Student’s Student’s tt DistributionDistribution

Lecture 35Lecture 35

Section 10.2Section 10.2

Mon, Mar 28, 2005Mon, Mar 28, 2005

IntroductionIntroduction

I told you I was losing my voice!I told you I was losing my voice!

IntroductionIntroduction

I told you I was losing my voice!I told you I was losing my voice! Get ready to help me out.Get ready to help me out.

Remember This?Remember This?

Is known?yes no

Is the population normal?

yes no

n

XZ

/

Is n 30?

yes no

n

XZ

/

ns

Xnt

/)1(

Give

up

Is the population normal?

yes no

Is n 30?

yes no

ns

XZ

/

Is n 30?

yes no

Zntns

Xnt

1/

)1( Give

up

Remember This?Remember This?

Is known?yes no

Is the population normal?

yes no

n

XZ

/

Is n 30?

yes no

n

XZ

/

ns

Xnt

/)1(

Give

up

Is the population normal?

yes no

Is n 30?

yes no

ns

XZ

/

Is n 30?

yes no

Zntns

Xnt

1/

)1( Give

up

The only case where thet distribution is needed

Table IV – Table IV – tt Percentiles Percentiles

Table IV (p. 945) gives certain percentiles Table IV (p. 945) gives certain percentiles of of tt for certain degrees of freedom. for certain degrees of freedom.

Open your book to page 945.Open your book to page 945. The columns give specific areas for The columns give specific areas for

upper-tails:upper-tails: 0.40, 0.30, 0.20, 0.10, 0.05, 0.025, 0.01, 0.40, 0.30, 0.20, 0.10, 0.05, 0.025, 0.01,

0.005.0.005. The rows give specific degrees of The rows give specific degrees of

freedom:freedom: 1, 2, 3, …, 30, 40, 60, 120.1, 2, 3, …, 30, 40, 60, 120.

Table IV – Table IV – tt Percentiles Percentiles

Look in the table at row 10, column Look in the table at row 10, column 0.05:0.05: The entry 1.812 meansThe entry 1.812 means

PP((tt > 1.812) = 0.05, > 1.812) = 0.05,

when when dfdf = 10. = 10. Draw a picture of the Draw a picture of the tt distribution distribution

and this upper tail.and this upper tail.

Table IV – Table IV – tt Percentiles Percentiles

Since the Since the tt distribution is symmetric, distribution is symmetric, we can also use the table for lower we can also use the table for lower tails by making the tails by making the tt values values negative.negative.

So, what is So, what is PP((tt < –1.812), when < –1.812), when dfdf = = 10?10?

Draw the picture.Draw the picture.

Table IV – Table IV – tt Percentiles Percentiles The table allows us to look up certain The table allows us to look up certain

percentiles, but it will percentiles, but it will notnot allow us to find allow us to find probabilities (in general).probabilities (in general).

For example, what is For example, what is PP((tt > 1.5) when > 1.5) when dfdf = = 18?18?

We can’t tell exactly.We can’t tell exactly. Look at row 18. You see 1.330 and 1.734, Look at row 18. You see 1.330 and 1.734,

from columns 0.10 and 0.05, respectively.from columns 0.10 and 0.05, respectively. Therefore, the best we can say isTherefore, the best we can say is

0.05 < 0.05 < PP((tt > 1.5) < 0.10. > 1.5) < 0.10.

tt-Table Practice-Table Practice

Find the following values of Find the following values of tt : : 9090thth percentile when percentile when dfdf = 25. = 25. 1010thth percentile when percentile when dfdf = 12. = 12. 2020thth percentile when percentile when nn = 5. = 5.

Find the following probabilities:Find the following probabilities: PP((tt > 2.518) when > 2.518) when dfdf = 21. = 21. PP((tt < 0.851) when < 0.851) when nn = 41. = 41. PP((tt > 2) when > 2) when dfdf = 8 (approx.). = 8 (approx.).

tt-Table Practice-Table Practice

Find the following values of Find the following values of tt : : 9090thth percentile when percentile when dfdf = 25. = 25. Answer: Answer:

1.316.1.316. 1010thth percentile when percentile when dfdf = 12. = 12. 2020thth percentile when percentile when nn = 5. = 5.

