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# Example 9.1 Gasoline Prices in the United States - PowerPoint PPT Presentation

Example 9.1 Gasoline Prices in the United States. Sampling Distributions. Objective. To use Excel’s TDIST function to analyze differences between a sample mean and a population mean for gasoline prices. Background Information.

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### Example 9.1Gasoline Prices in the United States

Sampling Distributions

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

• 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?

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

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.

• 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.

• 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).

• 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.