Learning Outcomes

Calculate and interpret trust intervals because that estimating a population mean and also a populace proportion

A confidence interval for a populace mean v a recognized standard deviation is based upon the truth that the sample method follow an approximately normal distribution. Expect that ours sample has actually a mean of displaystyleoverlinex=10 and we have created the 90% trust interval (5, 15) wherein EBM = 5.

You are watching: To ensure that excel interprets year values the way you intend, use ____.

Calculating the to trust Interval

To build a confidence interval for a single unknown populace mean μ, where the populace standard deviation is known, we require overlinex is the point estimate of the unknown populace mean μ.

The to trust interval estimate will have the form:

(point calculation – error bound, point estimate + error bound) or, in symbols,displaystyle(overlinex-EBM,overlinex+EBM)

The margin that error (EBM) counts on the confidence level (abbreviated CL). The trust level is often considered the probability that the calculate confidence interval estimate will contain the true populace parameter. However, it is much more accurate come state that the trust level is the percent of trust intervals that contain the true population parameter as soon as repeated samples room taken. Many often, that is the choice of the person creating the confidence interval to choose a confidence level that 90% or greater because that person wants to be reasonably details of his or her conclusions.

There is one more probability dubbed alpha (α). α is related to the confidence level, CL. α is the probability that the expression does no contain the unknown populace parameter.

Mathematically, α + CL = 1.


Example

Suppose us have accumulated data from a sample. We understand the sample mean however we carry out not recognize the average for the whole population.

The sample mean is seven, and the error bound because that the mean is 2.5.

displaystyleoverlinex=7quad extandquadEBM=2.5

The to trust interval is (7 – 2.5, 7 + 2.5), and also calculating the values provides (4.5, 9.5).

If the to trust level (CL) is 95%, then we say that, “We estimate with 95% confidence the the true value of the population mean is between 4.5 and also 9.5.”


try it

Suppose we have actually data from a sample. The sample average is 15, and also the error bound because that the median is 3.2.

What is the to trust interval calculation for the populace mean?

(11.8, 18.2)


A trust interval because that a populace mean v a well-known standard deviation is based on the reality that the sample means follow an roughly normal distribution. Intend that ours sample has a median of displaystyleoverlinex=10, and we have created the 90% to trust interval (5, 15) whereby EBM = 5.

To obtain a 90% confidence interval, us must encompass the central 90% of the probability of the regular distribution. If we incorporate the main 90%, us leave the end a full of α = 10% in both tails, or 5% in each tail, of the typical distribution.

*

To catch the central 90%, we have to go the end 1.645 “standard deviations” on either next of the calculated sample mean. The value 1.645 is the z-score from a conventional normal probability circulation that put an area that 0.90 in the center, one area of 0.05 in the much left tail, and an area of 0.05 in the far right tail.

It is crucial that the “standard deviation” used should be proper for the parameter we are estimating, therefore in this ar we must use the standard deviation that applies to sample means, i m sorry isdisplaystylefracsigmasqrtn. The fraction displaystylefracsigmasqrtn, is frequently called the “standard error the the mean” in order come distinguish clearly the conventional deviation because that a mean from the population standard deviation σ.

In summary, as a result of the main limit theorem:

displaystyleoverlineX, that is, displaystyleoverlineXsimNleft(mu_x, fracsigmasqrtn ight)When the population standard deviation σ is known, we usage a normal circulation to calculation the error bound.

Calculating the to trust Interval

To construct a confidence interval calculation for one unknown populace mean, we require data native a arbitrarily sample. The steps to construct and also interpret the confidence interval are:

Calculate the sample average displaystyleoverlinex from the sample data. Remember, in this ar we currently know the population standard deviation σ.Find the z-score that corresponds to the to trust level.Calculate the error bound EBM.Construct the to trust interval.Write a sentence that interprets the estimate in the context of the case in the problem. (Explain what the to trust interval means, in the native of the problem.)

We will an initial examine each step in much more detail, and also then show the procedure with some examples.