Find the following probabilities:Find the following probabilities: PP((tt > 2.518) when > 2.518) when dfdf = 21. = 21. PP((tt < 0.851) when < 0.851) when nn = 41. = 41. PP((tt > 2) when > 2) when dfdf = 8 (approx.). = 8 (approx.).

tt-Table Practice-Table Practice

Find the following values of Find the following values of tt : : 9090thth percentile when percentile when dfdf = 25. = 25. Answer: Answer:

1.316.1.316. 1010thth percentile when percentile when dfdf = 12. = 12. Answer: -Answer: -

1.356.1.356. 2020thth percentile when percentile when nn = 5. = 5.

Find the following probabilities:Find the following probabilities: PP((tt > 2.518) when > 2.518) when dfdf = 21. = 21. PP((tt < 0.851) when < 0.851) when nn = 41. = 41. PP((tt > 2) when > 2) when dfdf = 8 (approx.). = 8 (approx.).

tt-Table Practice-Table Practice

Find the following values of Find the following values of tt : : 9090thth percentile when percentile when dfdf = 25. = 25. Answer: 1.316.Answer: 1.316. 1010thth percentile when percentile when dfdf = 12. = 12. Answer: -1.356.Answer: -1.356. 2020thth percentile when percentile when nn = 5. = 5. Answer: -0.941. (Answer: -0.941. (dfdf

= 4)= 4) Find the following probabilities:Find the following probabilities:

PP((tt > 2.518) when > 2.518) when dfdf = 21. = 21. PP((tt < 0.851) when < 0.851) when nn = 41. = 41. PP((tt > 2) when > 2) when dfdf = 8 (approx.). = 8 (approx.).

tt-Table Practice-Table Practice

Find the following values of Find the following values of tt : : 9090thth percentile when percentile when dfdf = 25. = 25. Answer: 1.316.Answer: 1.316. 1010thth percentile when percentile when dfdf = 12. = 12. Answer: -1.356.Answer: -1.356. 2020thth percentile when percentile when nn = 5. = 5. Answer: -0.941. Answer: -0.941.

((dfdf = 4) = 4) Find the following probabilities:Find the following probabilities:

PP((tt > 2.518) when > 2.518) when dfdf = 21. = 21. Answer: 0.01.Answer: 0.01. PP((tt < 0.851) when < 0.851) when nn = 41. = 41. PP((tt > 2) when > 2) when dfdf = 8 (approx.). = 8 (approx.).

tt-Table Practice-Table Practice

Find the following values of Find the following values of tt : : 9090thth percentile when percentile when dfdf = 25. = 25. Answer: 1.316.Answer: 1.316. 1010thth percentile when percentile when dfdf = 12. = 12. Answer: -1.356.Answer: -1.356. 2020thth percentile when percentile when nn = 5. = 5. Answer: -0.941. Answer: -0.941.

((dfdf = 4) = 4) Find the following probabilities:Find the following probabilities:

PP((tt > 2.518) when > 2.518) when dfdf = 21. = 21. Answer: 0.01.Answer: 0.01. PP((tt < 0.851) when < 0.851) when nn = 41. = 41. Answer: 0.80. (Answer: 0.80. (dfdf = =

40)40) PP((tt > 2) when > 2) when dfdf = 8 (approx.). = 8 (approx.).

tt-Table Practice-Table Practice

Find the following values of Find the following values of tt : : 9090thth percentile when percentile when dfdf = 25. = 25. Answer: 1.316.Answer: 1.316. 1010thth percentile when percentile when dfdf = 12. = 12. Answer: -1.356.Answer: -1.356. 2020thth percentile when percentile when nn = 5. = 5. Answer: -0.941. (Answer: -0.941. (dfdf

= 4)= 4) Find the following probabilities:Find the following probabilities:

PP((tt > 2.518) when > 2.518) when dfdf = 21. = 21. Answer: 0.01.Answer: 0.01. PP((tt < 0.851) when < 0.851) when nn = 41. = 41. Answer: 0.80. (Answer: 0.80. (dfdf = =

40)40) PP((tt > 2) when > 2) when dfdf = 8 (approx.). = 8 (approx.). A:A: 0.025 < 0.025 < PP < <

0.05.0.05.

