Example 9.1 Gasoline Prices in the United States Sampling Distributions.

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Presentation transcript:

Example 9.1 Gasoline Prices in the United States Sampling Distributions

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | Objective To use Excel’s TDIST function to analyze differences between a sample mean and a population mean for gasoline prices.

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | Background Information n Suppose a government agency randomly samples 30 gas stations from the population of all gas stations in the United States. n Its goal is to estimate the mean price for a gallon of premium unleaded gasoline. n What is the probability mean price? n What is the probability that the sample mean price will differ by at least two standard errors from the population mean price?

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | The t Distribution n 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. n Then the standardized value in this equation has a t distribution with n-1 degrees of freedom.

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | The t Distribution -- continued n The t distribution looks very much like the standard normal distribution. It is bell shaped and is centered at 0. n The only difference is that it is slightly more spread out, and this increase in spread is greater for small degrees of freedom. n A t-value indicates the number of standard errors by which a sample mean differs from a population mean.

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | Solution n 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. n 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. n The figure on the next slide shows a one-tailed probability, where we are interested in whether a t- value exceeds some positive value. n 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.

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | One-Tailed Probability

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | Two-Tailed Probability

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | TDIST.XLS n The calculations from this spreadsheet appear on the next slide. n 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. n 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).

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | Solution -- continued

| 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 9.10 | 9.11 | 9.12 | 9.13 | 9.14 | Solution -- continued n We see that the probability of the sample mean exceeding the population mean by this much – two standard errors – is only n The answer to the second question is exactly twice this probability. n We find it with the formula in row 12. The only difference is that the third argument is now 2.