M32 Margin of Error 1 Department of ISM, University of Alabama, 1995-2002 Lesson Objectives Learn...

M32 Margin of Error 1 Department of ISM, University of Alabama, 1995-2002

Lesson Objectives

Learn the meaning of “margin of error,” or “m.o.e.”

Learn how to calculate them.o.e. for two situations: when the true population std. dev., ,is known and when its unknown.

Learn how to use the m.o.e. toconstruct a confidence interval.


“Margin of Error” for estimating

the True Mean of a population.

“Margin of Error” for estimating

the True Mean of a population.

m.o.e. at 95% confidence =“The amount that when added and subtracted to the true population meanwill define a region that will includethe middle 95% of all possible X-bar values.”


= true population mean.

Find the interval around the mean where 95% of all possible sample means will lie.

X

.95

? ? -1.96 0 +1.96 Z

.025.025

Illustration of “Margin of Error”


? - Z = n

+1.96 =? - n

? = + 1.96 n

Margin of ErrorMargin of Error


1.96 n x

standard error of the mean

Margin of Error for 95% confidence:

m.o.e. =


General form for “margin of error”when is known:

m.o.e. = Z n

2

Z 2

appropriate percentile from the standard normal distribution,i.e., the Z table.

where is the


Explanation of symbol:

Z ~ N(0,1)

/ 2

0 Z / 2

Z / 2 cuts off the top tail at area = /2

.5- /2


Amount ofconfidence

Area ineach tail

Table value

.95

.90

.80

.98

.0250

.0500

.1000

.0100

1.96

1.645

1.28

2.33

1 - / 2 Z / 2

Examples


(1-)100% Confidence Interval:

Point estimate m.o.e.

(1-)100% is the amount of confidence desired.

.95 confidence = .05 risk


m.o.e. = Z n

2

= 2.33 10 4 = 5.825 gal.

= Z.01 10

16

Estimate with 98% confidence the mean gallons of water used per shower for Dallas Cowboys after a game if the true standard deviation is known to be 10 gallons. The sample mean for 16 showers is 30.00 gal.

Example 1


30.00 5.825


or24.175 to 35.825 gallons

Example 1, continued . . .

98% confidence interval:


I am 98% confident that the true mean gallons

of water used per shower for Dallas Cowboys after a game will fall within the interval 24.175 to 35.825 gallons.

Example 1, Statement in the L.O.P.

A statement in A statement in L.O.P.L.O.P. must contain four parts: must contain four parts: 1. amount of 1. amount of confidenceconfidence.. 2. the 2. the parameterparameter being estimated in L.O.P. being estimated in L.O.P. 3. the 3. the populationpopulation to which we generalize to which we generalize in L.O.P. in L.O.P. 4. the 4. the calculated intervalcalculated interval. .


• Replace with s.

• replace z with t.

What do we do if the true population standard deviation is unknown?


General form for “margin of error”when is UN-known:

m.o.e. = t sn

2,n–1

sx

“estimated” standard error of the mean

t 2

,n–1where is the

appropriate percentile from the t-distribution.


Explanation of symbol:

t-distribution

/ 2

0 t / 2 , n–1

t / 2, n-1 cuts off the top tail at area = /2

Use the t-table to find the value.

0 t

Table gives right-tail area.(e.g., for a right-tail areaof 0.025 and d.f. = 15,the t value is 2.131.)

