Example 9.2Customer Response to a New Sandwich

Confidence Interval for a Mean

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Objective

To use StatPro’s one-sample procedure to obtain a 95% confidence for the mean satisfaction rating of a new sandwich.

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SANDWICH1.XLS This file contains the results of a survey done by a fast

food restaurant. The restaurant recently added a new sandwich to its menu. They conducted a survey to estimate the popularity of this sandwich.

A random sample of 40 customers who ordered the sandwich were surveyed. They were each asked to rate the sandwich from 1 to 10, 10 being best.

The manager wants to estimate the mean satisfaction rating over the entire population of customers by using a 95% confidence interval. How should she proceed?

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New Sandwich Data

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The t Distribution

The t distribution is a close relative of the normal distribution that appears in a variety of statistical applications.

The “degrees of freedom” is a numerical parameter of the t distribution that defines the precise shape of the distribution.

The only difference between the t distribution and the normal distribution is that it is a little more spread out and this increase in spread is greater for the small degrees of freedom.

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This file contains the sample calculations that illustrate the TDIST and TINV functions.

Details to using the TDIST function

– Its first argument must be nonnegative.

– Unlike the NORMDIST function, it returns the probability to the right of the first argument.

– Its third argument is either 1 or 2 and indicates the number of tails. By using 1 for this argument, we get the probability in the right-hand tail only.

TDIST.XLS

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The t Distribution -- continued Details of using the TINV function:

– The first argument is the total probability we want in both tails - half of this goes in the right-hand tail and half goes in the left-hand tail.

– Unlike the TDIST function, there is no third argument for the TINV function.

The t distribution is used when we want to make inferences about a population mean and the population standard deviation is unknown.

Two other close relatives of the normal distribution are the chi-square and F distribution.

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Confidence Intervals and Levels

To obtain a confidence interval for the population mean, we first specify a confidence level, usually 90%, 95%, or 99%.

We then use the sampling distribution of the point estimate to determine the multiple of the standard error we need to go out on either side of the point estimate to achieve the given confidence level.

To estimate confidence intervals we use the One-Sample procedure in Excel’s StatPro add-in.

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Calculation

The manager must use StatPro’s One-Sample procedures on the Satisfaction variable.

To use the procedure place the cursor anywhere in the data set and select StatPro/Statistical Inference/One-Sample Analysis menu item.

In the succeeding dialog boxes, select Satisfaction as the variable that you want to analyze, check that you want a confidence interval to the mean, and then accept the defaults from there on.

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Results

The results of the calculation are:

– The best guess for the population mean rating is 6.250, the sample average in cell F7.

– A 95% confidence interval for the population mean rating extends from 5.739 to 6.761.

Thus, the manager can be 95% confident that the true mean rating over all customers who might try the sandwich is within this interval.

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Results -- continued As the confidence level increases, the length of the

confidence interval increases.

You can convince yourself of this by entering different confidence levels in cell F11 of the SANDWICH1.XLS.

The lower and upper limits of the confidence interval in cells F15 and F16 will change automatically, getting closer together for 90% level and farther apart for the 99% level.

Just remember, you the analyst can choose the level, although 95% level is most commonly chosen.

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Assumptions We might question whether the sample is really a random

sample matters.

– The manager may have selected random customers but more likely she selected 40 consecutive customer. This type of sample is called a convenience sample. If there isn’t a reason to assume these 40 differ in any way from the entire population, then it is safe to assume them as a random sample.

Another assumption is that the population distribution is normal.

– This is probably not a problem because confidence intervals based on the t distribution are robust to violations of normality. This means that the resulting confidence intervals are valid for any populations that are approximately normal.

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Conclusions Finally, it is important to recognize what this confidence

interval tells us and what it doesn’t.

In the entire population of customers for this sandwich, there is a distribution of satisfaction ratings. All we are trying to determine is the average of all these ratings, and based on our analysis, we can be 95% confident that this average is between 5.739 and 6.761.

However, this confidence interval doesn’t tell us other characteristics of the population that might be of interest.