REVIEW Confidence Intervals for Means

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# REVIEW Confidence Intervals for Means - PowerPoint PPT Presentation

REVIEW Confidence Intervals for Means. When to use z and When to use t. USE z Large n or sampling from a normal distribution σ is known. USE t Large n or sampling from a normal distribution σ is unknown. z and t distributions are used in confidence intervals.

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REVIEW

Confidence Intervals for Means

When to use z and When to use t
• USEz
• Large n or sampling from a normal distribution
• σ is known
• USEt
• Large n or sampling from a normal distribution
• σ is unknown

z and t distributions are used in confidence intervals.

_ These are determined by the distribution of X.

General Form ofConfidence Intervals

The general form of a confidence interval is:

or

(Point Estimate) ± (Margin of Error)

(Point Estimate) ± (zα/2 or tα/2) (Appropriate Standard Error)

Example

The average cost of all required texts for introductory college English courses seems to have gone up substantially as the professors are assigning several texts.

• A sample of 41 courses was taken
• The average cost of texts for these 41 courses is \$86.15
• Construct a 95% confidence interval for the average costs of texts for these courses assuming:
• The standard deviation is \$22.
• The standard deviation is unknown, but the sample standard deviation of the sample is \$24.77.
Case 1
• Because the sample size > 30, it is not necessary to assume that the costs follow a normal distribution to construct a confidence interval.
• And because it is assumed that σ is known (to be \$22), this will be a z-interval.

\$86.15 ± \$6.73

(\$79.42\$92.92)

Case 2
• Because the sample size > 30, it is not necessary to assume that the costs follow a normal distribution to construct a confidence interval.
• Because it is assumed that σ is unknown, this will be a t-interval with 40 degrees of freedom and s = 24.77.

\$86.15 ± \$7.82

(\$78.33\$93.97)

EXCELz-Intervals

AVERAGE (data set)

CONFIDENCE(α,σ,n)

The z-interval =

(Sample Mean) ± (Margin of Error)

Thus,

the Lower Confidence Limit (LCL) =

AVERAGE(data set) - CONFIDENCE(α,σ,n)

And,

the Upper Confidence Limit (UCL) =

AVERAGE(data set) + CONFIDENCE(α,σ,n)

=AVERAGE(A2:A42)

=CONFIDENCE(.05,D1,41)

=AVERAGE(A2:A42)

=CONFIDENCE(.05,D1,41)

=D3-D5

=D3+D5

EXCELt-Intervals

The t-interval =

(Sample Mean) ± (Margin of Error)

Go to Tools/Data Analysis/Descriptive Statistics

From the output, find Mean and Confidence Level

Thus,

the Lower Confidence Limit (LCL) =

MEAN – CONFIDENCE LEVEL

and,

the Upper Confidence Limit (UCL) =

MEAN + CONFIDENCE LEVEL

=D3-D16

=D3+D16

REVIEW
• To construct a confidence interval
• Must have a large sampleOR assume you are sampling from a normal distribution
• Known σ -- z-interval Unknown σ – t-interval
• Form of interval: (Sample Mean) ± (Margin of Error)
• Calculating margin of error by hand:
• Excel:
• z-interval – Use CONFIDENCE function
• t-interval – Use DESCRIPTIVE STATISTICS in DATA ANALYSIS