Finding the z-score for the proclaimed Confidence Level

When we understand the population standard deviation σ, we use a conventional normal distribution to calculation the error tied EBM and construct the confidence interval. We need to find the worth of z that puts one area same to the trust level (in decimal form) in the center of the typical normal circulation Z ~ N(0, 1).

The trust level, CL, is the area in the middle of the traditional normal distribution. CL = 1 – α, for this reason α is the area that is separation equally in between the two tails. Each of the tails includes an area same to α2.

The z-score that has actually an area to the ideal of α2 is denoted through zα2.

For example, as soon as CL = 0.95, α = 0.05 and also α2 = 0.025; we create zα2=z0.025.

The area to the best of z0.025 is 0.025 and the area to the left the z0.025 is 1 – 0.025 = 0.975.

zα2=z0.025=1.96, making use of a calculator, computer system or a standard normal probability table.

invNorm(0.975, 0, 1) = 1.96

 Note

Remember to usage the area come the LEFT that ; in this thing the last 2 inputs in the invNorm command space 0, 1, due to the fact that you room using a typical normal distribution Z ~ N(0, 1).

Calculating the Error bound (EBM)

The error bound formula because that an unknown population mean μ as soon as the population standard deviation σ is recognized is

EBM = (displaystylez_fracalpha2)(fracsigmasqrtn)

Constructing the trust Interval

The to trust interval estimate has actually the style (

displaystyleoverlinex–EBM,displaystyleoverlinex+EBM).

The graph offers a snapshot of the whole situation.

CL + displaystylefracalpha2+fracalpha2= extCL+alpha=1



Writing the Interpretation

The translate should plainly state the to trust level ( CL), describe what populace parameter is being approximated (here, a population mean), and also state the confidence interval (both endpoints). “We estimate with ___% confidence the the true population mean (include the context of the problem) is between ___ and ___ (include ideal units).”


Example

Suppose scores ~ above exams in statistics room normally spread with one unknown populace mean and a population standard deviation of 3 points. A random sample the 36 scores is taken and gives a sample median (sample average score) that 68. Discover a confidence interval estimate for the population mean exam score (the average score on all exams).

Find a 90% confidence interval for the true (population) median of statistics test scores.

You have the right to use modern technology to calculation the trust interval directly.The an initial solution is presented step-by-step (Solution A).The 2nd solution offers the TI-83, 83+, and also 84+ calculators (Solution B).

Solution A:

To discover the to trust interval, you require the sample mean, and the EBM.

overlinex=68EBM=(z_fracalpha2)(fracsigmasqrtn)

displaystylesigma=3

n = 6

The trust level is 90% ( CL = 0.90)

CL = 0.90 so α = 1 – CL = 1 – 0.90 = 0.10

displaystylefracalpha2=0.05, displaystylez_fracalpha2=z_0.05

The area come the best of z0.05 is 0.05 and the area come the left of z0.05is 1 – 0. displaystylefracz_alpha2=z_0.05=1.645

Using invNorm(0.95, 0, 1) top top the TI-83,83+, and 84+ calculators. This can additionally be discovered using proper commands on various other calculators, using a computer, or utilizing a probability table for the standard normal distribution.

EBM = (1.645)(displaystylefrac3sqrt36)= 0.8225

displaystyleoverlinex– EBM = 68 – 0.8225 = 67.1775

displaystyleoverlinex+EBM = 68 + 0.8225 = 68.8225

The 90% to trust interval is (67.1775, 68.8225).

Solution B:

Press STAT and also arrow end toTESTS.

Arrow under to 7:ZInterval.

Press ENTER.

Arrow to Stats and press ENTER.

Arrow down and enter 3 for σ, 68 fordisplaystyleoverlineX, 36 because that n, and .90 because that C-level.

Arrow down to Calculate and press ENTER.

The to trust interval is (to 3 decimal places)(67.178, 68.822).


Interpretation

We estimate with 90% confidence that the true population mean test score for every statistics students is in between 67.18 and 68.82.

