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What if it is impossible or impractical to use a large sample?

Learn how to construct a confidence interval for the mean of a small sample using the Student's t distribution.

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What if it is impossible or impractical to use a large sample?

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  1. What if it is impossible or impractical to use a large sample? Apply the Student’s t distribution.

  2. Student’s t Variable

  3. The shape of the t distribution depends only only the sample size, n, if the basic variable x has a normal distribution. When using the t distribution, we will assume that the x distribution is normal.

  4. Table 6 in Appendix II gives values of the variable t corresponding to the number of degrees of freedom (d.f.)

  5. Degrees of Freedom d.f. = n – 1 where n = sample size

  6. The t Distribution has a Shape Similar to that of the the Normal Distribution A Normal distribution A “t” distribution

  7. d.f.=7 Find the critical value tc for a 95% confidence interval if n = 8.

  8. Confidence Interval for the Mean of Small Samples (n < 30) from Normal Populations • c = confidence level (0 < c < 1) • tc = critical value for confidence level c, and degrees of freedom = n - 1

  9. The mean weight of eight fish caught in a local lake is 15.7 ounces with a standard deviation of 2.3 ounces. Construct a 90% confidence interval for the mean weight of the population of fish in the lake.

  10. Mean = 15.7 ounces Standard deviation = 2.3 ounces. • n = 8, so d.f. = n – 1 = 7 • For c = 0.90, Table 6 in Appendix II gives t0.90 = 1.895.

  11. Mean = 15.7 ounces Standard deviation = 2.3 ounces. • E = 1.54 • The 90% confidence interval is: • 15.7 - 1.54 <  < 15.7 + 1.54 • 14.16 <  < 17.24

  12. The 90% Confidence Interval:14.16 <  < 17.24 We are 90% sure that the true mean weight of the fish in the lake is between 14.16 and 17.24 ounces.

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