TI-83 – Student’s TI-83 – Student’s tt DistributionDistribution

However, the TI-83 However, the TI-83 willwill find probabilities find probabilities for the for the tt distribution (but it will distribution (but it will notnot find find percentiles!).percentiles!). Press Press DISTRDISTR.. Select Select tcdftcdf and press and press ENTERENTER.. tcdf( appears in the display.tcdf( appears in the display. Enter the lower endpoint.Enter the lower endpoint. Enter the upper endpoint.Enter the upper endpoint. Enter the number of degrees of freedom (Enter the number of degrees of freedom (nn – –

1).1). Press Press ENTERENTER..

ExampleExample

Enter tcdf(1.812, E99, 10).Enter tcdf(1.812, E99, 10).

ExampleExample

Enter tcdf(1.812, E99, 10).Enter tcdf(1.812, E99, 10). ScottScott, what did you get?, what did you get?

ExampleExample

Enter tcdf(1.812, E99, 10).Enter tcdf(1.812, E99, 10). ScottScott, what did you get?, what did you get? BenBen, use the TI-83 to find P(, use the TI-83 to find P(tt > 2) > 2)

when when dfdf = 8. = 8.

ExampleExample

Enter tcdf(1.812, E99, 10).Enter tcdf(1.812, E99, 10). ScottScott, what did you get?, what did you get? BenBen, use the TI-83 to find P(, use the TI-83 to find P(tt > 2) > 2)

when when dfdf = 8. = 8. NathanNathan, is the answer between , is the answer between

0.025 and 0.05 (as we estimated 0.025 and 0.05 (as we estimated earlier)?earlier)?

Hypothesis Testing with Hypothesis Testing with tt

We should use the We should use the tt distribution if distribution if The population is normal (or nearly The population is normal (or nearly

normal), normal), andand is unknown, so we use is unknown, so we use ss in its place, in its place,

andand The sample size is small (The sample size is small (nn < 30). < 30).

Otherwise, we should not use Otherwise, we should not use tt.. Either use Either use ZZ or “give up.” or “give up.” See the decision tree.See the decision tree.

Remember that!

Hypothesis Testing with Hypothesis Testing with tt

The hypothesis testing procedure is the The hypothesis testing procedure is the same except for two steps.same except for two steps.

Step 3: Find the value of the test Step 3: Find the value of the test statistic.statistic. The test statistic is nowThe test statistic is now

Step 4: Find the Step 4: Find the pp-value.-value. We must look it up (estimate) in the We must look it up (estimate) in the tt table, table,

or use tcdf on the TI-83.or use tcdf on the TI-83.

ns

xt

/0

ExampleExample

Re-do Example 10.1 (by hand) under Re-do Example 10.1 (by hand) under the assumption that the assumption that is unknown. is unknown.

ExampleExample

Re-do Example 10.1 (by hand) under Re-do Example 10.1 (by hand) under the assumption that the assumption that is unknown. is unknown.

Mike D.Mike D., what is step 1?, what is step 1?

ExampleExample

Re-do Example 10.1 (by hand) under Re-do Example 10.1 (by hand) under the assumption that the assumption that is unknown. is unknown.

Mike D.Mike D., what is step 1?, what is step 1? WinchesterWinchester, what is step 2?, what is step 2?

ExampleExample

Re-do Example 10.1 (by hand) under Re-do Example 10.1 (by hand) under the assumption that the assumption that is unknown. is unknown.