0.1 0.05 0.025 0.01 0.005

d.f. = 1 3.078 6.314 12.706 31.821 63.6562 1.886 2.920 4.303 6.965 9.9253 1.638 2.353 3.182 4.541 5.8414 1.533 2.132 2.776 3.747 4.6045 1.476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.7077 1.415 1.895 2.365 2.998 3.4998 1.397 1.860 2.306 2.896 3.3559 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.16911 1.363 1.796 2.201 2.718 3.10612 1.356 1.782 2.179 2.681 3.05513 1.350 1.771 2.160 2.650 3.01214 1.345 1.761 2.145 2.624 2.97715 1.341 1.753 2.131 2.602 2.94716 1.337 1.746 2.120 2.583 2.92117 1.333 1.740 2.110 2.567 2.89818 1.330 1.734 2.101 2.552 2.87819 1.328 1.729 2.093 2.539 2.86120 1.325 1.725 2.086 2.528 2.84521 1.323 1.721 2.080 2.518 2.83122 1.321 1.717 2.074 2.508 2.81923 1.319 1.714 2.069 2.500 2.80724 1.318 1.711 2.064 2.492 2.79725 1.316 1.708 2.060 2.485 2.78726 1.315 1.706 2.056 2.479 2.77927 1.314 1.703 2.052 2.473 2.77128 1.313 1.701 2.048 2.467 2.76329 1.311 1.699 2.045 2.462 2.75630 1.310 1.697 2.042 2.457 2.75031 1.309 1.696 2.040 2.453 2.74432 1.309 1.694 2.037 2.449 2.73833 1.308 1.692 2.035 2.445 2.73334 1.307 1.691 2.032 2.441 2.72835 1.306 1.690 2.030 2.438 2.724

0.1 0.05 0.025 0.01 0.005

36 1.306 1.688 2.028 2.434 2.71937 1.305 1.687 2.026 2.431 2.71538 1.304 1.686 2.024 2.429 2.71239 1.304 1.685 2.023 2.426 2.70840 1.303 1.684 2.021 2.423 2.70441 1.303 1.683 2.020 2.421 2.70142 1.302 1.682 2.018 2.418 2.69843 1.302 1.681 2.017 2.416 2.69544 1.301 1.680 2.015 2.414 2.69245 1.301 1.679 2.014 2.412 2.690

97 1.290 1.661 1.985 2.365 2.62798 1.290 1.661 1.984 2.365 2.62799 1.290 1.660 1.984 2.365 2.626

100 1.290 1.660 1.984 2.364 2.6261.282 1.645 1.960 2.326 2.576

Want 95% CI,n = 20,

/2 = .025d.f. = 19

t.025, 19= 2.0932.093

0 t

0.1 0.05 0.025 0.01 0.005

d.f. = 1 3.078 6.314 12.706 31.821 63.6562 1.886 2.920 4.303 6.965 9.9253 1.638 2.353 3.182 4.541 5.8414 1.533 2.132 2.776 3.747 4.6045 1.476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.7077 1.415 1.895 2.365 2.998 3.4998 1.397 1.860 2.306 2.896 3.3559 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.16911 1.363 1.796 2.201 2.718 3.10612 1.356 1.782 2.179 2.681 3.05513 1.350 1.771 2.160 2.650 3.01214 1.345 1.761 2.145 2.624 2.97715 1.341 1.753 2.131 2.602 2.94716 1.337 1.746 2.120 2.583 2.92117 1.333 1.740 2.110 2.567 2.89818 1.330 1.734 2.101 2.552 2.87819 1.328 1.729 2.093 2.539 2.86120 1.325 1.725 2.086 2.528 2.84521 1.323 1.721 2.080 2.518 2.83122 1.321 1.717 2.074 2.508 2.81923 1.319 1.714 2.069 2.500 2.80724 1.318 1.711 2.064 2.492 2.79725 1.316 1.708 2.060 2.485 2.78726 1.315 1.706 2.056 2.479 2.77927 1.314 1.703 2.052 2.473 2.77128 1.313 1.701 2.048 2.467 2.76329 1.311 1.699 2.045 2.462 2.75630 1.310 1.697 2.042 2.457 2.75031 1.309 1.696 2.040 2.453 2.74432 1.309 1.694 2.037 2.449 2.73833 1.308 1.692 2.035 2.445 2.73334 1.307 1.691 2.032 2.441 2.72835 1.306 1.690 2.030 2.438 2.724

