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Chapter 7 Estimation Chapter 7 ESTIMATION Section 2 Estimating m When s Is Unknown What if it is impossible or impractical to use a large sample? Apply the Student’s t distribution. The shape of the t distribution depends 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. Confidence Interval for the Mean of Small Samples (n < 30) Table 6 in Appendix II gives values of the variable t corresponding to the number of degrees of freedom (d.f.) Degrees of Freedom d.f. = n – 1 where n = sample size The t Distribution has a Shape Similar to that of the the Normal Distribution A Normal distribution A “t” distribution Find the critical value tc for a 95% confidence interval if n = 7. c 0.900 0.050 0.100 0.950 0.025 0.050 0.980 0.010 0.020 0.990 0.005 0.010 6 ...1.9432 2.4469 3.1427 3.7074 7 ...1.8946 2.3646 2.9980 3.4995 8 ...1.8595 2.3060 2.8965 3.3554 ’ ’’ d.f. ... Confidence Interval for the Mean of Small Samples (n < 30) from Normal Populations x-E<m<x+E where x = Sample Mean E = tc s n c = confidence level (0 < c < 1) tc = critical value for confidence level c, and degrees of freedom = n - 1 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. Key Information • Mean = 15.7 ounces • Standard deviation = 2.3 ounces • n = 8, so d.f. = n – 1 = 7 • For c = 0.90, Table t chart • gives t0.90 = 1.8946 The 90% confidence interval is: 2.3 2.3 15.7 - 1.8946 < m < 15.7 + 1.8946 8 8 4.3576 4.3576 15.7 < m < 15.7 + 2.8284 2.8284 15.7- < 1.5406 < m < 15.7 + 1.5406 14.1594 < m < 17.2406 We can say with 90% confidence that the population mean weight of the fish in the lake is between 14.1594 and 17.2406 ounces. The 90% confidence interval is: Calculator Computation VARS Statistics TEST H: lower 14.1594 I: upper 17.2406 We can say with 90% confidence that the population mean weight of the fish in the lake is between 14.1594 and 17.2406 ounces. THE END OF SECTION 2