Mike D.Mike D., what is step 1?, what is step 1? WinchesterWinchester, what is step 2?, what is step 2? EverybodyEverybody, do step 3., do step 3.

ExampleExample

Re-do Example 10.1 (by hand) under Re-do Example 10.1 (by hand) under the assumption that the assumption that is unknown. is unknown.

Mike D.Mike D., what is step 1?, what is step 1? WinchesterWinchester, what is step 2?, what is step 2? EverybodyEverybody, do step 3., do step 3. PatrickPatrick, what did you get?, what did you get?

ExampleExample

Re-do Example 10.1 (by hand) under Re-do Example 10.1 (by hand) under the assumption that the assumption that is unknown. is unknown.

Mike D.Mike D., what is step 1?, what is step 1? WinchesterWinchester, what is step 2?, what is step 2? EverybodyEverybody, do step 3., do step 3. PatrickPatrick, what did you get?, what did you get? EverybodyEverybody, do step 4., do step 4.

ExampleExample

Re-do Example 10.1 (by hand) under Re-do Example 10.1 (by hand) under the assumption that the assumption that is unknown. is unknown.

Mike D.Mike D., what is step 1?, what is step 1? WinchesterWinchester, what is step 2?, what is step 2? EverybodyEverybody, do step 3., do step 3. PatrickPatrick, what did you get?, what did you get? EverybodyEverybody, do step 4., do step 4. NickNick, using the TI-83, what did you , using the TI-83, what did you

get?get?

ExampleExample

GlenGlen, using the tables, what did you , using the tables, what did you get?get?

ExampleExample

GlenGlen, using the tables, what did you , using the tables, what did you get?get?

MladenMladen, what is the decision?, what is the decision?

ExampleExample

GlenGlen, using the tables, what did you , using the tables, what did you get?get?

MladenMladen, what is the decision?, what is the decision? MattMatt, what is the conclusion?, what is the conclusion?

TI-83 – Hypothesis TI-83 – Hypothesis Testing When Testing When is is

UnknownUnknown The TI-83 will perform a The TI-83 will perform a tt-test.-test. EverybodyEverybody, perform the following steps , perform the following steps

as we work Example 10.1 on the TI-83.as we work Example 10.1 on the TI-83. Press Press STATSTAT.. Select Select TESTSTESTS.. Select Select T-TestT-Test.. A window appears requesting information.A window appears requesting information. Choose Choose StatsStats..

TI-83 – Hypothesis TI-83 – Hypothesis Testing When Testing When is is

UnknownUnknown Enter Enter 00.. EnterEnterxx.. Enter Enter ss. (Remember, . (Remember, is unknown.) is unknown.) Enter Enter nn.. Select the correct alternative Select the correct alternative

hypothesis and press hypothesis and press ENTERENTER.. Select Calculate and press Select Calculate and press ENTERENTER..

TI-83 – Hypothesis TI-83 – Hypothesis Testing When Testing When is is

UnknownUnknown A window appears with the following A window appears with the following information.information. The title “T-Test”The title “T-Test” The alternative hypothesis.The alternative hypothesis. The value of the test statistic The value of the test statistic tt.. The The pp-value.-value. The sample mean.The sample mean. The sample standard deviation.The sample standard deviation. The sample size.The sample size.

TI-83 – Hypothesis TI-83 – Hypothesis Testing When Testing When is is

UnknownUnknown JoshJosh, what does the TI-83 report as , what does the TI-83 report as

the value of the value of t t ?? Michael K.Michael K., what does the TI-83 , what does the TI-83

report as the report as the pp-value?-value?

Let’s Do It!Let’s Do It!

Let’s do it! 10.3, p. 582 – Study Let’s do it! 10.3, p. 582 – Study Time.Time. Check:Check:

Is Is known? known? Is the population normal?Is the population normal? What about the sample size?What about the sample size?

Let’s do it! 10.4, p. 583 – pH Levels.Let’s do it! 10.4, p. 583 – pH Levels.