0.1 0.05 0.025 0.01 0.005

36 1.306 1.688 2.028 2.434 2.71937 1.305 1.687 2.026 2.431 2.71538 1.304 1.686 2.024 2.429 2.71239 1.304 1.685 2.023 2.426 2.70840 1.303 1.684 2.021 2.423 2.70441 1.303 1.683 2.020 2.421 2.70142 1.302 1.682 2.018 2.418 2.69843 1.302 1.681 2.017 2.416 2.69544 1.301 1.680 2.015 2.414 2.69245 1.301 1.679 2.014 2.412 2.690

97 1.290 1.661 1.985 2.365 2.62798 1.290 1.661 1.984 2.365 2.62799 1.290 1.660 1.984 2.365 2.626

100 1.290 1.660 1.984 2.364 2.6261.282 1.645 1.960 2.326 2.576

Want 98% CI,n = 33,

/2 = .01d.f. = 32

t.01, 32= 2.4492.449


0 t

0.1 0.05 0.025 0.01 0.005

d.f. = 1 3.078 6.314 12.706 31.821 63.6562 1.886 2.920 4.303 6.965 9.9253 1.638 2.353 3.182 4.541 5.8414 1.533 2.132 2.776 3.747 4.6045 1.476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.7077 1.415 1.895 2.365 2.998 3.4998 1.397 1.860 2.306 2.896 3.3559 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.16911 1.363 1.796 2.201 2.718 3.10612 1.356 1.782 2.179 2.681 3.05513 1.350 1.771 2.160 2.650 3.01214 1.345 1.761 2.145 2.624 2.97715 1.341 1.753 2.131 2.602 2.94716 1.337 1.746 2.120 2.583 2.92117 1.333 1.740 2.110 2.567 2.89818 1.330 1.734 2.101 2.552 2.87819 1.328 1.729 2.093 2.539 2.86120 1.325 1.725 2.086 2.528 2.84521 1.323 1.721 2.080 2.518 2.83122 1.321 1.717 2.074 2.508 2.81923 1.319 1.714 2.069 2.500 2.80724 1.318 1.711 2.064 2.492 2.79725 1.316 1.708 2.060 2.485 2.78726 1.315 1.706 2.056 2.479 2.77927 1.314 1.703 2.052 2.473 2.77128 1.313 1.701 2.048 2.467 2.76329 1.311 1.699 2.045 2.462 2.75630 1.310 1.697 2.042 2.457 2.75031 1.309 1.696 2.040 2.453 2.74432 1.309 1.694 2.037 2.449 2.73833 1.308 1.692 2.035 2.445 2.73334 1.307 1.691 2.032 2.441 2.72835 1.306 1.690 2.030 2.438 2.724

0.1 0.05 0.025 0.01 0.005

36 1.306 1.688 2.028 2.434 2.71937 1.305 1.687 2.026 2.431 2.71538 1.304 1.686 2.024 2.429 2.71239 1.304 1.685 2.023 2.426 2.70840 1.303 1.684 2.021 2.423 2.70441 1.303 1.683 2.020 2.421 2.70142 1.302 1.682 2.018 2.418 2.69843 1.302 1.681 2.017 2.416 2.69544 1.301 1.680 2.015 2.414 2.69245 1.301 1.679 2.014 2.412 2.690

97 1.290 1.661 1.985 2.365 2.62798 1.290 1.661 1.984 2.365 2.62799 1.290 1.660 1.984 2.365 2.626

100 1.290 1.660 1.984 2.364 2.6261.282 1.645 1.960 2.326 2.576

Want 90% CI,n = 600,

/2 = .05d.f. = 599

t.05, = 1.6451.645Same as Normal



t-dist.Z

Bell shaped, symmetric

Mean

Std. dev.

As “n–1” increases, “tn–1” approaches “Z”

Degrees of freedom

= 0

> 1

Yes Yes

= 0

= 1

“n-1”“”

Comparison of “Z” and “t”


= 2.602 10.4 4 = 6.765 gal.

