Example 9.1Gasoline Prices in the United States

Sampling Distributions

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Objective

To use Excel’s TDIST function to analyze differences between a sample mean and a population mean for gasoline prices.

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Background Information

Suppose a government agency randomly samples 30 gas stations from the population of all gas stations in the United States.

Its goal is to estimate the mean price for a gallon of premium unleaded gasoline.

What is the probability mean price?

What is the probability that the sample mean price will differ by at least two standard errors from the population mean price?

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The t Distribution We are interested in estimating a population mean

with a sample of size n. We assume the population distribution is normal with unknown standard deviation . We intend to base inferences on the standard value of X-bar, where is replaced by the sample standard deviation s.

Then the standardized value

in this equation has a t distribution with n-1 degrees of freedom.

nst

/

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The t Distribution -- continued

The t distribution looks very much like the standard normal distribution. It is bell shaped and is centered at 0.

The only difference is that it is slightly more spread out, and this increase in spread is greater for small degrees of freedom.

A t-value indicates the number of standard errors by which a sample mean differs from a population mean.

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Solution First, we note that the answers to these questions do not depend on

the values of the sample and population means or the standard error of the mean.

They depend only on finding the probability that a “standardized” t-value is beyond some value as shown in the figures of the next two slides.

The figure on the next slide shows a one-tailed probability, where we are interested in whether a t-value exceeds some positive value.

The second figure shows a two-tailed probability, where we are interested in whether the magnitude of a t-value, positive or negative, exceeds 2.

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One-Tailed Probability

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Two-Tailed Probability

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TDIST.XLS The calculations from this spreadsheet appear on the

next slide.

We answer the first question in rows 7 and 8. We want the probability that a t-value with 29 degrees of freedom exceeds 2. We find this with the formula in row 8.

The first argument of TDIST is the value we want to exceed, the second is the degrees of freedom, and the third is the number of tails (1 or 2).

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Solution -- continued

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Solution -- continued

We see that the probability of the sample mean exceeding the population mean by this much – two standard errors – is only 0.0275.

The answer to the second question is exactly twice this probability.

We find it with the formula in row 12. The only difference is that the third argument is now 2.