Sections 7-1 and 7-2
Review and Preview
and
Estimating a Population Proportion
INFERENTIAL STATISTICS
This chapter presents the beginnings of inferentialstatistics. The two major applications ofinferential statistics involve the use of sampledata to:
1.estimate the value of a populationparameter, and
2.test some claim (or hypothesis) about apopulation.
INFERENTIAL STATISTICS(CONTINUED)
This chapter deals with the first of these.
1.We introduce methods for estimating values ofthese important population parameters:proportions, means, and variances.
2.We also present methods for determiningsample sizes necessary to estimate thoseparameters.
DEFINITIONS
•Estimator is a formula or process for usingsample data to estimate a population parameter.
•Estimate is a specific value or range of valuesused to approximate a population parameter.
•Point estimate is a single value (or point) usedto approximate a population parameter.
ASSUMPTIONS FORESTIMATING A PROPORTION
We begin this chapter by estimating a populationproportion. We make the following assumptions:
1.The sample is simple random.
2.The conditions for the binomial distribution aresatisfied. (See Section 5-3.)
3.There are at least 5 successes and 5 failures.
NOTATION FOR PROPORTIONS
p =
population proportion
sample proportion of x successesin a sample of size n.
sample proportion of failures in asample of size n.
POINT ESTIMATE
A point estimate is a single value (or point) usedto approximate a population parameter.
The sample proportion is the best pointestimate of the population proportion p.
CONFIDENCE INTERVALS
A confidence interval (or interval estimate) is arange (or an interval) of values used to estimatethe true value of a population parameter. Aconfidence interval is sometimes abbreviated asCI.
CONFIDENCE LEVEL
A confidence level is the probability 1 − α (oftenexpressed as the equivalent percentage value) thatconfidence interval actually does contain thepopulation parameter, assuming that the estimationprocess is repeated a large number of times. (Theconfidence level is also called the degree ofconfidence, or the confidence coefficient.) Somecommon confidence levels are:
90%,
95%, or
99%
(α = 10%)
(α = 5%)
(α = 1%)
A REMARK ABOUTCONFIDENCE INTERVALS
Do not use the overlapping of confidenceintervals as the basis for making final conclusionsabout the equality of proportions.
CRITICAL VALUES
1.Under certain conditions, the sampling distributionof sample proportions can be approximated by anormal distribution. (See Figure 7-2.)
2.A z score associated with a sample proportion has aprobability of α/2 of falling in the right tail of Figure7-2.
3.The z score separating the right-tail is commonlydenoted by zα/2, and is referred to as a critical valuebecause it is on the borderline separating z scoresthat are likely to occur from those that are unlikely tooccur.
z =0
Figure 7-2
Found from Table A-2.
(corresponds to an area of
1 − α/2.)
−zα/2
zα/2
α/2
α/2
CRITICAL VALUE
A critical value is the number on the borderlineseparating sample statistics that are likely tooccur from those that are unlikely to occur. Thenumber zα/2 is a critical value that is a z scorewith the property that it separates an area of α/2in the right tail of the standard normaldistribution. (See Figure 7-2).
NOTATION FOR CRITICALVALUE
The critical value zα/2 is the positive z value that isat the vertical boundary separating an area of α/2in the right tail of the standard normaldistribution. (The value of –zα/2 is at the verticalboundary for the area of α/2 in the left tail). Thesubscript α/2 is simply a reminder that the z scoreseparates an area of α/2 in the right tail of thestandard normal distribution.
FINDING zα/2 FOR 95% DEGREEOF CONFIDENCE
−zα/2 = −1.96
zα/2 = 1.96
α = 5% = 0.05
α/2 = 2.5% = 0.025
α/2 = 0.025
α/2 = 0.025
Confidence Level: 95%
critical values
MARGIN OF ERROR
When data from a simple random sample are usedto estimate a population proportion p, the marginof error, denoted by E, is the maximum likelydifference (with probability 1 – α, such as 0.95)between the observed proportion p and the truevalue of the population proportion p. The margin oferror E is also called the maximum error of theestimate and can be found using the formula on thefollowing slide.
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MARGIN OF ERROR OF THEESTIMATE FOR p
NOTE: n is the size of the sample.
The confidence interval is often expressed in thefollowing equivalent formats:
or
CONFIDENCE INTERVAL FOR THEPOPULATION PROPORTION p
ROUND-OFF RULE FORCONFIDENCE INTERVALS
Round the confidenceinterval limits to
three significant digits.
PROCEDURE FOR CONSTRUCTINGA CONFIDENCE INTERVAL
1.Verify that the required assumptions are satisfied. (Thesample is a simple random sample, the conditions for thebinomial distribution are satisfied, and the normaldistribution can be used to approximate the distribution ofsample proportions because there are at least 5 successes andat least 5 failures.)
2.Refer to Table A-2 and find the critical value zα/2 thatcorresponds to the desired confidence level.
3.Evaluate the margin of error
4.Using the calculated margin of error, E and thevalue of the sample proportion, p, find the values ofp – E and p + E. Substitute those values in thegeneral format for the confidence interval:p − E < p < p + E
5.Round the resulting confidence interval limits tothree significant digits.
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CONFIDENCE INTERVAL LIMITS
The two values are calledconfidence interval limits.
FINDING A CONFIDENCEINTERVAL USING TI-83/84
1.Select STAT.
2.Arrow right to TESTS.
3.Select A:1–PropZInt….
4.Enter the number of successes as x.
5.Enter the size of the sample as n.
6.Enter the Confidence Level.
7.Arrow down to Calculate and press ENTER.
NOTE: If the proportion is given, you must first compute numberof successes by multiplying the proportion (as a decimal) by thesample size. You must round to the nearest integer.
SAMPLE SIZES FORESTIMATING A PROPORTION p
When an estimate p is known:
When no estimate p is known:
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ROUND-OFF RULE FORDETERMINING SAMPLE SIZE
In order to ensure that the required sample size isat least as large as it should be, if the computedsample size is not a whole number,
round up to the next higher whole number.
FINDING THE POINT ESTIMATE ANDE FROM A CONFIDENCE INTERVAL
Point estimate of p:
Margin of error:
CAUTION
Do not use the overlapping of confidenceintervals as the basis for making final conclusionsabout the equality of proportions.