Section 8.3

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Section 8.3

Estimating Population Means ( Unknown)

And some valuable added stuff by D.R.S., University of Cordele

Should I use z or t for my confidence interval?

• We have two bell-shaped distributions, z and t.• Which one to use?

I need more advanced techniques that are beyond the scope of this course.

Do I have a “big” sample?

Is the population

approximately normally

distributed?

no no

Do I know the value of σ, the population’s

standard deviation?

yes

yes

Use z Use tyes no

The formula for E in a t problem isvery much like the formula for E in a z problem

They have the same arithmetic structure.

Difference: using a t critical value instead of a z critical value.

Difference: using sample standard deviation s instead of population standard deviation σ.

Same: square root of the sample size,

, the margin of error:






Estimating Population Means ( Unknown)

Confidence Interval for a Population Mean The confidence interval for a population mean is given by

Where is the sample mean, which is the point estimate for the population mean, and E is the margin of error.

or

,

x E x E

x E x E

x

The train crossing at Highway 92 was observed one day. Thirteen trains passed and the length of time the roadway was blocked was recorded for each. The mean was 282 sec (4 min 42 sec) and the standard deviation was 100 sec (1 min 40 sec). The distribution of all trains’ lengths is thought to be normal. Construct the 90% confidence interval of the time it takes for a train to pass. Is this a z problem or a t problem? ______

Why is it legitimate to use these formulas?

Because even though ___________, the

population has ___________ ____________.

Make note of variables and values:

_____ = 13_____ = 282_____ = 100

c = _____α = _____α / 2 = _____t α / 2 = _____

To replay the entire example, go to http://2205.drscompany.com,click on Examples, click on Chapter 8,click on Section 8.3,click on “Confidence interval with t”

Compute the Margin of Error:

Compute the Confidence Interval:

or

,

x E x E

x E x E

Verify by doing the problem with the TI-84 Tinterval.






Example 8.16: Constructing a Confidence Interval for a Population Mean ( Unknown)

A marketing company wants to know the mean price of new vehicles sold in an up-and coming area of town. Marketing ‑strategists collected data over the past two years from all of the dealerships in the new area of town. From previous studies about new car sales, they believe that the population distribution looks like the following graph.Valid sample? _______Normal distribution? ________Use z or t? _______Need a big sample, 30 or more? _____






Example 8.16: Constructing a Confidence Interval for a Population Mean ( Unknown) (cont.)

The simple random sample of 756 cars has a mean of $27,400 with a standard deviation of $1300. Construct a 95% confidence interval for the mean price of new cars sold in this area.

Make some notes as you read:

756 = _____ $27400 = _______ $1300 = ______

95% = ______ so _____ = ______ and ____ / 2 = ______Should we use z or use t or do we need a technique that’snot part of this course? __________________________

Example 8-16 – Car Prices, continued

Compute the margin of error, E. You’ll need d.f. = _____.

Determine the confidence interval, :

State a conclusion in plain English: “We are _____% confident that __________________________________is between $__________ and $__________

Example 8-16 – Car Prices, continued

Do the same problem again, but this time use TI-84 STAT, TESTS, 8:TIntervalWhat does the TI-84 givefor the confidence interval?

It does not tell you the margin of error, E, directly.But could you figure out E from the information shown?






Further words about ZInterval and TInterval

• If you’re asked for a confidence interval,• Use ZInterval for a normal distrib. situation.• Use TInterval for a t-distribution situation.

• If the problem asks for a critical value of z or t, too, • Then you have to use invNorm( or invT( or a

printed table to answer that question.• Make the right choice between

• Stats, if you’re given the mean, etc.• Data, if you’re given a list of raw data






Example 8.18: Constructing a Confidence Interval for a Population Mean ( Unknown) from Original Data

Given the following sample data from a study on the average amount of water used per day by members of a household while brushing their teeth, calculate the 99% confidence interval for the population mean using a TI-83/84 Plus calculator. Assume that the sample used in the study was a simple random sample.

Should we use z or t or some other advanced technique? And why?






Example 8.18: Constructing a Confidence Interval for a Population Mean ( Unknown) from Original Data (cont.)

* Household Water Used for Brushing Teeth (in Gallons per Day)

0.485 0.428 0.39 0.308 0.231

0.587 0.516 0.465 0.370 0.282

0.412 0.367 0.336 0.269 0.198

0.942 0.943 0.940 0.941 0.946

0.868 0.898 0.889 0.910 0.927

0.925 0.950 0.959 0.948 0.956

0.805 0.810 0.839 0.860 0.861

0.515 0.463 0.420 0.326 0.243






Example 8.18: Constructing a Confidence Interval for a Population Mean ( Unknown) from Original Data (cont.)

To begin with, since we are given the raw data and not the sample statistics, we need to enter the data in the calculator list, like in L1. Then use TInterval, but this time highlight the Data option, not the Stats option!You’ll see some differences in the prompts.Confidence interval result is( _________, ___________ )Conclusion: _____ % confidentthat __________________________________________

Stuff you need to know about the Practice and Certify problems

• They come in pairs.• First part asks you for “the critical value”.• Second part asks you for the confidence

interval.

• For the critical value, the easiest thing to do is to use your printed tables.

• invT can be used but the tables area easier.

Stuff you need to know about the Practice and Certify problems

• For the Confidence Interval, TInterval with the calculator is the best way. The Tutor probably does it the long way. There are some multipart problems that give you the raw data. Put the data into a TI-84 list.

• Use 1-Var Stats to answer any preliminary questions about mean and standard deviation.

• Use T-Interval with “Data”, not “Stats”, to get the confidence interval.

Documents

Section 8.3