Point EstimatesPoint Estimates

A point estimatepoint estimate is a single number determined from a sample that is used to estimate the corresponding population parameter.

Confidence IntervalsConfidence Intervals

Confidence IntervalConfidence Interval Developed from a sample. Provides a range of likely values for a parameter.Expresses the confidence level that the true population parameter is included.

Confidence IntervalsConfidence Intervals

Point EstimateLower Confidence

LimitUpper Confidence

Limit

95% Confidence Intervals95% Confidence Intervals(Figure 7-3)(Figure 7-3)

0.95

z.025= -1.96 z.025= 1.96

Confidence IntervalConfidence Interval- General Format -- General Format -

Point Estimate (Critical Value)(Standard Error)

Confidence IntervalsConfidence Intervals

The confidence levelconfidence level refers to a percentage greater than 50 and less than 100. For a given size sample it is the percentage that the interval will contain the true population value.

Confidence Interval Confidence Interval EstimatesEstimates

CONFIDENCE INTERVAL CONFIDENCE INTERVAL ESTIMATE FOR ESTIMATE FOR ( ( KNOWN) KNOWN)

where:z = Critical value from

standard normal table

= Population standard deviation

n = Sample size

nzx

Example of a Confidence Example of a Confidence Interval Estimate for Interval Estimate for

A sample of 100 cans, from a population with = 0.20, produced a sample mean equal to 12.09. A 95% confidence interval would be:

039.009.1210020.096.109.12

n

zx

12.051 ounces

12.129 ounces

Margin of ErrorMargin of Error

The margin of errormargin of error is the largest possible sampling error at the specified level of confidence.

Margin of ErrorMargin of ErrorMARGIN OF ERROR (ESTIMATE FOR MARGIN OF ERROR (ESTIMATE FOR

WITH WITH KNOWN) KNOWN)

where:e = Margin of errorz = Critical value = Standard error of the

sampling distributionn

nze

Example of Impact of Example of Impact of Sample Size on Sample Size on

Confidence IntervalsConfidence IntervalsIf instead of sample of 100 cans, suppose a sample of 400 cans, from a population with = 0.20, produced a sample mean equal to 12.09. A 95% confidence interval would be:

0196.009.1240020.096.109.12

n

zx

12.051 ounces

12.129 ounces

12.0704 ounces

12.1096 ouncesn=400

n=100

Student’s t-DistributionStudent’s t-Distribution

The t-distributiont-distribution is a family of distributions:Bell-shaped and symmetric Greater area in the tails than the normal.Defined by its degrees of freedom. The t-distribution approaches the normal distribution as the degrees of freedom increase.

Confidence Interval Confidence Interval EstimatesEstimates

CONFIDENCE INTERVAL CONFIDENCE INTERVAL (( UNKNOWN) UNKNOWN)

where:t = Critical value from t-

distribution with n-1 degrees of freedom

= Sample means = Sample standard deviationn = Sample size

nstx

x

Confidence Interval Confidence Interval EstimatesEstimates

CONFIDENCE INTERVAL-LARGE CONFIDENCE INTERVAL-LARGE SAMPLE WITH SAMPLE WITH UNKNOWN UNKNOWN

where:z =Value from the standard

normal distribution = Sample means = Sample standard deviationn = Sample size

nszx

x