Inference about Two Population Standard Deviations

Hypothesis Test for Two Variances: Raw Data in Two Columns

Let’s work through Example 3 in Section 11.4. This is the same data we analyzed in Table 3 from Section 11.3.

Table5 <- read.csv("https://sullystats.github.io/Statistics6e/Data/Chapter11/Table3.csv")
head(Table5,n=3)

##   Flight Control
## 1   8.59    8.65
## 2   6.87    7.62
## 3   7.00    7.33

The syntax for the test is

var.test(data_frame\(*x*, *data_frame*\)y, alternative = “less”,“greater”,“two.sided”)

var.test(Table5$Flight,Table5$Control,alternative="two.sided")

## 
##  F test to compare two variances
## 
## data:  Table5$Flight and Table5$Control
## F = 1.024, num df = 13, denom df = 13, p-value = 0.9666
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##  0.3287201 3.1897184
## sample estimates:
## ratio of variances 
##           1.023975

The test statistic is \(F_0 = 1.024\) and the P-value is 0.9666.

Hypothesis Test for Two Variances: Raw Data – One Column with Quantitative Variable; One with Qualitative Variable

Let’s use the Tornado_2017 data and determine if the variability in the length of Georgia tornadoes is greater than that of Louisiana.

Tornado <- read.csv("https://sullystats.github.io/Statistics6e/Data/Tornadoes_2017.csv")
Data_LA_GA <- subset(Tornado,State=="LA"|State=="GA")  # The | means "or" in R
head(Data_LA_GA)

##   Month Day     Time State F.Scale Injuries Fatalities PropLoss Length Width
## 3     1   2 10:06:00    LA       1        0          0    25000   0.30    20
## 4     1   2 10:17:00    LA       1        0          0    50000   1.20    50
## 5     1   2 10:30:00    LA       1        0          0    20000   4.64   100
## 6     1   2 10:30:00    LA       1        0          0   150000   2.74   100
## 7     1   2 11:06:00    LA       1        0          0    50000   0.54    50
## 8     1   2 11:30:00    LA       0        0          0    75000   0.54    25
##   NumberStates  F0
## 3            1  No
## 4            1  No
## 5            1  No
## 6            1  No
## 7            1  No
## 8            1 Yes

install.packages("mosaic")

library(mosaic)
sd(Length~State,data=Data_LA_GA)

##       GA       LA 
## 7.725883 4.048813

The sample standard deviation for length of tornadoes in Georgia is 7.73 miles; the sample standard deviation for length of tornadoes in Louisiana is 4.05 miles.

var.test(Length ~ State,data=Data_LA_GA,alternative="greater")  #Mosaic not needed for this command. 

## 
##  F test to compare two variances
## 
## data:  Length by State
## F = 3.6412, num df = 117, denom df = 86, p-value = 8.42e-10
## alternative hypothesis: true ratio of variances is greater than 1
## 95 percent confidence interval:
##  2.599724      Inf
## sample estimates:
## ratio of variances 
##           3.641169

The test statistic is \(F_0 = 3.64\) and the P-value is 8.42e-10 (which is 0.000000000842).