Explanation that 90% to trust Level

Ninety percent of all confidence intervals created in this means contain the true average statistics exam score. Because that example, if we created 100 of this confidence intervals, we would expect 90 of them come contain the true populace mean test score.


try it

Suppose average pizza distribution times room normally spread with one unknown population mean and also a population standard deviation of six minutes. A arbitrarily sample that 28 pizza distribution restaurants is taken and also has a sample mean delivery time of 36 minutes.

Find a 90% trust interval calculation for the populace mean shipment time.

(34.1347, 37.8653)


Example

The details Absorption rate (SAR) for a cell phone actions the amount of radio frequency (RF) energy took in by the user’s body when using the handset. Every cell phone emits RF energy. Various phone models have different SAR measures. To obtain certification from the Federal communications Commission (FCC) for sale in the unified States, the SAR level for a cell phone have to be no an ext than 1.6 watts per kilogram. This table reflects the highest SAR level because that a random selection of mobile models together measured through the FCC.

Phone ModelSARPhone ModelSARPhone ModelSAR
Apple iphone phone 4S1.11LG Ally1.36Pantech Laser0.74
BlackBerry Pearl 81201.48LG AX2751.34Samsung Character0.5
BlackBerry tour 96301.43LG Cosmos1.18Samsung epos 4G Touch0.4
Cricket TXTM81.3LG CU5151.3Samsung M2400.867
HP/Palm Centro1.09LG Trax CU5751.26Samsung Messager III SCH-R7500.68
HTC One V0.455Motorola Q9h1.29Samsung Nexus S0.51
HTC Touch pro 21.41Motorola Razr2 V80.36Samsung SGH-A2271.13
Huawei M835 Ideos0.82Motorola Razr2 V90.52SGH-a107 GoPhone0.3
Kyocera DuraPlus0.78Motorola V195s1.6Sony W350a1.48
Kyocera K127 Marbl1.25Nokia 16801.39T-Mobile Concord1.38

Find a 98% to trust interval because that the true (population) average of the certain Absorption rates (SARs) for cell phones. Assume that the population standard deviation is σ = 0.337.

Solution A:

To discover the trust interval, begin by recognize the suggest estimate: the sample mean.

displaystyleoverlinex = 1.024

Next, discover the EBM. Due to the fact that you are creating a 98% to trust interval, CL = 0.98.



You require to discover z0.01 having the residential or commercial property that the area under the normal thickness curve to the right of z0.01 is 0.01 and the area come the left is 0.99. Usage your calculator, a computer, or a probability table for the traditional normal circulation to find z0.01 = 2.326.

EBM = (displaystylez_0.01fracsigmasqrtn=(2.236)frac0.337sqrt30=0.1431

To discover the 98% confidence interval, finddisplaystyleoverlinexpmEBM

displaystyleoverlinex– EBM = 1.024 – 0.1431 = 0.8809

displaystyleoverlinex+EBM = 1.024 +0.1431 = 1.1671

We estimate through 98% confidence the the true SAR average for the population of cabinet phones in the United states is between 0.8809 and also 1.1671 watt per kilogram.

Solution B:

Press STAT and arrow end to TESTS.Arrow under to 7: ZInterval.Press ENTER.Arrow to Stats and press ENTER.Arrow down and also enter the complying with values:σ: 0.337displaystyleoverlinex: 1.024n: 30C-level: 0.98Arrow down to Calculate and press ENTER.The to trust interval is (to 3 decimal places) (0.881, 1.167).

try it

This table shows a different random sampling the 20 mobile models. Use this data to calculate a 93% confidence interval because that the true average SAR for cell phones certified for use in the united States. As previously, assume that the population standard deviation is σ = 0.337.