= t.01, 15 10.4

16

Estimate with 98% confidence the mean gallons of water used per shower at the UA Student Rec Center. For a sample of 16 showers the mean was 30.00 gallons and the standard deviation was 10.4 gallons.

Example 2

m.o.e. sn

t 2

, n-1


30.00 6.765


or23.235 to 36.765 gallons

Example 2, continued . . .

98% confidence interval:


I am 98% confident that the true mean gallons

of water used per shower for at the UA Student Rec Center will fall within the interval 23.235 to 36.765 gallons.

Example 2, Statement in the L.O.P.

A statement in A statement in L.O.P.L.O.P. must contain four parts: must contain four parts: 1. amount of 1. amount of confidenceconfidence.. 2. the 2. the parameterparameter being estimated in L.O.P. being estimated in L.O.P. 3. the 3. the populationpopulation to which we generalize to which we generalize in L.O.P. in L.O.P. 4. the 4. the calculated intervalcalculated interval. .


To get a smaller Margin of Error:

Increase “n”.

Decrease the amount of confidence desired.

= 1.96 n

Margin of Error for 95% confidence:


Confidence vs. Probability

BEFORE a sample is collected, there is a 95% probability that the future to be computed sample mean, will fall within m.o.e. units of .

AFTER the sample is collected, the computed sample mean either fell within m.o.e. units of or it did not. After the event, it does not make sense to talk about probability.

Analogy: Suppose you own 95 tickets in a 100-ticket lottery. The drawing was held one hour ago, but you don’t know the result.

P(win) = 0 or 1, but you are very CONFIDENT that you have won the lottery.


What sample size is needed to estimate the mean mpg of Toyota Camrys with an m.o.e. of 0.2 mpg at 90%confidence if the pop. std. dev. is 0.88 mpg?

m.o.e. = Z n

2

0.2 = 1.645 0.88n

n =1.6452 0.882

0.22 = 52.39

Round

UP,use 53

Round

UP,use 53


Interpretations of theInterpretations of theConfidence Interval for Confidence Interval for

• If we took many, many samples of size n, and calculated a confidence interval for each, then I would expect that 95% of all these many intervals would contain the true mean, and 5% would not.


Example 3: A car rental agency wants to estimate the average mileage driven by its customers. A sample of 225 customer receipts, selected at random, yields an average of 325 miles. Assuming that the population standard deviation is 120 miles, construct a 95% confidence interval for the population average mileage.

325 1.96 * 120 / 15325 15.680

309.32, 340.68 miles


Example 4: An insurance company collected data to estimate the mean value of personal property held by apartment renters in Tuscaloosa. In a random sample of 45 renters, the average value of personal property was $14,280 and the standard deviation was $6,540.

Example 4:


a. At 95% confidence, find the margin of error for estimating the true population mean?

Example 4:

t(.025,44) = 2.015m.o.e. = 1964.833


b. Construct a 95% confidence interval for the true mean value of personal property owned by renters in Tuscaloosa.

Example 4:

( 12,315, 16,245)


c. Three years ago the true mean value of personal property owned by renters in Tuscaloosa was $13,050.

Based on your confidence interval,is there evidence that the true mean has changed?

Example 4:


m.o.e. =

= 2.602 10 4

= 6.505

t sn

2 ,n-1

Find the “margin of error” at 98% confidence for estimating the mean gallons of water used to wash cars using 16 observations from a normal population if the sample standard deviation is calculated to be 10.0.


Definition

A (1– ) 100% confidence interval for

X + t / 2, n–1 sX

X + m.o.e.or

when is unknown and X ~ Normal, i.e., either original pop. ~ Normal or by CLT if n > 30,

is given bym.o.e.

M32 Margin of Error 1 Department of ISM, University of Alabama, 1995-2002 Lesson Objectives Learn...

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