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Review: Confidence Intervals

• What was their purpose again?

• How did we calculate them again?

Review: Confidence Intervals

• Our sample mean ( 𝑋) is our best estimate of the true population mean (μ)

• We know however that because of sampling error 𝑋 is likely to deviate from μ

• The standard error of the mean tells us on average how far off 𝑋 is likely to be from μ

• We can use this to construct a range of mean values around our sample mean 𝑋 that contains μ with some level of probability • This is called a confidence interval

Review: Confidence Intervals

Standard Error of the Mean Estimated Standard Error of the Mean

Formula 𝜎 𝑋 = 𝜎

𝑁 𝑆 𝑋 =

𝑆

𝑁 − 1

Distribution Z Distribution T Distribution (small samples) Z Distribution (large samples)

Review: The 95% Confidence Interval

• An interval around our sample mean ( 𝑋) that contains our population mean (μ) 95% of the time

• Formula:

95% 𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 𝑋 +

− (1.96)(𝜎 𝑋)

Exercises

• See textbook page 194-195 for example

• See textbook page 212, question 21 and 22 for practice

Review: Confidence Intervals with the T Distribution • We use the T distribution if:

• We estimate the standard error of the mean using the 𝑆 𝑋 = 𝑆

𝑁 −1 formula

• AND if our sample size is small

• There is actually a family of T distributions • The number of degrees of freedom you have will tell you which distribution is

applicable

• Formula: 𝐷𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝐹𝑟𝑒𝑒𝑑𝑜𝑚 𝑑𝑓 = 𝑁 − 1

Review: Using the T Table

• Alpha (α)- the area in the tails of the distribution • Formula:

𝛼 = 1 − 𝑙𝑒𝑣𝑒𝑙 𝑜𝑓 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒

• Degrees of freedom (df)- the number of free observations • Formula:

𝑑𝑓 = 𝑁 − 1

Review: Calculating Confidence Intervals with the T table

• Confidence intervals based on the T distribution

• Formula:

𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 𝑋 +

− 𝑡 (𝑆 𝑋)

Exercises

• See textbook page 201-203 for example

• See textbook page 212, question 26 and 27 for practice

Review: Difference of Means Hypothesis Testing • Assume that the null hypothesis is correct (i.e. the difference

between population means is 0) • 𝜇1 − 𝜇2 = 0

• Calculate the observed difference of means between the samples • 𝑋1 − 𝑋2

• Calculate the probability that we would obtain a difference as extreme as the one we found if the null hypothesis is true

• Reject the null hypothesis if this probability is small enough • P < .05

Difference of Means Testing- Overview

• Calculate the sample means 𝑋1 𝑎𝑛𝑑 𝑋2 • Calculate the difference between sample means

• 𝑋1 − 𝑋2

• Translate the mean difference into a T score

• 𝑇 = 𝑋1 − 𝑋2

𝑆 𝑋1 − 𝑋2 𝑆 𝑋1 − 𝑋2 =

𝑁1 𝑆1 2+ 𝑁2𝑆2

𝑁1+𝑁2 −2

𝑁1+ 𝑁2

𝑁1𝑁2

• Compare the observed T value to the table T value whereby df = N1 + N2 – 2 and alpha = 0.05 • If observed T > Table T, you can reject the null

• If observed T < Table T, you fail to reject the null

Exercises

• See textbook page 239-242 for example

• See textbook page 265, question 17, 18, and 19 for practice

• Calculate expected frequencies for each category. • Divide total number of observations by K, the number of categories.

• Calculate Chi-Square statistic • Consult textbook page 323

Review: The One Way Chi Square Test

• Look up the critical Chi Square value in the table (p.557) • Degrees of Freedom = K-1

• Compare critical Chi Square to calculated Chi square • If your calculated Chi Square is > your critical Chi square you reject the null

hypothesis

• If your calculated Chi Square is < you critical Chi Square you fail to reject the null hypothesis

Exercises

• See textbook page 323-325 for example

• See textbook page 350-351, question 10 and 11 for practice

Review: The Two Way Chi Square Test

The Two Way Chi Square Test – Example

Democrat Republican Total

Obama 20 10 30

Romney 5 65 70

Total 25 75 100

Row Marginal Totals

Column Marginal Totals

Exercises

• See textbook page 331-333 for example

• See textbook page 357, question 31 and 32 for practice

The Correlation Coefficient (r)

• A coefficient that tells us about the strength and direction of a relationship

• Always ranges from -1 to 1

• Direction: • Positive numbers indicate a positive relationship

• Negative numbers indicate a negative relationship

• Strength:

-1 Perfect Neg. Correlation

1 Perfect Pos. Correlation

-.6 Strong Neg. Correlation

.6 Strong Pos. Correlation

-.3 Moderate Neg. Correlation

.3 Moderate Pos. Correlation

-.1 Weak Neg. Correlation

.1 Weak Pos. Correlation

No Correlation 0

The Correlation Coefficient (r)

The Computational Formula for r

Testing the Significance of r

Correlation – Specific Steps

Correlation – Specific Steps cont.