Phone ModelSARPhone ModelSAR
Blackberry Pearl 81201.48Nokia E71x1.53
HTC Evo design 4G0.8Nokia N750.68
HTC Freestyle1.15Nokia N791.4
LG Ally1.36Sagem Puma1.24
LG Fathom0.77Samsung Fascinate0.57
LG Optimus Vu0.462Samsung infuse 4G0.2
Motorola Cliq XT1.36Samsung Nexus S0.51
Motorola Droid Pro1.39Samsung Replenish0.3
Motorola Droid Razr M1.3Sony W518a Walkman0.73
Nokia 7705 Twist0.7ZTE C790.869

displaystyleoverlinex = 0.940

displaystylefracalpha2=frac1-CL2=frac1-0.932=0.035

displaystylez_0.05=1.812

EBM=displaystyle(z_0.05)(fracsigmasqrtn)=(1.182)(frac0.337sqrt20=0.1365

displaystyleoverlinex– EBM = 0.940 – 0.1365 = 0.8035

displaystyleoverlinex+EBM = 0.940 + 0.1365 = 1.0765

We estimate with 93% confidence that the true SAR average for the population of cell phones in the United claims is between 0.8035 and also 1.0765 watt per kilogram.


Notice the difference in the trust intervals calculated in instance 3 and also the try It simply completed. These intervals are different for number of reasons: they to be calculated from different samples, the samples were different sizes, and also the intervals were calculated for various levels that confidence. Also though the intervals are different, they carry out not productivity conflicting information. The effects of these kinds of alters are the subject of the following section in this chapter.

Changing the trust Level or Sample Size


Example

Suppose we readjust the original trouble in example 2 by making use of a 95% trust level. Discover a 95% to trust interval for the true (population) average statistics test score.

Solution:

To uncover the to trust interval, you need the sample mean,displaystyleoverlinex, and the EBM.

displaystyleoverlinex = 68

EBM =(displaystylez_fracalpha2)(fracsigmasqrtn)

displaystylesigma=3

n = 36

CL = 0.95 so α = 1 – CL = 1 – 0.95 = 0.05

The area to the right of z0.025 is 0.025 and also the area to the left of z0.025 is 1 – 0.025 = 0.975.

when utilizing invnorm(0.975,0,1) on the TI-83, 83+, or 84+ calculators. (This can additionally be found using ideal commands on other calculators, using a computer, or making use of a probability table for the conventional normal distribution.)

Notice that the EBM is bigger for a 95% confidence level in the original problem.


Interpretation

We estimate v 95% confidence the the true populace mean for all statistics exam scores is between 67.02 and also 68.98.

Explanation of 95% to trust Level

Ninety-five percent of all confidence intervals built in this method contain the true worth of the population mean statistics exam score.

Comparing the Results

The 90% to trust interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If friend look at the graphs, due to the fact that the area 0.95 is larger than the area 0.90, it provides sense that the 95% confidence interval is wider. To be more confident that the confidence interval actually does save on computer the true value of the populace mean for every statistics exam scores, the trust interval necessarily needs to it is in wider.



Summary: effect of transforming the confidence LevelIncreasing the confidence level boosts the error bound, making the trust interval wider.Decreasing the confidence level decreases the error bound, making the to trust interval narrower.

try it

Refer ago to the pizza-delivery try It exercise. The population standard deviation is six minutes and the sample mean provide time is 36 minutes. Use a sample size of 20. Discover a 95% to trust interval estimate for the true mean pizza delivery time.

(33.37, 38.63)


Example

Suppose we readjust the original difficulty in instance 2 to view what wake up to the error bound if the sample size is changed.

Leave whatever the same other than the sample size. Usage the initial 90% confidence level. What happens to the error bound and also the trust interval if we increase the sample size and use n = 100 rather of n = 36? What happens if us decrease the sample dimension to n = 25 instead of n = 36?

= 68EBM =σ = 3; The trust level is 90% (CL=0.90); .

Solution A:

If us increase the sample size n come 100, us decrease the error bound.

Solution B:

If us decrease the sample size n come 25, we increase the error bound.

Summary: result of changing the Sample SizeIncreasing the sample size reasons the error bound to decrease, making the confidence interval narrower.Decreasing the sample size causes the error bound come increase, make the confidence interval wider.

Try It

Refer back to the pizza-delivery shot It exercise. The mean delivery time is 36 minutes and also the populace standard deviation is 6 minutes. Assume the sample dimension is changed to 50 restaurants v the exact same sample mean. Discover a 90% confidence interval calculation for the population mean delivery time.

