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8.3 The t-Distribution and Testing Hypothesis about the mean for small Sample

Properties of the t-distribution

▪ The t-distribution has a symmetric bell-shaped density centered at 0, similar to the N(0,1) distribution.

▪ The t-distribution is “flatter” and has “heavier tails” than the N(0,1) distribution.

▪ As the sample size increases, the t-distribution approaches the N(0,1) distribution.

Example

Find the critical t value for [pic] =0.05 with d.f. = 16 for a right-tailed t test.

Solution

Find the 0.05 column in the top row and 16 in the left-hand column. Where the row and column meet, the appropriate critical value is found; it is +1.746

[pic]

Let’s Do It!

Find the critical t value for [pic]0.01 with d.f. = 22 for a left-tailed test.

Find the critical values for [pic] 0.10 with d.f. =18 for a two-tailed t test.

[pic]

Example Hospital Infections

A medical investigation claims that the average number of infections per week at a hospital in southwestern Pennsylvania is 16.3. A random sample of 10 weeks had a mean number of 17.7 infections. The sample standard deviation is 1.8. Is there enough evidence to reject the investigator’s claim at [pic] 0.05?

Solution

[pic]

Let’s Do It! An educator claims that the average salary of substitute teachers in school districts in Allegheny County, Pennsylvania, is less than $60 per day. A random sample of eight school districts is selected, and the daily salaries (in dollars) are shown. Is there enough evidence to support the educator’s claim at [pic] 0.10?

[pic]

Let’s Do It! State and Local Taxes

The U.S. average for state and local taxes for a family of four is $4172. A random sample of 20 families in a northeastern state indicates that they paid an annual amount of $4560 with a standard deviation of $1590. At [pic] 0.05, is there sufficient evidence to conclude that they pay more than the national average of $4172?

Let’s Do It! Doctor Visits

A report by the Gallup Poll stated that on average a woman visits her physician 5.8 times a year. A researcher randomly selects 20 women and obtained these data. At [pic] 0.05 can it be concluded that the average is different from 5.8 visits per year?

[pic]

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The Decision Tool (THE TEST STATISTIC) In order to perform the hypothesis test, we first determine the the value of the observed statistic of our sample. Many hypotheses are tested using a statistical test based on the following general formula:

[pic]

There are two cases:

• If the sample used is large, then the distribution used for testing the hypothesis is Z. The standardized observed test statistic is given by:

[pic](we discussed this case in the previous section)

• If the sample is small (n ................
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