Warm-upCh.11 Inference for Linear Regression (Day 1)

1. The following is from a particular region’s mortality table.

What is the probability that a 20-year-old will survive to be 60? (A) 0.1959 (B) 0.4419 (C) 0.7800 (D) 0.8041 (E) 0.9700

2. Fifty-three percent of adults say they have trouble sleeping. If adoctor contacts an SRS of 85 adults, what is the probability that over55% will say they have trouble sleeping? (A) 0.3109 (B) 0.3558 (C) 0.3650 (D) 0.4000 (E) 0.6442

Age 0 20 40 60 80

Number Surviving

10,000

9,700 9,240 7,800 4,300

Answers to 3 and 4 of χ2 worksheet3. The expected cell frequency for all 5 countries would be616 for Necessary 234 for Unnecessary and 30 forUndecided.Step 1: The randomization condition is not met because the

1,000 person sample for each country is not specified to be random. It is questionable whether the 1,000 sampled are representative of each country. The count condition is met because there are counts for all cells of each category. The expected cell frequency is greater than 5 as shown above.

The conditions are somewhat met for a Chi-Square model with 8 degrees of freedom for a test of homogeneity.

Step 2 and 3Step 2: Ho: The proportion of opinions is distributed uniformly

for all five countries.HA: The proportion of opinions is not uniformly distributed for

all five counties. Step 3: χ2 = 1110.93 p-value ≈ 0 .

Step 4 and #4 Step 1Step 4: The p-value is approximately zero. I reject the Ho. There is no

similarity in the distribution of opinions of the five countries. The Chisquare value of 1110.93 for 8 degrees of freedom demonstratesthere is a drastic difference between expected observe.#4 Step 1:

The sample of 5,387 is not random, but the sample is big enough to suspect of bias. The expected cell frequency is shown to be greater than five. There are counts for all cells of the categories.

The conditions are met for a X2 model with 12 degrees of freedom with a test of independence.

Remaining steps are for #4Step 2: Ho: Eye color is independent of hair color.

HA: Eye color is not independent of hair color.

Step 3: χ2 = 1241.06 p-value ≈ 0 .

Step 4: I reject the Ho in favor of the HA. Eye color is notindependent of hair color. With a p-value of approximatelyzero and a chi-square of 1241.06 there is strong evidence that thesefactors are not related.

Inference for Regression• Remember when we covered scatter plotswas the least squares regression line. It included andthe LSRL describes the set of numerical data in hopes ofpredicted the response variable for the given explanatoryvariable.• When sampling for different sets of data we know that

there will be different that will affect the LSRL.• For inference of a regression line we will use called line of means or line of averages

• Since there are two estimates, we will work with n – 2 degrees of freedom.

• For the A.P. Statistics we are only interested in inference for

slope (β).

bxay ˆ),( yx

),( yx

xy

Problem 1 (checking conditions only…for now)

With the help of satellite images of Earth, craters frommeteor impact, have been identified. Now more than 180 areknown. These are only a small sample because many of thecraters have been uncovered or eroded away. Astronomershave recognized roughly 35 million year cycle of cratering.Here’s a scatter plot of known impacts.

Any ideas why both x and y

axis are in log?

Checking Conditions• Linearity Assumption. The data on the scatter plot needs to demonstrate a

linear relationship.• Randomization• *Equal Variance Condition. Looking at the residual plot the points need to

be scattered constant throughout the line. No outliers or fan-shapes.• *Nearly Normal Condition. A histogram of the residuals and/or a normal

plot needs to be evaluated.

* There must be a residual plot and at least a residual plot to check the conditions.

Problem 2Checking conditions

Enter data in L1 and L2, Then Stat, Calc select Lin Reg y = a +bx, LinReg L1, L2, Y1 Graph using Statplot function L1 and L2 and sketch the graph.For residual graph: Statplot, under YList go to List, Names and find ResidGraph and sketch plot. Then write if it meets all the conditions.To get the histogram of the residuals: Statplot, Select Histogram, X-List: Resid