T 11 df 1 2 3 4 25 1000 0816 0765 0741 20 1376 1061

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Unformatted text preview: et K310 Statistical Techniques Inference for Distributions 3 / 26 The t distribution Critical values for the t distribution are in Table D (p. T-11) df 1 2 3 4 . . . .25 1.000 0.816 0.765 0.741 . . . .20 1.376 1.061 0.978 0.941 1000 z∗ 0.675 0.674 50% 0.842 0.841 60% Example: Upper-tail probability p .15 .10 .05 1.963 3.078 6.314 1.386 1.886 2.920 1.250 1.638 2.353 1.190 1.533 2.132 . . . . . . 1.037 1.282 1.646 1.036 1.282 1.645 70% 80% 90% Confidence level C .025 12.71 4.303 3.182 2.776 .02 15.89 4.849 3.482 2.999 . . . 1.962 1.960 95% 2.056 2.054 96% df = 3 22 19 15 12 24 23 21 20 18 17 16 14 13 11 10 9 8 7 6 5 4 2 C = 0.90, two-tailed t −−−21132 0 12.746920 2−21...860 − .353 .−1.015 2 . 943 − 895 920833 813 796 782 771 761 753 746 740 734 729 725 721 717 714 711 1..711 12771 714 717 721 725 729 734 740 753 2015 761 782 796 .895 813 833 860 .132 943 .2. 353 K310 Statistical Techniques Inference for Distributions 4 / 26 The one-sample t confidence interval Suppose we draw a sample of size n from a population having unknown mean µ. A level C confidence interval for µ is s x ± t∗ √ ¯ n where t ∗ is the value for the t (n − 1) density curve with area C between s −t ∗ and t ∗ . Then, the margin of error is t ∗ √ . n K310 Statistical Techniques Inference for Distributions 5 / 26 One-sample t confidence interval example Measurements of vitamin C (mg/100 g) in a dry corn soy blend in one production run: 26, 31, 23, 22, 11, 22, 14, 31 Find a 95% confidence interval for µ K310 Statistical Techniques Inference for Distributions 6 / 26 One-sample t confidence interval example Measurements of vitamin C (mg/100 g) in a dry corn soy blend in one production run: 26, 31, 23, 22, 11, 22, 14, 31 Find a 95% confidence interval for µ x = 22.50, s = 7.19 ¯ K310 Statistical Techniques Inference for Distributions 6 / 26 The one-sample t test Null hypothesis: H0 : µ = µ0 Draw a sample of size n and compute the t statistic: t= Alternative P -value Ha : µ > µ0 P (T ≥ t ) Ha : µ < µ0 P (T ≤ t ) Ha : µ = µ...
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