Section 6.5

Finding t-Values Using the Student t-Distribution

with TI-84 info added by D.R.S.

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Similar to the normal distribution in shape but with more area under the tails and is defined by the number of degrees of freedom.

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Student t-Distribution:

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

1. A t-curve is symmetric and bell-shaped, centered about 0.

2. A t-curve is completely defined by its number of degrees of freedom, d.f.

3. The total area under a t-curve equals 1.

4. The x-axis is a horizontal asymptote for a t-curve.

Properties of a Student t-Distribution:

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Comparison of the Normal and Student t-Distributions:

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

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Student t-Distribution Table:

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

1 3.078 6.314 12.706 31.821 63.657

2 1.886 2.920 4.303 6.965 9.925

3 1.638 2.353 3.182 4.541 5.841

4 1.533 2.132 2.776 3.747 4.604

5 1.476 2.015 2.571 3.365 4.032

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

TI-84 (but not available on TI-83/Plus):invT(area to left, degrees of freedom)Example: invT(0.025, 3)=-3.182446305 and fix up the sign - or use invT(0.975, 3)

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Student t-Distribution Table (continued):

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

When calculating the t-values, round your answers to three decimal places.

1. The numbers across the top row represent an area to the right of t, known as .

2. The numbers down the first column represent the degrees of freedom, d.f. = n – 1.

3. Where the appropriate row and column intersect, we find the t-value associated with the particular area and degrees of freedom.

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Find the value of t0.025 with 25 degrees of freedom.

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

t0.025 = 2.060

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

23 1.319 1.714 2.069 2.500 2.807

24 1.318 1.711 2.064 2.492 2.797

25 1.316 1.708 2.060 2.485 2.787

26 1.315 1.706 2.056 2.479 2.779

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

23 1.319 1.714 2.069 2.500 2.807

24 1.318 1.711 2.064 2.492 2.797

25 1.316 1.708 2.060 2.485 2.787

26 1.315 1.706 2.056 2.479 2.779

TI-84 invT(0.025, 25)=-2.059538532

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How many degrees of freedom make t0.010 0.005 = 4.604?

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

1 3.078 6.314 12.706 31.821 63.657

2 1.886 2.920 4.303 6.965 9.925

3 1.638 2.353 3.182 4.541 5.841

4 1.533 2.132 2.776 3.747 4.604

5 1.476 2.015 2.571 3.365 4.032

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

1 3.078 6.314 12.706 31.821 63.657

2 1.886 2.920 4.303 6.965 9.925

3 1.638 2.353 3.182 4.541 5.841

4 1.533 2.132 2.776 3.747 4.604

5 1.476 2.015 2.571 3.365 4.032

d.f. = 4

TI-84 can’t do this – must use the table

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Find the value of t such that the shaded area to the right is 0.1 for 17 degrees of freedom.

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

t0.100 = 1.333

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

15 1.341 1.753 2.131 2.602 2.947

16 1.337 1.746 2.120 2.583 2.921

17 1.333 1.740 2.110 2.567 2.898

18 1.330 1.734 2.101 2.552 2.878

19 1.328 1.729 2.093 2.539 2.861

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

15 1.341 1.753 2.131 2.602 2.947

16 1.337 1.746 2.120 2.583 2.921

17 1.333 1.740 2.110 2.567 2.898

18 1.330 1.734 2.101 2.552 2.878

19 1.328 1.729 2.093 2.539 2.861

TI-84 invT(0.1, 17)=-1.33337939

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Find the value of t such that the shaded area to the left is 0.05 for 11 degrees of freedom.

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

t0.050 = -1.796

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

11 1.363 1.796 2.201 2.718 3.106

12 1.356 1.782 2.179 2.681 3.055

13 1.350 1.771 2.160 2.650 3.012

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

11 1.363 1.796 2.201 2.718 3.106

12 1.356 1.782 2.179 2.681 3.055

13 1.350 1.771 2.160 2.650 3.012

t0.050 = 1.796, however the table assumes that the area is to the right of t. Since the t-curve is symmetric at t = 0, we can simply change the sign of the t-value to obtain the correct answer.

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Find the value of t such that the shaded area to the left is 0.05 for 11 degrees of freedom.

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

TI-84 invT(0.05, 11)=-1.795884781

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Find the value of t such that the shaded area in the tails is 0.02. Assume there are 7 degrees of freedom.

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

t0.010 = 2.998

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

7 1.415 1.895 2.365 2.998 3.499

8 1.397 1.860 2.306 2.896 3.355

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

11 1.363 1.796 2.201 2.718 3.106

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

7 1.415 1.895 2.365 2.998 3.499

8 1.397 1.860 2.306 2.896 3.355

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

11 1.363 1.796 2.201 2.718 3.106

This type of problem is called two-tailed.If the area in both tails is 0.02, then the area in one tail would be 0.01.

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Find the value of t such that the shaded area in the tails is 0.02. Assume there are 7 degrees of freedom.

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

This type of problem is called two-tailed.If the area in both tails is 0.02, then the area in one tail would be 0.01.

TI-84 invT(0.01, 7)=-2.997951566

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Find the value of t such that the shaded area between –t and t is 99%. Assume 24 degrees of freedom.

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

t = 2.797.

Since 99% of the area of the curve is in the middle, that leaves 1%, or 0.01 of the area on the outside.Because of symmetry each tail will only have half of 0.01 in its area, 0.005.

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

23 1.319 1.714 2.069 2.500 2.807

24 1.318 1.711 2.064 2.492 2.797

25 1.316 1.708 2.060 2.485 2.787

26 1.315 1.706 2.056 2.479 2.779

Student t-Distribution Table

d.f. 0.100 0.050 0.025 0.010 0.005

23 1.319 1.714 2.069 2.500 2.807

24 1.318 1.711 2.064 2.492 2.797

25 1.316 1.708 2.060 2.485 2.787

26 1.315 1.706 2.056 2.479 2.779

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Find the value of t such that the shaded area between –t and t is 99%. Assume 24 degrees of freedom.

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

Since 99% of the area of the curve is in the middle, that leaves 1%, or 0.01 of the area on the outside.Because of symmetry each tail will only have half of 0.01 in its area, 0.005.

TI-84 invT(0.005, 24)=-2.796939498