(34.6041, 37.3958)


Working Backwards to discover the Error tied or Sample Mean

When we calculate a confidence interval, we discover the sample mean, calculate the error bound, and use them to calculation the trust interval. However, occasionally when we review statistical studies, the study may state the trust interval only. If we recognize the confidence interval, we can work backwards to find both the error bound and also the sample mean.

Finding the Error Bound

From the upper value for the interval, subtract the sample mean,OR, indigenous the top value because that the interval, subtract the reduced value. Then divide the distinction by two.

Finding the Sample Mean

Subtract the error bound indigenous the upper value of the confidence interval,OR, median the upper and lower endpoints that the trust interval.

Example

Notice the there space two methods to do each calculation. You can choose the method that is less complicated to use v the details you know.

Suppose we understand that a to trust interval is (67.18, 68.82) and also we desire to find the error bound. Us may understand that the sample median is 68, or possibly our source only offered the trust interval and also did not tell united state the worth of the sample mean.

Calculate the Error Bound:

If we understand that the sample median is 68: EBM = 68.82 – 68 = 0.82.If us don’t know the sample mean: .

Calculate the Sample Mean:

If we know the error bound: = 68.82 – 0.82 = 68If we don’t know the error bound: .

Try it

Suppose we recognize that a confidence interval is (42.12, 47.88). Discover the error bound and the sample mean.

Sample typical is 45, error bound is 2.88


Calculating the Sample size n

If researchers desire a details margin that error, climate they deserve to use the error bound formula to calculation the required sample size.

The error bound formula for a population mean as soon as the populace standard deviation is well-known is

The formula because that sample size is , discovered by addressing the error tied formula for n.

In this formula, z is , equivalent to the desired confidence level. A researcher planning a examine who desires a stated confidence level and also error bound have the right to use this formula to calculation the dimension of the sample essential for the study.


Example

The population standard deviation for the age of Foothill university students is 15 years. If we want to it is in 95% confident that the sample mean period is within 2 years the the true populace mean age of Foothill university students, how countless randomly selected Foothill university students have to be surveyed?

From the problem, we understand that σ = 15 and EBM = 2.z = z0.025 = 1.96, since the confidence level is 95%.using the sample size equation.Use n = 217: always round the answer up to the next higher integer to ensure that the sample dimension is large enough.

Therefore, 217 Foothill college students need to be surveyed in order to it is in 95% confident that we room within 2 years the the true populace mean period of Foothill college students.


try it

The population standard deviation for the height of high college basketball players is 3 inches. If we want to it is in 95% confident the the sample mean height is in ~ one inch of the true population mean height, how many randomly selected students have to be surveyed?

35 students


References

“American reality Finder.” U.S. Census Bureau. Easily accessible online in ~ http://factfinder2.census.gov/faces/nav/jsf/pages/searchresults.xhtml?refresh=t (accessed July 2, 2013).

“Disclosure Data Catalog: Candidate an introduction Report 2012.” U.S. Federal Election Commission. Obtainable online at http://www.fec.gov/data/index.jsp (accessed July 2, 2013).

“Headcount Enrollment patterns by student Demographics Ten-Year loss Trends to many Recently perfect Fall.” Foothill De Anza neighborhood College District. Accessible online at http://research.fhda.edu/factbook/FH_Demo_Trends/FoothillDemographicTrends.htm (accessed September 30,2013).

Kuczmarski, Robert J., Cynthia L. Ogden, Shumei S. Guo, Laurence M. Grummer-Strawn, Katherine M. Flegal, Zuguo Mei, Rong Wei, Lester R. Curtin, Alex F. Roche, Clifford L. Johnson. “2000 CDC growth Charts because that the united States: Methods and Development.” Centers for condition Control and Prevention. Easily accessible online in ~ http://www.cdc.gov/growthcharts/2000growthchart-us.pdf (accessed July 2, 2013).

La, Lynn, Kent German. “Cell phone Radiation Levels.” c|net component of CBX interactive Inc. Available online at http://reviews.cnet.com/cell-phone-radiation-levels/ (accessed July 2, 2013).

