Question

1. You have the following data of wind speed (in mph) in Ithaca, NY at 3 pm each day:

                  10, 7, 3, 9, 7, 9, 12, 13, 1, 12.    

Evaluate whether this data is a better match for the normal or lognormal distribution using the Chi Square test, the Maximum Likelyhood test, and the Kalmogorov-Smirnoff test.  Use the winning distribution to predict the probability of getting wind speeds over 15 mph.   It is recommended you use R for these calculations, but show what you are doing in your hwk submission.

USING R studio

Answer 1

install.packages("EnvStats")
library(EnvStats)
x=c(10,7,3,9,7,9,12,13,1,12)

############# chi-square test (for normal) #########################

R code is as follows:

gofTest(x,distribution="norm",est.arg.list=list(method="mle"),test="chisq")

##################And the output is:#################

Results of Goodness-of-Fit Test
-------------------------------

Test Method: Chi-square GOF

Hypothesized Distribution: Normal

Estimated Parameter(s): mean = 8.300000
sd = 3.716181

Estimation Method: mle/mme

Data: x

Sample Size: 10

Test Statistic: Chi-square = 3.2

Test Statistic Parameter: df = 3

P-value: 0.361805

Alternative Hypothesis: True cdf does not equal the
Normal Distribution.
Warning message:
In chisqGofTest(x = c(10, 7, 3, 9, 7, 9, 12, 13, 1, 12), n.classes = NULL, :
Expected counts < 5. Chi-squared approximation may not
be appropriate.

############# KS- test (for normal) #########################

R code is as follows:

gofTest(x,distribution="norm",est.arg.list=list(method="mle"),test="ks")

##################And the output is:#################

Results of Goodness-of-Fit Test
-------------------------------

Test Method: Kolmogorov-Smirnov GOF

Hypothesized Distribution: Normal

Estimated Parameter(s): mean = 8.300000
sd = 3.716181

Estimation Method: mle/mme

Data: x

Sample Size: 10

Test Statistic: ks = 0.1747049

Test Statistic Parameter: n = 10

P-value: 0.9203193

Alternative Hypothesis: True cdf does not equal the
Normal Distribution.
Warning messages:
1: In ksGofTest(x = c(10, 7, 3, 9, 7, 9, 12, 13, 1, 12), alternative = "two.sided", :
The standard Kolmogorov-Smirnov test is very conservative (Type I error smaller than assumed; high Type II error) for testing departures from the Normal distribution when you have to estimate the distribution parameters.

2: In ks.test(x = c(10, 7, 3, 9, 7, 9, 12, 13, 1, 12), y = "pnorm", :
ties should not be present for the Kolmogorov-Smirnov test

############# chi-square test (for lognormal) #########################

R code is as follows:

gofTest(x,distribution="lnorm",est.arg.list=list(method="mle"),test="chisq")

##################And the output is:#################

Results of Goodness-of-Fit Test
-------------------------------

Test Method: Chi-square GOF

Hypothesized Distribution: Lognormal

Estimated Parameter(s): meanlog = 1.9222229
sdlog = 0.7560852

Estimation Method: mle/mme

Data: x

Sample Size: 10

Test Statistic: Chi-square = 11.6

Test Statistic Parameter: df = 3

P-value: 0.008886889

Alternative Hypothesis: True cdf does not equal the
Lognormal Distribution.
Warning message:
In chisqGofTest(x = c(10, 7, 3, 9, 7, 9, 12, 13, 1, 12), n.classes = NULL, :
Expected counts < 5. Chi-squared approximation may not
be appropriate.

############# KS- test (for lognormal) #########################

R code is as follows:

gofTest(x,distribution="lnorm",est.arg.list=list(method="mle"),test="ks")

##################And the output is:#################

Results of Goodness-of-Fit Test
-------------------------------

Test Method: Kolmogorov-Smirnov GOF

Hypothesized Distribution: Lognormal

Estimated Parameter(s): meanlog = 1.9222229
sdlog = 0.7560852

Estimation Method: mle/mme

Data: x

Sample Size: 10

Test Statistic: ks = 0.3124963

Test Statistic Parameter: n = 10

P-value: 0.2828932

Alternative Hypothesis: True cdf does not equal the
Lognormal Distribution.
Warning messages:
1: In ksGofTest(x = c(10, 7, 3, 9, 7, 9, 12, 13, 1, 12), alternative = "two.sided", :
The standard Kolmogorov-Smirnov test is very conservative (Type I error smaller than assumed; high Type II error) for testing departures from the Lognormal distribution when you have to estimate the distribution parameters.

2: In ks.test(x = c(10, 7, 3, 9, 7, 9, 12, 13, 1, 12), y = "plnorm", :
ties should not be present for the Kolmogorov-Smirnov test

From p-values of output we conclude that the data is good fit to normal distribtion
with parameter mean= 8.3 and S.D=3.716181

Now to probability of wind speed over 15 mph is given by

1-pnorm(15,8.3,3.716181)

Output is : 0.03569