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Question: Problem 1: Karl Pearson (a famous statistician) organized the collection of data on over 1100 fam...

Problem 1:
Karl Pearson (a famous statistician) organized the collection of data on over 1100 families in
England in the period 1893 to 1898. This particular data set gives the Heights in inches of
mothers and their daughters, with up to two daughters per mother. All daughters are at least
age 18, and all mothers are younger than 65.

We want to build a model that uses a mother’s height to predict her daughter’s height. Import data from the R package alr4 and answer the following questions. You may use the code below to install and load the R package (R packages are also called R libraries. They are created and published by R users, and include custom-made functions and sometimes datasets.)

install.packages(‘alr4’) # Note: you only need to install the package once for your laptop
library(alr4) # You must load the R library/package every time you start an R session
if you want to use the R functions or datasets in the library/package.
# You can now use the dataset called Heights
View(Heights) # For example, you can use this code to view the data,

(d) Run a hypothesis test for H0 : 0 = 0 vs. HA : 0 =/ 0 Obtain the test statistic and draw
a conclusion for the test.

(e) Run a hypothesis test for H0 : 1 = 1 vs. HA : 1 =/ 1 Obtain the test statistic and draw
a conclusion for the test.

(f) Based on (d) and (e), do you think the data support the idea that a daughter is expected
to have the same height as her mother.

Problem 2:
Prove that using the least square estimation method will give you . Hint:
n
i=1
eˆi = 0 x ˆ 0 = y ˆ 1

Problem 3:
Question 3 in Chapter 2.8 of the textbook.
Note: You don’t need to import data into R to solve this problem. Utilize the R output provided
on P40 of the textbook to answer (a), (b), and (c).

Explanation / Answer

install.packages("alr4")
library(alr4)
View(Heights)
y = Heights$mheight
x= Heights$dheight
model = lm (y~x)
summary (model)

Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q Median      3Q     Max
-5.6635 -1.4635 0.0254 1.3476 7.6478

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 34.1167     1.3590   25.10   <2e-16 ***
x             0.4445     0.0213   20.87   <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.053 on 1373 degrees of freedom
Multiple R-squared: 0.2408,    Adjusted R-squared: 0.2402
F-statistic: 435.5 on 1 and 1373 DF, p-value: < 2.2e-16

(d) Run a hypothesis test for H0 : 0 = 0 vs. HA : 0 =/ 0 Obtain the test statistic and draw
a conclusion for the test.

TS = 25.10     see the summary , row (intercept) column t-value

p-value = 2e-16 << 0.05

we reject the null hypothesis

(e) Run a hypothesis test for H0 : 1 = 1 vs. HA : 1 =/ 1 Obtain the test statistic and draw
a conclusion for the test.

TS = (b1^ - 1)/se(b1^)

= ( 0.4445      - 1 ) / 0.0213

= -26.0798

|TS| > critical value

we reject the null hypothesis

(f) Based on (d) and (e), do you think the data support the idea that a daughter is expected
to have the same height as her mother.

since we rejectd the null hypothesis that b1 = 1

we conclude that the data does not support the idea that a daughter is expected
to have the same height as her mother.

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