“Mean income in the previous 12 month (in 2011 Inflaction-Adjusted Dollars): 2011 American neighborhood Survey 1-Year Estimates.” American fact Finder, U.S. Census Bureau. Accessible online at http://factfinder2.census.gov/faces/tableservices/jsf/pages/productview.xhtml?pid=ACS_11_1YR_S1902&prodType=table (accessed July 2, 2013).

“Metadata description of Candidate an introduction File.” U.S. Federal Election Commission. Easily accessible online in ~ http://www.fec.gov/finance/disclosure/metadata/metadataforcandidatesummary.shtml (accessed July 2, 2013).

“National Health and Nutrition examination Survey.” Centers for an illness Control and Prevention. Easily accessible online at http://www.cdc.gov/nchs/nhanes.htm (accessed July 2, 2013).

Concept Review

In this module, us learned just how to calculation the to trust interval for a single populace mean where the populace standard deviation is known. As soon as estimating a populace mean, the margin the error is dubbed the error bound for a populace mean ( EBM). A to trust interval has actually the basic form:

(lower bound, top bound) = (point estimate – EBM, suggest estimate + EBM)

The calculation of EBM relies on the dimension of the sample and also the level of confidence desired. The trust level is the percent that all possible samples that have the right to be intended to incorporate the true populace parameter. Together the trust level increases, the corresponding EBM increases as well. Together the sample dimension increases, the EBM decreases. By the main limit theorem,

Given a to trust interval, you have the right to work backwards to find the error bound ( EBM) or the sample mean. To uncover the error bound, find the distinction of the upper bound the the interval and also the mean. If you execute not understand the sample mean, friend can find the error bound by calculating fifty percent the difference of the upper and also lower bounds. To discover the sample mean given a trust interval, uncover the distinction of the upper bound and the error bound. If the error bound is unknown, then typical the upper and lower bounds of the to trust interval to find the sample mean.

Sometimes researchers recognize in breakthrough that they want to estimate a populace mean in ~ a specific margin of error for a offered level of confidence. In the case, fix the EBM formula because that n to discover the dimension of the sample the is essential to accomplish this goal:

Formula Review

displaystyleoverlineXsimNleft(mu_x, fracsigmasqrtn ight). The distribution of sample way is normally distributed with mean equal to the populace mean and also standard deviation given by the population standard deviation split by the square root of the sample size.

The general type for a trust interval because that a single populace mean, well-known standard deviation, normal distribution is offered by

(lower bound, upper bound) = (point calculation – EBM, allude estimate + EBM)

=(displaystyleoverlinex – EBM, displaystyleoverlinex+EBM)

=(displaystyleoverlinex-z_fracalphasqrtn, overlinex+z_fracalphasqrtn)

EBM = displaystylez_fracalphasqrtn= the error bound for the mean, or the margin the error for a single populace mean; this formula is supplied when the populace standard deviation is known.

See more: Transformers Revenge Of The Fallen Game Cheats, Transformers: Revenge Of The Fallen

CL = confidence level, or the relationship of confidence intervals developed that space expected come contain the true populace parameter

α = 1 – CL = the ratio of to trust intervals that will certainly not save the population

displaystylez_fracalphasqrtn= the z-score with the residential or commercial property that the area come the ideal of the z-score is ∝2 this is the z-score supplied in the calculation of “EBM where α = 1 – CL.

n = displaystylefracz^2sigma^2EBM^2 = the formula provided to determine the sample dimension (n) necessary to achieve a wanted margin the error in ~ a provided level the confidence

General form of a trust interval

(lower value, top value) = (point estimate−error bound, suggest estimate + error bound)

To discover the error bound once you recognize the confidence interval

error tied = upper value−point estimate OR error bound =displaystylefrac extuppervalue-lowervalue2

Single populace Mean, recognized Standard Deviation, normal Distribution

Use the Normal circulation for Means, population Standard Deviation is well-known EBM=displaystylez_fracalpha2cdotfracsigmasqrtn

The confidence interval has actually the format EBM = (displaystyleoverlinex – EBM, displaystyleoverlinex+EBM)