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.