Example 8.11

Controlling Confidence Interval Length

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Background Information

The fast-food manager in Example 8.1 surveyed 40 customers, each of whom rated a new sandwich on a scale of 1 to 10.

Based on the data, a 95% confidence interval for the mean rating of all potential customers extended from 5.739 to 6.761, for a half-length of (6.761-5.379)/2 = 0.511.

How large a sample would be needed to reduce this half-length to approximately 0.3?

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Confidence Intervals Confidence intervals are a function of three things:

– the data in the sample

• We have control over the data by using the various random sampling plans to reduce variability.

• An area of statistics called experimental design suggests how to perform experiments to obtain the most information from a given amount of sample data.

– the confidence level

• This effect is clear as the confidence level increases, the length of the confidence interval increases as well.

– the sample size(s)

• The most obvious way to control confidence interval length is to choose the size of the sample appropriately.

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Sample Size

Sample size selection must be done before a sample is observed.

Sample size estimation formula:

2

3.0

597.196.1

n

2

B

xmultiplezn

est

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Calculations

The formula for n uses three inputs:

– the z multiple, which is 1.96 for a 95% confidence level;

– the prescribed confidence interval half-length B, which is 0.3;

– and an estimate sigmaest of the standard deviation

The final input must be guessed, but for this example with a sample size 40 we can use the observed sample standard deviation of 1.597 (which can be determined with the STDEV function).

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Calculations -- continued The formula yields a rounded result of n = 109.

The claim, then is that the manager surveys 109 customers with a 95% confidence interval will have the approximate half length of 0.3.

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Calculations -- continued

The same calculations can be done using the Sample Size Selection procedure of Excel’s StatPro add-in.

– Just select the menu item and select the parameter to analyze and enter the requested values.

– A message telling you the required sample size is displayed.

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Question What if the manager was at the planning stage and didn’t

have a “preliminary” sample of size 40? What standard deviation estimate should she use for est ?

The manager basically has three choices:

– she can base her estimate of the standard deviation on historical data assuming that relevant historical data are available,

– she can take a small preliminary sample (of size 20, say) just to get an estimate of the standard deviation,

– she can simply guess a value for the standard deviation.

• not recommended but there are some cases where it is the only feasible option.