We wish to test of hypothesisH0 : μ1 = μ2 = μ3vsH1 : Not all group means are the same.Use the significance level α = 0.01.a. Find the value of the test statistic and its distribution.b. Find the rejection region.c. Find the p-value of the test.d. Make a conclusionall subparts please! thank you Regression modelThe regression equation for ANOVA is given byy = Bo + B₁ + region South + B + regionWest +where ~ N(0,0²). The dummy variables regionSouth and regionWest are created by the 1m () functionand appear in the output below.Fitting the ANOVA model and output from summary() and anova () functionsThe following R code fit the ANOVA model with 1m () function and print outputs from summary() functionand anova () function.anov_out <- lm (balance - region, data = dat)summary (anov_out)**## Call:## 1m (formula = balance - region, data = dat)## Residuals:Min1Q Median3QMax## -531.00 -457.08 -63.25 339.25 1480.50## Coefficients:#### (Intercept)## regionSouth -12.50## regionWest -18.69Estimate Std. Error t value Pr(>1t|)531.0046.32 11.464 <2e-16 ***56.68 -0.22165.02 -0.287## Signif. codes: 0*** 0.001 0.01 0.05 0.11#### Response: balance#### Residual standard error: 460.9 on 397 degrees of freedom## Multiple R-squared: 0.0002188, Adjusted R-squared: -0.004818## F-statistic: 0.04344 on 2 and 397 DF, p-value: 0.9575anova (anov_out)## Analysis of Variance Table##Box plot0.8260.774Df2## region## Residuals 397 84321458 212397Sum Sq Mean Sq F value Pr(>F)18454 9227 0.0434 0.9575boxplot (balance - region, data = dat) 500 1000 1500 2000balance0EastOSouthregionWestWe wish to compare Balance in 3 different Region.• Region: A factor with levels East, South, and West indicating the individual's geographical location.Balance: Average credit card balance in $.The mean balance for East, South and West are denoted, respectively, by 1, 2 and 3.Data setlibrary (ISLR2)head (Credit)2 3411 No3 8215 Yes## Income Limit Rating Cards Age Education Own Student Married Region Balance## 1 14.891 3606## 2 106.025 6645## 3 104.593 7075## 4 148.924 9504## 5 55.882 4897## 6 80.180 80474 713 362 684 77Yes SouthYesWestNo WestNo WestYes SouthNo South283483514681357569balance <- Credit$Balanceregion

Given information:k= 3 Groups n= 400 Total observation Inference One way ANOVA

Answered: We wish to test of hypothesis H0 : μ1 =…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

We wish to test of hypothesis
H0 : μ1 = μ2 = μ3
vs
H1 : Not all group means are the same.
Use the significance level α = 0.01.
a. Find the value of the test statistic and its distribution.
b. Find the rejection region.
c. Find the p-value of the test.
d. Make a conclusion

all subparts please! thank you

Regression model
The regression equation for ANOVA is given by
y = Bo + B₁ + region South + B + regionWest +
where ~ N(0,0²). The dummy variables regionSouth and regionWest are created by the 1m () function
and appear in the output below.
Fitting the ANOVA model and output from summary() and anova () functions
The following R code fit the ANOVA model with 1m () function and print outputs from summary() function
and anova () function.
anov_out <- lm (balance - region, data = dat)
summary (anov_out)
**
## Call:
## 1m (formula = balance - region, data = dat)
## Residuals:
Min
1Q Median
3Q
Max
## -531.00 -457.08 -63.25 339.25 1480.50
## Coefficients:
##
## (Intercept)
## regionSouth -12.50
## regionWest -18.69
Estimate Std. Error t value Pr(>1t|)
531.00
46.32 11.464 <2e-16 ***
56.68 -0.221
65.02 -0.287
## Signif. codes: 0*** 0.001 0.01 0.05 0.11
##
## Response: balance
##
## Residual standard error: 460.9 on 397 degrees of freedom
## Multiple R-squared: 0.0002188, Adjusted R-squared: -0.004818
## F-statistic: 0.04344 on 2 and 397 DF, p-value: 0.9575
anova (anov_out)
## Analysis of Variance Table
##
Box plot
0.826
0.774
Df
2
## region
## Residuals 397 84321458 212397
Sum Sq Mean Sq F value Pr(>F)
18454 9227 0.0434 0.9575
boxplot (balance - region, data = dat)

500 1000 1500 2000
balance
0
East
O
South
region
West
We wish to compare Balance in 3 different Region.
• Region: A factor with levels East, South, and West indicating the individual's geographical location.
Balance: Average credit card balance in $.
The mean balance for East, South and West are denoted, respectively, by 1, 2 and 3.
Data set
library (ISLR2)
head (Credit)
2 34
11 No
3 82
15 Yes
## Income Limit Rating Cards Age Education Own Student Married Region Balance
## 1 14.891 3606
## 2 106.025 6645
## 3 104.593 7075
## 4 148.924 9504
## 5 55.882 4897
## 6 80.180 8047
4 71
3 36
2 68
4 77
Yes South
Yes
West
No West
No West
Yes South
No South
283
483
514
681
357
569
balance <- Credit$Balance
region <as.factor (Credit$Region)
dat <- data.frame (region, balance)
str(dat)
##
region
## East 99
## South: 199
## West 102
##
##
##
11 No
11 Yes.
balance
:
Min.
1st Qu.:
Median
Mean
3rd Qu.: 863.00
0.00
68.75
459.50
: 520.01
Max. :1999.00
16 No
10 No
## 'data.frame':
400 obs. of 2 variables:
## $ region Factor w/ 3 levels "East", "South",..: 2 3 3 3 2 2 1 3 2 1 ...
## $ balance: num
333 903 580 964 331
summary (dat)
No
Yes
No
No
No
No
333
903
580
964
331
1151

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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