AP Statistics Section 10.1 B

CI for Population Mean When is Known

Before calculating a confidence interval for or p, there are three

important conditions that you must check:

1. The data should come from an ______ of the population. a) Often times, the problem will simply state the sample is random but not specifically say it is an SRS. While the calculations we use technically require an SRS, the AP exam allows to simply state that you have a random sample. b) If the data does not come from a random sample of the population,

c) The margin of error in a confidence interval covers only chance variation due to random sampling. It does not account for mistakes in our sampling method such as undercoverage.

SRS

.population the togeneralizenot may results the

2. The sampling distribution of or must be at least approximately Normal.For means:a) If the population distribution is Normal, then the distribution of is Normal.

b) If the population distribution is not Normal, then

c) If neither a) nor b) is appropriate, look at the sample data. If the sample data does not show any striking deviations from Normality (outliers or strong skewness), we will assume that the population distribution is at least approximately Normal and therefore the distribution of is approximately Normal.

d) If the sampling distribution is not at least approximately Normal, then

Normal.ely approximat be willx ofon distributi

the30)(nenough large is size sample theas long as

x p̂

x

accurate benot may interval confidence resulting the

3. The individual observations in the random sample must be independent.

a) Since we almost always sample without replacement, we need to verify that

(_________)b) If the individual observations are not independent, our calculations may not be accurate.

sample theas large as times10least at is population the

10nN

nzx

Example: Find the value of z* for the following confidence levels.

88% b) 95%

z

88.06.

555.1

)1,0,94(.

z

invNorm

z

95.025.

960.1

)1,0,975(.

z

invNorm

These z-scores that “mark off” a specified area under the Standard

Normal curve are often called critical values.

The confidence levels at the right and their corresponding upper p critical value are so common that

they are worth memorizing so that you will not have to take the time to

find them each time.Confidence Level Tail Area z*

90% 0.05 1.64595% 0.025 1.96099% 0.005 2.576

When constructing a confidence interval you should use the tools from the Inference Toolbox.

Step 1: _______________ Identify the population of interest and the parameter you want to draw conclusions about.

Step 2: _______________ Identify the appropriate inference procedure and verify the conditions for using it.

Step 3: _______________ Carry out the inference procedure: Step 4: ___________________ State your conclusions in the context of the problem.

Parameter

Conditions

nsCalculatio

tionsInterpreta

Example: A manufacturer of high-resolution video terminals must control the tension on the mesh of fine wires that lies behind the surface of the viewing screen. Too much tension will tear the mesh and too little will

allow wrinkles. The tension is measured by an electrical device with output readings in millivolts (mV). Some

variation is inherent in the production process. Careful study has shown that when the process is operating

properly, the standard deviation of the tension readings is mV.

Construct and interpret a 90% confidence interval for the mean

tension of all such screens.

Parameter: The population of interest is

We want to estimate , the

high resolution terminals.

mean tension for the wire mesh in these screens.

Conditions: Since we know ____ use the CI for a population mean where is known.

SRS:

Normality: Since we do not know if the population distribution is Normal…. CLT?

screens 20 of SRS a from comes Data

small toois size sample the20n with NO,

Boxplot Normal probability plot?

d.distributeNormally is population

assume will weso Normal appears Sample

Independence:

20010(20)

least at is nalssuch termi of population theassume

must wet,replacemen without sampled weSince

Calculations:

32.306

645.1z 90%,CFor

x

82.1532.30620

43645.132.306

nzx

)14.322,5.290(

Interpretation:

.millivolts 322.14 and 290.5between is

terminals videoin the wiresfine ofmesh theof

on mean tensi that theconfident 90% are We

TI-83/84: interval - z :7 Tests STAT