• Find the critical r in the table • Calculate the degrees of freedom

• N - 2

• α = .05

• Compare calculated r to critical r • If our calculated r is > critical r we reject the null hypothesis

• If our calculated r is < critical r we fail to reject the null hypothesis

Exercises

• See textbook page 377-379 for example

• See textbook page 393, question 16, 17, and 18 for practice

Partial Correlation

Find the Partial Correlation Coefficient- Specific Steps

• Calculate the degrees of freedom • N – 3

• Look up the critical value in the Table H • alpha = .05

• Compare the calculated correlation to the critical value • If the calculated value is > critical value we reject the null hypothesis

• If the calculated value is < critical value we fail to reject the null hypothesis

Review- Regression

Exercises

• See textbook page 385-386 for example

• See textbook page 397, question 30 and 31 for practice

Review- Regression

Review- Regression Analysis Example

Regression Analysis- Example Cont.

Y = 1.56 + .86 X

(0, 1.56)

(4, 5)

Exercises

• See textbook page 403-406 for example

• See textbook page 435, question 6 and 7 for practice (a-d only)

Interpreting Regression Output

_cons 1.545455 1.420253 1.09 0.326 -2.105423 5.196332

x1 .8636364 .3246104 2.66 0.045 .0291988 1.698074

y1 Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 28 6 4.66666667 Root MSE = 1.5226

Adj R-squared = 0.5032

Residual 11.5909091 5 2.31818182 R-squared = 0.5860

Model 16.4090909 1 16.4090909 Prob > F = 0.0449

F( 1, 5) = 7.08

Source SS df MS Number of obs = 7

b a

T values P Values Confidence Intervals

Interpreting Regression Output

_cons 11 1.359676 8.09 0.004 6.672905 15.32709

x2 -.5 .2118296 -2.36 0.099 -1.174136 .1741364

y2 Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 10 4 2.5 Root MSE = 1.0801

Adj R-squared = 0.5333

Residual 3.5 3 1.16666667 R-squared = 0.6500

Model 6.5 1 6.5 Prob > F = 0.0994

F( 1, 3) = 5.57

Source SS df MS Number of obs = 5

Interpreting Regression Output

_cons .1500793 1.503553 0.10 0.921 -2.799602 3.099761

partyid7 11.81617 .3270595 36.13 0.000 11.17454 12.4578

obamatherm Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 1592752.78 1289 1235.64994 Root MSE = 24.783

Adj R-squared = 0.5029

Residual 791073.363 1288 614.187394 R-squared = 0.5033

Model 801679.413 1 801679.413 Prob > F = 0.0000

F( 1, 1288) = 1305.27

Source SS df MS Number of obs = 1290

. reg obamatherm partyid7

Interpreting Regression Output

_cons 52.86189 2.239573 23.60 0.000 48.46827 57.2555

poliinterest -2.110292 .9567143 -2.21 0.028 -3.987184 -.2333998

obamatherm Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 1592279.44 1287 1237.20236 Root MSE = 35.121

Adj R-squared = 0.0030

Residual 1586277.95 1286 1233.49763 R-squared = 0.0038

Model 6001.48622 1 6001.48622 Prob > F = 0.0276

F( 1, 1286) = 4.87

Source SS df MS Number of obs = 1288

Categorical/Ordinal

Dependent Variable

Interval Dependent

Variable

Categorical/ Ordinal

Independent Variable Chi Square Difference of Means

Interval Independent

Variable Not Covered! Correlation OR

Regression (bivariate)

Which method to use?

• You want to know whether there is a relationship between gender (male-female) and the vote (voted-did not vote). What test do you use?

• You want to know whether growing up in Long Island makes you more likely to be vote Republican (0 – not likely at all to 10 – very likely). What test do you use?

• You want to know whether there is a relationship between years of education and perceived competence of President Obama (0- not competent at all to 10 – very competent). What test do you use?