Question

Determine the t critical value(s) that will capture the desired t-curve area in each of the following cases. (Round your answers to three decimal places.)

(a) Central area = 0.95, df = 10

(b) Central area = 0.95, df = 20

(c) Central area = 0.99, df = 20

(d) Central area = 0.99, df = 60

(e) Upper-tail area = 0.01, df = 30

(f) Lower-tail area = 0.025, df = 5

Answer 1

Concepts and reason

The t critical value is a cut-off point on the t distribution which covers area from left hand side. It is almost identical to the z critical value. The difference is that the shape of the t distribution is a different from the shape of the normal distribution. The t-critical value is useful in finding the probability till that value. In testing of hypothesis, this value can be used to make the decisions.

Fundamentals

The critical value of t distribution is obtained from the t tables at a given degrees of freedom with $100\left( {1 - \alpha } \right)\%$ level of significance. Also, the same values can be obtained from Excel software.

The critical value of two-sided confidence interval can be found using Excel command $T.INV.2T\left( {{\rm{alpha, degrees of freedom}}} \right)$ .

The critical value for upper tail are can be found using Excel command $T.INV\left( {{\rm{alpha, degrees of freedom}}} \right)$ .

The critical value for lower tail area can be found using Excel command $T.INV\left( {{\rm{1}} - {\rm{alpha, degrees of freedom}}} \right)$ .

(a)

The objective is to determine the t-critical value(s) that will capture the desired t-curve area.

Central area = 0.95 and degrees of freedom = 10.

Here, the centre area must be 0.95. This means that the remaining 1 – 0.95=0.05 area must be equally spread in both the ends. As a result, $\frac{{0.05}}{2} = 0.025$ area must be from left hand side and 0.025 area from right hand side will be left and the remaining 95% area must be covered (as shown below)

Consider the following t-table and shade the required value according to the left hand side area.

The t critical values that capture the desired t curve area with central area 0.95 and having degrees of freedom 10 are, $\pm 2.228$ (Using excel function =t.inv.2t(0.05,10))

The t critical values are –2.228 and +2.228.

(b)

The objective is to determine the t-critical value(s) that will capture the desired t-curve area.

Central area = 0.95 and degrees of freedom = 20.

Here, the centre area must be 0.95. This means that the remaining 1 – 0.95=0.05 area must be equally spread in both the ends. As a result, $\frac{{0.05}}{2} = 0.025$ area must be from left hand side and 0.025 area from right hand side will be left and the remaining 95% area must be covered (as shown below)

Consider the following t-table and shade the required value according to the 0.025left hand side area.

The t critical values that capture the desired t curve area with central area 0.95 and having degrees of freedom 20 are, $\pm 2.086$ (Using excel function =t.inv.2t(0.05,20))

The t critical values are –2.086 and +2.086

(c)

The objective is to determine the t-critical value(s) that will capture the desired t-curve area.

Central area = 0.99 and degrees of freedom = 20.

Here, the centre area must be 0.99. This means that the remaining 1 – 0.99=0.01 area must be equally spread in both the ends. As a result, $\frac{{0.01}}{2} = 0.005$ area must be from left hand side and 0.005 area from right hand side will be left and the remaining 99% area must be covered (as shown below)

Consider the following t-table and shade the required value according to the 0.025left hand side area.

The t critical values that capture the desired t curve area with central area 0.99 and having degrees of freedom 20 are, $\pm 2.845$ (Using excel function =t.inv.2t(0.01,20))

The t critical values are –2.845 and +2.845.

(d)

The objective is to determine the t-critical value(s) that will capture the desired t-curve area.

Central area = 0.99 and degrees of freedom = 60.

Here, the centre area must be 0.99. This means that the remaining 1 – 0.99=0.01 area must be equally spread in both the ends. As a result, $\frac{{0.01}}{2} = 0.005$ area must be from left hand side and 0.005 area from right hand side will be left and the remaining 99% area must be covered (as shown below)

Consider the following t-table and shade the required value according to the 0.025left hand side area.

The t critical values that capture the desired t curve area with central area 0.99 and having degrees of freedom 60 are, $\pm 2.660$ (Using excel function =t.inv.2t(0.01,60))

The t critical values are –2.660 and +2.660.

(e)

The objective is to determine the t-critical value(s) that will capture the desired t-curve area.

Upper-tail area = 0.01 and degrees of freedom = 30

Here, the upper-tail area must be 0.01. This means that, the remaining 1–0.01= 0.99 area must be below the value. That is 0.99 area must be from left hand side will be left and the remaining 1% area must be covered (as shown below)

Consider the following t-table and shade the required value according to the 0.025left hand side area.

The t critical values that capture the desired t curve area with left side area 0.99 and having degrees of freedom 30 is 2.457(Using excel function =t.inv(0.99,30))

The t critical value is 2.457.

(f)

The objective is to determine the t-critical value(s) that will capture the desired t-curve area.

Lower-tail area = 0.025 and degrees of freedom = 5

Here, the lower-tail area must be 0.025. This means that, the remaining 1–0.025= 0.975 area must be above the value. That is 0.975 area must be towards right hand side (as shown below)

Consider the following t-table and shade the required value according to the 0.025left hand side area.

The t critical values that capture the desired t curve area with left side area 0.025 and having degrees of freedom 5 is 2.571(Using excel function =t.inv(0.01,5)). Since the value is towards left side, negative sign can be shown to the value.

The t critical value is –2.571.

Ans: Part a

The t critical values are –2.228 and +2.228.

Part b

The t critical values are –2.086 and +2.086.

Part c

The t critical values are –2.845 and +2.845.

Part d

The t critical values are –2.660 and +2.660.

Part e

The t critical value is 2.457.

Part f

The t critical value is –2.571.