Section 8.2

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Section 8.2

Student’s t-DistributionWith 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

Comparison of the Normal and Student t-Distributions:

Continuous Random Variables

6.5 Finding t-Values Using the Student

t-Distribution

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.

Observations about the t distribution

• In the middle, at the mean, the t distribution peaks { lower or higher } than the normal distribution..

• In the tails, the t distribution is { lower or higher } than the normal distribution.

• The differences between the t and the normal distributions are because there is more un__________ty built into the t distribution.

• As the sample size n gets larger,the degrees of freedom d.f. gets larger,the t distribution gets { closer to, farther from } the normal distribution bell curve we use in z problems.

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:1. This lesson – learn about t and areas and critical

values, much like we have done with z.2. Next lesson – doing confidence intervals with t.

The t and the z tables – the same but different

z, StandardNormal Distribution

.0#

#.# .####

Lookup zin row label and column header Read inward to find the area.

The t and the z tables – the same but different


ThetDistribution

.0#

#.# .####

1 tail 0.100 0.050 0.025 0.010

df 2 tails 0.200 0.100 0.050 0.020

n – 1 #.###

Lookup zin row label and column header Read inward to find the area.

Read inward to find the t value.

Choose which header row and column for Area:

Choose the df row = sample sizen, minus 1.

area in 1 tail or in 2 tails

The t and the z calculator functions


ThetDistribution

If I know AREA and want to work backward to find z1) What z has area = .0500 to its left?2) What z has area .9500 to its left?

If I know AREA and want to work backward to find tFind the t and –t values such that the area in two tails is .1000 for df = 20. Area in one tail is ________; Answer: t = _____ (positive)

If I know two z values and I want the area between themWhat is the area between z = -1.645 and z 1.645 ?

If I know two t values and I want the area between them (Less common, use tcdf(low t, high t, df) if it comes up.)






Example 8.9: Finding the Value of tα

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

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)

For finding critical values of t, using the table is the quickest and easiest way to find the needed value.

Even though the TI-84 has an invT function, we’ll emphasize the table method for t problems.Also - invT is not available with the TI-83 family.

Answer:t = ____________






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

*

df

Area in One Tail 0.100 0.050 0.025 0.010 0.005

Area in Two Tails 0.200 0.100 0.050 0.020 0.010

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

27 1.314 1.703 2.052 2.473 2.771

28 1.313 1.701 2.048 2.467 2.763

Seekingt0.025 …

Note TWO sets of headings!One Tail & Two Tails

USE THIS ONE

TI-84 only – not available on TI-83:invT(area to the left, degrees of freedom) it’s at 2ND DISTR 4:invT(

invT(area to the left, degrees of freedom)invT(0.025,25) gives -2.059538532then you have to fix up the sign & round






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.

Answer:

t = ____________






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

*

df

Area in One Tail 0.100 0.050 0.025 0.010 0.005

Area in Two Tails 0.200 0.100 0.50 0.020 0.010

15 1.341 1.753 2.131 2.602 2.94716 1.337 1.746 2.12 2.583 2.92117 1.333 1.740 2.110 2.567 2.89818 1.330 1.734 2.101 2.552 2.87819 1.328 1.729 2.093 2.539 2.86120 1.325 1.725 2.086 2.528 2.845

invT(area to the left, degrees of freedom)invT(0.100,17) gives -1.33337939then 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.

Answer:(careful of sign!!!)t = ____________






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

*

df

Area in One Tail 0.100 0.050 0.025 0.010 0.005

Area in Two Tails 0.200 0.100 0.050 0.020 0.010

9 1.383 1.833 2.262 2.821 3.2510 1.372 1.812 2.228 2.764 3.16911 1.363 1.796 2.201 2.718 3.10612 1.356 1.782 2.179 2.681 3.05513 1.350 1.771 2.160 2.65 3.01214 1.345 1.761 2.145 2.624 2.977

invT(area to the left, degrees of freedom)invT(0.050,11) gives -1.795884781then round (in this case, it is negative t)






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.

Answer:

t = ____________






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

.

df

Area in One Tail 0.100 0.050 0.025 0.010 0.005

Area in Two Tails 0.200 0.100 0.050 0.020 0.010

5 1.476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.7077 1.415 1.895 2.365 2.998 3.4998 1.397 1.860 2.306 2.896 3.3559 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

invT(area to the left, degrees of freedom)invT(______,7) gives -2.997951566then round and use the positive value

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 tailfirst, and then….






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 the table has a two tails heading.

If you’re using TI-84 invT instead of the table, you’ll need to divide by 2 and work with the area in one tail.

Area in middle = ______Area in tails = ______Answer:t = ____________






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

@

df

Area in One Tail 0.100 0.050 0.025 0.010 0.005

Area in Two Tails 0.200 0.100 0.050 0.020 0.010

27 1.314 1.703 2.052 2.473 2.77128 1.313 1.701 2.048 2.467 2.76329 1.311 1.699 2.045 2.462 2.75630 1.310 1.697 2.042 2.457 2.75031 1.309 1.696 2.040 2.453 2.74432 1.309 1.694 2.037 2.449 2.738

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 ____

Area in middle = ______Area in tails = ______Answer:t = ____________






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

#

df

Area in One Tail 0.100 0.050 0.025 0.010 0.005

Area in Two Tails 0.200 0.100 0.050 0.020 0.010

21 1.323 1.721 2.080 2.518 2.83122 1.321 1.717 2.074 2.508 2.81923 1.319 1.714 2.069 2.500 2.80724 1.318 1.711 2.064 2.492 2.79725 1.316 1.708 2.060 2.485 2.78726 1.315 1.706 2.056 2.479 2.779

invT(area to the left, degrees of freedom)invT(______,____) gives _____________then round and use the positive value

8.2 Practice and Certify Problems

• They come in pairs.• You are given some area. You find the t value.• The first part is easy – multiple choice from four

pictures: easy – left tail or right tail or two tails or between two tails.

• The second part asks for the “critical value of t”, what t value separates out the prescribed area.

• But the problem presentation is awkward and you need to be prepared for it.


• The second part asks for the “critical value of t”, what t value separates out the prescribed area.

• The way it’s set up, you have to click in the table. It doesn’t let you type in the box.

• There’s only one heading and it isn’t labeled. You just have to know that it’s for area in one tail, so if you have a 95% level of confidence, that means there is area ________ in two tails, so there is area ________ in one tail,and that’s the heading you need for these problems.


• The table the problem only has one header row and the row isn’t labeled. It’s an “Area In One Tail”.

Area in One Tail

scro

ll ba

r to

get

the

prop

er d

f


• When you click in the table, the t value is copied to the answer box.

• If you need to change it to a negative t value, click the +/- button.

Also – “scroll bar” isbarra di scorrimento

in Italian.

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 8.11: 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)

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Section 8.2

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