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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
REVIEW

Confidence Intervals for Means

when to use z and when to use t
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 of confidence intervals
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
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
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
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)

excel z intervals
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)

slide10

=AVERAGE(A2:A42)

=CONFIDENCE(.05,D1,41)

slide11

=AVERAGE(A2:A42)

=CONFIDENCE(.05,D1,41)

=D3-D5

=D3+D5

excel t intervals
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

slide16

=D3-D16

=D3+D16

review
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