prob. distribution func-

same

ables

tion

Discrete Random Vari-

X

X

the random variable X

ables

Discrete Random Vari-

X ∼

the distribution of X

same

ables

Discrete Random Vari-

B

binomial distribution

same

ables

Discrete Random Vari-

G

geometric distribution

same

ables

Discrete Random Vari-

H

hypergeometric dist.

same

ables

Discrete Random Vari-

P

Poisson dist.

same

ables

Discrete Random Vari-

λ

Lambda

average of Poisson dis-

ables

tribution

Discrete Random Vari-

greater than or equal to

same

ables

Discrete Random Vari-

less than or equal to

same

ables

Discrete Random Vari-

=

equal to

same

ables

Discrete Random Vari-

=

not equal to

same

ables

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653

Continuous

Random

f (x)

f of x

function of x

Variables

Continuous

Random

pd f

prob. density function

same

Variables

Continuous

Random

U

uniform distribution

same

Variables

Continuous

Random

Exp

exponential

distribu-

same

Variables

tion

Continuous

Random

k

k

critical value

Variables

Continuous

Random

f (x) =

f of x equals

same

Variables

Continuous

Random

m

m

decay rate (for exp.

Variables

dist.)

The Normal Distribu-

N

normal distribution

same

tion

The Normal Distribu-

z

z-score

same

tion

The Normal Distribu-

Z

standard normal dist.

same

tion

The Central Limit The-

CLT

Central Limit Theorem

same

orem

The Central Limit The-

X

X-bar

the random variable X-

orem

bar

The Central Limit The-

µ x

mean of X

the average of X

orem

The Central Limit The-

µ x

mean of X-bar

the average of X-bar

orem

The Central Limit The-

σ x

standard deviation of X

same

orem

The Central Limit The-

σ x

standard deviation of

same

orem

X-bar

The Central Limit The-

ΣX

sum of X

same

orem

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654

APPENDIX

The Central Limit The-

Σx

sum of x

same

orem

Confidence Intervals

CL

confidence level

same

Confidence Intervals

CI

confidence interval

same

Confidence Intervals

EBM

error bound for a mean

same

Confidence Intervals

EBP

error bound for a pro-

same

portion

Confidence Intervals

t

student-t distribution

same

Confidence Intervals

df

degrees of freedom

same

Confidence Intervals

t α

student-t with a/2 area

same

2

in right tail

^

Confidence Intervals

p’ p

p-prime; p-hat

sample proportion of

success

^

Confidence Intervals

q’ q

q-prime; q-hat

sample proportion of

failure

Hypothesis Testing

H0

H-naught, H-sub 0

null hypothesis

Hypothesis Testing

Ha

H-a, H-sub a

alternate hypothesis

Hypothesis Testing

H1

H-1, H-sub 1

alternate hypothesis

Hypothesis Testing

α

alpha

probability of Type I er-

ror

Hypothesis Testing

β

beta

probability of Type II

error

Hypothesis Testing

X1 − X2

X1-bar minus X2-bar

difference

in

sample