Question

Apply t-test to the following data sets and compare it with the value of u=4. Identify the test and find the critical value and state the conclusion

a) 2 3 4 5 6 7 b) 1 2 2 5 5 7

Answer 1

a)0.654

b)-0.34

Explanation:

a)2 3 4 5 6 7

`\mu = 4`

`Mean = \frac{\text{Sum of all observations}}{\text{No. of observations}}\\Mean = (2+3+4+5+6+7)/(6)\\Mean =4.5`

Standard deviation
`=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}`

Standard deviation
`=\sqrt{((2-4.5)^2+(3-4.5)^2+(4-4.5)^2+(5-4.5)^2+(6-4.5)^2+(7-4.5)^2)/(6-1)}=1.87`

`t=(x-\mu)/((\sigma)/(√(n)))\\t=(4.5-4)/((1.87)/(√(6)))\\t=0.654`

Df = n-1 = 6-1 = 5

Assume α=0.05

So
`t_((\alpha,df))=t_(0.05,5)=2.57`

So, t critical = 2.57

So, t calculated < t critical

b)1 2 2 5 5 7

`\mu = 4 \\Mean = \frac{\text{Sum of all observations}}{\text{No. of observations}}\\Mean = (1+2+2+5+5+7)/(6)\\Mean =3.67`

Standard deviation =
`\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}`

Standard deviation =
`\sqrt{((1-3.67)^2+(2-3.67)^2+(2-3.67)^2+(5-3.67)^2+(5-3.67)^2+(7-3.67)^2)/(6-1)}=2.338`

`t=(x-\mu)/((\sigma)/(√(n)))t=(3.67-4)/((2.338)/(√(6)))t=-0.34`

Df = n-1 = 6-1 = 5

Assume α=0.05

So,
`t_((\alpha,df))=t_(0.05,5)=2.57`

So, t critical = 2.57

So, t calculated < t critical