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### Section 8.2

Student’s t-Distribution

With the usual enthralling extra content you’ve come to expect,

by D.R.S., University of Cordele

Student’s t-Distribution

Properties of a t-Distribution

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

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

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

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

t-distribution curve.

math courseware specialists

Continuous Random Variables

6.5 Finding t-Values Using the Student t-Distribution

Comparison of the Normal and Student t-Distributions:

A t-distribution is pretty much the same as a normal distribution!

There’s this additional little wrinkle of “d.f.”, “degrees of freedom”. Slightly different t distributions for different d.f.; higher d.f. is closer & closer to the normal distribution.

Why bother with t ?

If you don’t know the population standard deviation, σ, but you still want to use a sample to find a confidence interval.

t builds in a little more uncertainty based on the lack of a trustworthy σ.

The plan:

This lesson – learn about t and areas and critical values, much like we have done with z.

Next lesson – doing confidence intervals with t.

Why bother with t ?

Observe in the picture how t isn’t quite as high and bold in the middle part of the bell curve.

The uncertainty shows up as extra area in the tails of the bell curve.

As the sample size n gets larger,the degrees of freedom d.f. gets larger,and the uncertainty becomes less uncertain,and the t bell curve gets very much closer to the normal distribution bell curve we use in z problems.

History of t : Q.A. at an Irish brewery circa 1900. See textbook or internet for all the details.

Example 8.9: Finding the Value of tα

Find the value of t0.025 for the t-distribution with 25 degrees of freedom.

Solution

The number of degrees of freedom is listed in the first column of the t-distribution table. Since the

t-distribution in our example has 25 degrees of freedom, the value we need lies on the row corresponding to df = 25.

tα means “what t value has area α to its right?

(And because of symmetry, α is also the area to the left of –tα)

Example 8.9: Finding the Value of tα(cont.)

Note TWO sets of headings!

One Tail &

Two Tails

*

USE THIS ONE

Seeking

t0.025 …

invT(area to the left, degrees of freedom)

invT(0.025,25) gives -2.059538532

then you have to fix up the sign & round

TI-84 only – not available on TI-83:

invT(area to the left, degrees of freedom)

it’s at 2ND DISTR 4:invT(

Example 8.10: Finding the Value of t Given the Area to the Right

Find the value of t for a t-distribution with 17 degrees of freedom such that the area under the curve to the right of t is 0.10.

Example 8.10: Finding the Value of t Given the Area to the Right (cont.)

*

invT(area to the left, degrees of freedom)

invT(0.100,17) gives -1.33337939

then you have to fix up the sign & round

Example 8.11: Finding the Value of t Given the Area to the Left

Find the value of t for a t-distribution with 11 degrees of freedom such that the area under the curve to the left of t is 0.05.

Example 8.11: Finding the Value of t Given the Area to the Left (cont.)

*

invT(area to the left, degrees of freedom)

invT(0.050,11) gives -1.795884781

then round (in this case, it is negative t)

Example 8.11: Finding the Value of t Given the Area to the Left , with Excel

Excel: T.INV(area to the left of t, df), same thing.

Excel special if you know area in two tails total:=T.INV.2T(area in two tails total, df)

Example 8.12: Finding the Value of t Given the Area in Two Tails

Find the value of t for a t-distribution with 7 degrees of freedom such that the area to the left of -t plus the area to the right of t is 0.02, as shown in the picture.

Example 8.12: Finding the Value of t Given the Area in Two Tails (cont.)

If using the table, just go directly to 0.020 in the Two Tails heading.

.

But if using the TI-84 invT(), you must divide 0.020 ÷ 2 = ______ area in one tail

first, and then….

invT(area to the left, degrees of freedom)

invT(______,7) gives -2.997951566

then round and use the positive value

Example 8.12: Finding the Value of t Given the Area in Two Tails (cont.) – with Excel

Recall: we seek t and –t such that two tails total area 0.02, d.f. = 7

Excel with convenient =T.INV.2T(total area, d.f.)

Or Excel with one-tailed version, manually divide area by 2: = T.INV(one tailed area, d.f.)

Example 8.13: Finding the Value of t Given Area between -t and t

Find the critical value of t for a t-distribution with 29 degrees of freedom such that the area between −t and t is 99%.

This is a two-tail problem.

The area in two tails is ____

and if using TI-84 invT,

you need to compute the

area in one tail which is ____

Example 8.13: Finding the Value of t Given Area between -t and t (cont.)

@

invT(area to the left, degrees of freedom)

invT(______,29) gives ______________

then round and use the positive value

Example 8.14: Finding the Critical t-Value for a Confidence Interval

Find the critical t-value for a 95% confidence interval using a t-distribution with 24 degrees of freedom.

Solution

This is a two-tail problem.

The area in two tails is ____

and if using TI-84 invT,

you need to compute the

area in one tail which is ____

Example 8.14: Finding the Critical t-Value for a Confidence Interval (cont.)

#

invT(area to the left, degrees of freedom)

invT(______,____) gives _____________

then round and use the positive value

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