Confidence IntervalsWeek 10

Chapter 6.1, 6.2

What is this unit all about?• Have you ever estimated something and

tossed in a “give or take a few” after it?

• Maybe you told a person a range in which you believe a certain value fell into.

• Have you ever see a survey or poll done, and at the end it says: +/- 5 points.

• These are all examples of where we are going in this section.

Chapter 6.1 - disclaimer• To make this unit as painless as possible, I will

show the formula but will teach this unit with the use of the TI – 83 graphing calculator whenever possible.

• It is not always possible to use the TI-83 for every problem.

• You can also follow along in Chp. 6.1 in the TEXT and use their examples in the book to learn how to do them by hand.

What is a Confidence Interval?• If I were to do a study or a survey, but

could not survey the entire population, I would do it by sampling.

• The larger the sample, the closer the results will be to the actual population.

• A confidence interval is a point of estimate (mean of my sample) “plus or minus” the margin of error.

What will we need to do these?

• Point of estimate – mean of the random sample used to do the study.

• Confidence Level – percentage of accuracy we need to have to do our study.

• Critical two-tailed Z value - (z-score) using table IV.

• Margin of Error – a formula used involving the Z value and the sample size.

Formula for Confidence Intervals* This formula is to be used when the Mean and

Standard Deviation are known:

nzx

nzx

22

meanpopulation

samplesizen

samplemeanx

inoferrormn

z arg2

Finding a Critical Z-value(Ex 1) – Find the critical two-tailed z value for a

90% confidence level:

* This means there is 5% on each tail of the curve, the area under the curve in the middle is 90%. Do Z (1-.05) = Z .9500

*We will be finding the z score to the left of .9500 in table IV. It lands in-between .9495-.9505, thus it is = +/- 1.645

(this is the 5% on each end)

(Ex 2) – Find the critical two-tailed z value for a 95% confidence level:

* This means there is 2.5% on each tail of the curve, the area under the curve in the middle is 95%. Do Z (1-.025) = Z .9750

*We will be finding the z score to the left of .9750 in table IV.

* It is = +/- 1.96

(this is the 2.5% on each end)

Finding a Critical Z-value

(Ex 3) – Find the critical two-tailed z value for a 99% confidence level:

* This means there is .005% on each tail of the curve, the area under the curve in the middle is 99%. Do Z (1-.005) = Z .9950

*We will be finding the z score to the left of .9950 in table IV.

* It is = +/- 2.575

(this is the .005% on each end)

Finding a Critical Z-value

Finding a Critical Z-value(Ex 4) – Find the critical two-tailed z value

for a 85% confidence level:

Margin of Error

• The confidence interval is the sample mean, plus or minus the margin of error.

nZE

2

Find the MoE:

Ex (5) – After performing a survey from a sample of 50 mall customers, the results had a standard deviation of 12. Find the MoE for a 95% confidence level.

Special features of Confidence IntervalsAs the level of confidence (%) goes up, the

margin of error also goes up!As you increase the sample size, the margin of

error goes down.To reduce the margin of error, reduce the

confidence level and/or increase the sample size. If you were able to include the ENTIRE

population, the would not be a margin of error.The magic number is 30 samples to be

considered an adequate sample size.

Finding Confidence Intervals:(Ex 6) – After sampling 30 Statistics students

at NCCC, Bob found a point estimate of an 81% on Test # 3, with a standard deviation of 8.2. He wishes to construct a 90% confidence interval for this data.

How did we get that?

nzx

nzx

22

30

2.8645.181

30

2.8645.181

5.835.78

Using TI-83 to do this:• Click STAT

• go over to TESTS

• Click ZInterval

• Using the stats feature, input S.D., Mean, sample size, and confidence level.

• arrow down, and click enter on calculate.

Finding Confidence Intervals:(Ex 7) – After sampling 100 cars on the I-90,

Joe found a point estimate speed 61 mph and a standard deviation of 7.2 mph. He wishes to construct a 99% confidence interval for this data.

Finding an appropriate sample size• This will be used to achieve a specific

confidence level for your study.2

2

*

E

z

n

Find a sample size:(Ex 8) – Bob wants to get a more accurate idea of

the average on Stats Test # 3 of all NCCC stats class students . How large of a sample will he need to be within 2 percentage points (margin of error), at a 95% confidence level, assuming we know the σ = 9.4?

How did we get this?

2

2

*

E

z

n

2

2

4.9*96.1

n

8586.84 n

Finish Bob’s Study:Ex (9) - Now lets say Bob wants to perform

his study, finds the point of estimate for Test # 3 = 83, with a SD of 9.4 and confidence level of 95%. Find the confidence interval for this study.

nzx

nzx

22

What about an interval found with a small sample size? (chp 6.2)

To do these problems we will need: TABLE 5: t-Distribution.Determine from the problem: n, x, s.

Sample, mean, sample standard deviation.

Use the MoE formula for small samples:

n

stE c

ct t-value from Table 5

d.f. = n-1 (degrees of freedom)

Small Sample Confidence Int.(Ex 10) – Trying to determine the class

average for Test # 3, Janet asks 5 students their grade on the test. She found a mean of 78% with a σ = 7.6. Construct a confidence interval for her data at a 90% confidence level.

What did we do?

n

stx

n

stx cc

4.876.68

d.f. = 5-1 = 4; .90 Lc = 2.132

5

6.7132.278

5

6.7132.278

Or with TI-83/84

STAT

TESTS

8:TInterval

Stats

Input each value, hit calculate.