Question

Suppose the environmentalist wanted to check if the air quality had improved in recent years. A more comprehensive study was thus conducted in 2018. This time the concentration of NO2 was measured hourly at thirty randomly selected monitoring stations.

The data: NO2_2018 0.095 0.099 0.099 0.083 0.092 0.087 0.098 0.096 0.098 0.104 0.093 0.082 0.079 0.098 0.097 0.082 0.07 0.09 0.089 0.096 0.097 0.094 0.089 0.076 0.078 0.097 0.096 0.076 0.059 0.112

(a) Using Welch’s two-sample t test, determine if the NO2 levels have significantly lowered. Test using the 5% level of significance. [6 marks]

The other set of data is NO2_2013 0.108 0.122 0.165 0.194 0.116 0.084 0.095 0.141 0.15 0.146 0.126 0.126 0.132 0.068 0.101 0.061 0.304 0.116 0.052 0.165

[Hint: you will need to combine the two datasets in a particular format in order to perform the test using R Commander.

(b) State the statistical assumptions underlying the Welch’s two-sample t test. Demonstrate if the assumptions are valid or not for these data. [3 marks]

(c) Is it appropriate to use the paired t test to conduct the test here? Explain your answer. [2 marks]

Answer 1

a)

no2_2013 <- c(0.108, 0.122, 0.165, 0.194, 0.116, 0.084, 0.095, 0.141, 0.15, 0.146, 0.126, 0.126, 0.132, 0.068, 0.101, 0.061, 0.304, 0.116, 0.052, 0.165)

no2_2018 <- c(0.095, 0.099, 0.099, 0.083, 0.092, 0.087, 0.098, 0.096, 0.098, 0.104, 0.093, 0.082, 0.079, 0.098, 0.097, 0.082, 0.07, 0.09, 0.089, 0.096, 0.097, 0.094, 0.089, 0.076, 0.078, 0.097, 0.096, 0.076, 0.059, 0.112)

Null hypothesis for Welch's test is that the NO2 levels in 2018 are greater than or equal to the levels in 2013.

Alternate hypothesis is the NO2 levels in 2018 is less than the levels in 2013.

t.test(no2_2018, no2_2013, var.equal = FALSE, paired = FALSE, conf.level = 0.95, alternative = "less")

Thus we reject the null hypothesis and accept alternate hypothesis.

b) Assumptions in welch's test

variances are unequal , also observed in the data

var.test(no2_2013,no2_2018)

This shows that variances are unequal.

c) It is appropriate to conduct a paired t-test for this provided we ensure to measure the observations at same stations at different periods of time. Else it is not appropriate.