Confidence Intervals Confidence Intervals for the for the Population Mean Population Mean ( ( Known) Known)
Confidence Intervals for theConfidence Intervals for thePopulation Mean Population Mean
(( Known) Known)
Point Estimators
• There are many ways one could estimate – Take a sample of size n and
• Average all but the upper 5% and the lower 5%
• Take the median
• Take the mode
• Average all n observations -- x
Unbiased Estimators/Consistency
• An unbiased estimateunbiased estimate is one where the long run average value equals the quantity you are trying to estimate
• A consistent estimatorconsistent estimator is one that gets closer and closer to the quantity you are trying to estimate as the sample size increases
is an unbiased and consistent estimator for
x
CONFIDENCE INTERVALCONCEPTS
• We do not know the true mean, • To estimate , we take a random sample and
calculate • is a point estimate for
– But the chances that = are slim at best
• So a margin of error can be included with the
estimate to give a Confidence IntervalConfidence Interval estimate for μ
x
xx
Confidence IntervalConfidence IntervalError) of (Margin x
A (1-)x100%Confidence Interval for μ
( known) is the probability that a confidence interval
does notdoes not contain μ
– Thus (1-) is the probability that a confidence interval doesdoes contain μ
• Confidence levels are usually expressed as percentages– Thus a 95% confidence interval would have = .05
If samples of size n were repeatedly
taken, and 95% confidence intervals
according to the correct formula,
in the long run 95% of these intervals
would contain .
Precise Meaning of aConfidence Interval
A 95% confidence interval for A 95% confidence interval for means means
zzpp Notation• zzpp is the z value that puts probability p in the
upper tailupper tail of the standard normal distribution
• Note that -z-zpp is the z value that puts probability p in the lower taillower tail
ZZ
EXCEL -z-zpp = NORMSINV(p)
Probability = pProbability = p
zzpp
EXCEL zzpp = NORMSINV(1-p)
- z- zp p
Probability = pProbability = p
Interval Centered Around • Consider the random variable that has:
mean =mean = and standard deviation =standard deviation =
• Although we don’t know we don’t know , with probability (1-) we will draw a sample of size n, we will get an that lies between
X
n
σ
x
n
σzμ and
n
σzμ /2α/2α
Let’s illustrate this.Let’s illustrate this.
n
σ σ
X
X
n
σzμ and
n
σzμ
Interval The
/2α/2α
/2/2/2/2
n
σzμ α/2
n
σzμ /2α
Interval Width
Constructing (1-) x 100% Intervals
• Now suppose we want to estimate – Calculate – Construct interval
x
n
σzx α/2
As long as we get an in the yellow area (of the previous figure), this interval will contain
This occurs (1-) percent of the time.
x
Let’s illustrate this also.Let’s illustrate this also.
X
n
σ σ
X
time theof )%α-(1 happens This
μ Contains Interval The
Area Yellow TheIn Is xWhen
xn
σzx /2α
n
σzx /2α
Interval Contains
X
n
σ σ
X
time theof )%α-(1 happens This
μ Contains Interval The
Area Yellow TheIn Is xWhen
x
Interval Contains
n
σzx /2α
n
σzx /2α
X
n
σ σ
X
time theof α% happens This
μContain Not Does Interval The
Area Purple TheIn Is xWhen
x
Interval Does Not Contain
n
σzx /2α
n
σzx /2α
(1- )x100% Confidence Intervals
• The (1-)% Confidence Interval is then:
• z/2 = the z value that puts probability /2 in the upper tail
n
σzx
σzx
/2α
x/2α
Important values for z/2
Confidence /2 z/2
90% .10 .05 1.645
95% .05 .025 1.960
99% .01 .005 2.576
LCL (Lower Confidence Limit) =C1-CONFIDENCE(.05,12,22)
UCL (Upper Confidence Limit) =C1+CONFIDENCE(.05,12,22)
CONFIDENCE INTERVALS USING EXCEL
• Excel calculates by the command:
• Generating a 95% confidence interval for μ in Excel based on n = 22 observations assuming: – Data comes from a normal population with = 12– Sample mean is in cell C1
n
σz /2α
=CONFIDENCE(,,n)
=AVERAGE(A2:A23)=AVERAGE(A2:A23)
=C1 – CONFIDENCE(.05,C2,C3)
=C1 + CONFIDENCE(.05,C2,C3)
Example 1
Construct a 95% confidence interval for average battery life if the average of 100 batteries is 408 hours and the known standard deviation is 40 hours.
By EXCELBy EXCEL
LCL: =408 - CONFIDENCE(.05,40,100)
UCL: =408 + CONFIDENCE(.05,40,100)
100
4096.1408
Example 2
• Construct a 90% confidence interval for average time of a flight from LAX to NYC if the average time of 64 flights is 330 minutes and the known standard deviation is 20 minutes.
64
20645.1330
By EXCELBy EXCEL
LCL: =330 - CONFIDENCE(.10,20,64)
UCL: =330 + CONFIDENCE(.10,20,64)
Example 3
• Construct a 99% confidence interval for the time to play football games if the average time it took to play 16 games was 190 minutes with a known standard deviation of 10 minutes, assuming times follow a normal distribution
16
10576.2190
By EXCELBy EXCEL
LCL: =190 - CONFIDENCE(.01,10,16)
UCL: =190 + CONFIDENCE(.01,10,16)
Example 4• Construct an 92% confidence interval for
Example 3 = .08 Thus /2 = .04 and we need z.04
– From table, the z value that puts .0400 in the upper tail (the z value that puts .9600 to its left) is z = 1.75 (approximately).
16
1075.1190
By EXCELBy EXCEL
LCL: =190 - CONFIDENCE(.08,10,16)
UCL: =190 + CONFIDENCE(.08,10,16)
What If σ Is Unknown?
• We have assumed that σ is known.
• If σσ is unknown is unknown– If the sample size is smallsample size is small (normally (normally n < 30n < 30))
• Must assume that we are sampling from a normal distribution
– This will be discussed later.
• If we cannot assume that we are sampling from a normal distribution we cannot construct confidence intervals in this manner.
– If the sample size is largesample size is large (normally n n 30 30)• s can be used as a good approximation for σ for doing
hand calculations of confidence intervals– As shown later, we need not do this when using Excel
Approximate (1-α)x100% Confidence Intervals for μ
When σσ is Unknown is Unknown, But n is Largen is Large
• An approximate confidence interval for μ is:
nzx
zx
/2
x/2
s
s
α
α
Example 5
Construct an approximatte 95% confidence interval for average fill of liter bottles if a sample of 142 bottles is taken and the standard deviation of bottle fills is unknown.
142
55.96.15.33
oz. .55 s oz., 33.5 x sample, thisFor
REVIEW• A (1-)x 100% confidence interval for when is known is:
• A (1-)x100% confidence interval means that if we repeatedly took samples of size n and constructed intervals using the above formula, in the long run (1-)% would contain
• zp is the z -value that puts probability p in the upper tail of the standard normal distribution
• Confidence Intervals Using Excel’s CONFIDENCE function
n
σzx /2α