ttest2

Load the data set. Create vectors containing the first and second columns of the data matrix to represent students’ grades on two exams.

load examgrades
x = grades(:,1);
y = grades(:,2);

Test the null hypothesis that the two data samples are from populations with equal means.

[h,p,ci,stats] = ttest2(x,y)

h = 
0

p = 
0.9867

ci = 2×1

   -1.9438
    1.9771

stats = struct with fields:
    tstat: 0.0167
       df: 238
       sd: 7.7084

The returned value of h = 0 indicates that ttest2 does not reject the null hypothesis at the default 5% significance level.

Load the data set. Create vectors containing the first and second columns of the data matrix to represent students’ grades on two exams.

load examgrades
x = grades(:,1);
y = grades(:,2);

Test the null hypothesis that the two data vectors are from populations with equal means, without assuming that the populations also have equal variances.

[h,p] = ttest2(x,y,'Vartype','unequal')

h = 
0

p = 
0.9867

The returned value of h = 0 indicates that ttest2 does not reject the null hypothesis at the default 5% significance level even if equal variances are not assumed.

Load the sample data. Create a categorical vector to label the vehicle mileage data according to the vehicle year.

load carbig.mat;

decade = categorical(Model_Year < 80,[true,false],["70s","80s"]);

Create box plots of the mileage data for each decade.

boxchart(decade,MPG)
xlabel("Decade")
ylabel("Mileage")

Create vectors from the mileage data for each decade. Use a left-tailed, two-sample t-test to test the null hypothesis that the data comes from populations with equal means. Use the alternative hypothesis that the population mean for the mileage of cars made in the 1970s is less than the population mean for the mileage of cars made in the 1980s.

MPG70s = MPG(decade == "70s");
MPG80s = MPG(decade == "80s");

[h,~,~,stats] = ttest2(MPG70s,MPG80s,"Tail","left")

h = 
1

stats = struct with fields:
    tstat: -14.0630
       df: 396
       sd: 6.3910

The returned value of h = 1 indicates that ttest2 rejects the null hypothesis at the default significance level of 5%, in favor of the alternative hypothesis that the population mean for the mileage of cars made in the 1970s is less than the population mean for the mileage of cars made in the 1980s.

Plot the corresponding Student's t-distribution, the returned t-statistic, and the critical t-value. Calculate the critical t-value at the default confidence level of 95% by using tinv.

nu = stats.df;
k = linspace(-15,15,300);
tdistpdf = tpdf(k,nu);
tval = stats.tstat

tval = 
-14.0630

tvalpdf = tpdf(tval,nu);
tcrit = -tinv(0.95,nu)

tcrit = 
-1.6487

plot(k,tdistpdf)
hold on
scatter(tval,tvalpdf,"filled")
xline(tcrit,"--")
legend(["Student's t pdf","t-statistic", ...
    "Critical Cutoff"])

The orange dot represents the t-statistic and is located to the left of the dashed black line that represents the critical t-value.