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Ch7 - The t distribution Before We knew the populations...

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The t distribution Before Now We knew the population’s standard deviation σ K310 Statistical Techniques Inference for Distributions 1 / 26
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The t distribution Before Now We knew the population’s standard deviation σ Standard error: σ ¯ x = σ n K310 Statistical Techniques Inference for Distributions 1 / 26
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The t distribution Before Now We knew the population’s standard deviation σ Standard error: σ ¯ x = σ n Test statistic: z = ¯ x - μ σ/ n K310 Statistical Techniques Inference for Distributions 1 / 26
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The t distribution Before Now We knew the population’s standard deviation σ We don’t know σ Standard error: σ ¯ x = σ n Test statistic: z = ¯ x - μ σ/ n K310 Statistical Techniques Inference for Distributions 1 / 26
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The t distribution Before Now We knew the population’s standard deviation σ We don’t know σ Standard error: σ ¯ x = σ n Standard error: SE ¯ x = s n Test statistic: z = ¯ x - μ σ/ n K310 Statistical Techniques Inference for Distributions 1 / 26
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The t distribution Before Now We knew the population’s standard deviation σ We don’t know σ Standard error: σ ¯ x = σ n Standard error: SE ¯ x = s n Test statistic: z = ¯ x - μ σ/ n Test statistic: t = ¯ x - μ s / n where s = 1 n - 1 n i =1 ( x i - ¯ x ) 2 K310 Statistical Techniques Inference for Distributions 1 / 26
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The t distribution We draw a sample of size n from a N ( μ, σ ) population. Then, the one-sample t statistic t = ¯ x - μ s / n has the t distribution with n - 1 degrees of freedom , denoted t ( n - 1). t 0 df = (normal distribution) df = 10 df = 5 df = 2 df = 1 K310 Statistical Techniques Inference for Distributions 2 / 26
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William S. Gosset William S. Gosset (1876–1937), developed t distributions in 1908 while working for Guinness brewing company. Guinness forbid its employees to publish scientific discoveries, so he wrote a series of papers under the pseudonym “Student”. Often the t distribution is called Student’s t distribution. http://en.wikipedia.org/wiki/William Sealey Gosset K310 Statistical Techniques Inference for Distributions 3 / 26
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The t distribution Critical values for the t distribution are in Table D (p. T-11) Upper-tail probability p df .25 .20 .15 .10 .05 .025 .02 1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 . . . . . . . . . . . . . . . 1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 z * 0.674 0.841 1.036 1.282 1.645 1.960 2.054 50% 60% 70% 80% 90% 95% 96% Confidence level C K310 Statistical Techniques Inference for Distributions 4 / 26
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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 ¯ x ± t * s n where t * is the value for the t ( n - 1) density curve with area C between - t * and t * . Then, the margin of error is t * s n . K310 Statistical Techniques Inference for Distributions 5 / 26
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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
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One-sample t confidence interval example Measurements of vitamin C (mg/100 g) in a dry corn soy blend in one
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