Least Square Estimate and Fitted Equation

Review 2 2004.6.9

Chapter 12

Least square estimate and fitted equation

Testing hypothesis (F statistic and t statistic) and confidence interval



prediction

residual plots

Example:

Given are 5 observations for two variables x and y.

/ 2 / 3 / 5 / 1 / 8
/ 25 / 25 / 20 / 30 / 16

Suppose the model is

(a)Find the least square estimate and the fitted regression equation

(b)Provide an ANOVA table and use F statistic to test at .

(d)Find the 95% confidence interval for .and use the confidence interval to test .

(e)Find the ANOVA table for and use F statistic to test the hypothesis at .

(f)Determine and the sample correlation coefficient.

(g)Find the 90% confidence interval for .

(h)Develop a plot of the residuals.

[solutions:]

(a)Since

thus,

Then, the least square estimate is

The fitted regression equation is

(b)

Since

The ANOVA table is

Source / df / SS / MS / F
Regression / 1 / SSR=108.467 / /
Residual (Error) / n-2=3 / SSE=6.333 /
Total (corrected) / n-1=4 / SST=114.8

Since , we reject .

(c)

Since

we do not reject .

(d)

The 95% confidence interval for is

Since , we do not reject.

(e)

Since , the ANOVA table for is

Source / df / SS / MS / F
Regression / 2 / 2799.667 / /
Residual (Error) / n-2=3 / SSE=6.333 /
Total / 5 / 2806

Since , we reject

(f)

(g)

Since , the 90% confidence interval for is

(h)

The residuals are

-1.577922 / 0.2987013 / -0.9480519 / 1.545455 / 0.6818182

Chapter 11

Interval estimate, hypothesis tests, and p-value of the population proportion difference .

test for proportions of a multinomial population and for the independence (contingency table).

Example:

The following are the number of wrong answers for the number of the students.

Number of wrong answers / 0 / 1 / 2 / 3
Number of the students / 21 / 31 / 12 / 0

Suppose X is the random variable representing the number of wrong answers. Please test X is distributed as Binomial(3,0.25) with .

(Note: the distribution function for Binomial(3,0.25) is

[solutions:]

As is true, the distribution for the number of wrong answers is

Since the sample size , the expected numbers under are

Therefore,

Since

we do not reject .

Chapter 10

Interval estimate, hypothesis tests, and p-value of the population mean difference .

Two methods, independent samples and matched samples, can be used for the population mean difference .

ANOVA for testing the equality of k population means.

Example:

The following data are from 4 different populations.

Population 1 / 8.2 / 8.7 / 9.4 / 9.2
Population 2 / 7.7 / 8.4 / 8.6 / 8.1 / 8.0
Population 3 / 6.9 / 5.8 / 7.2 / 6.8 / 7.4 / 6.1
Population 4 / 6.8 / 7.3 / 6.3 / 6.9 / 7.1

Let , , and be the mean number of products of the 4 production lines.

(a) Provide the ANOVA table.

(b) Please test the hypothesis with .

[solution:]

(a)

Thus,

The ANOVA table is

Source / SS / df / MS / F
Between / SSB=15.462 / / /
Within / SSW=3.888 / /
Total / 19.35 /

(b)

Since

we reject .