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Chapter 9

Chapter 9 - Hypothesis Testing So far Ho = some value...

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Hypothesis Testing So far: H o : μ = ‘some value’ assume σ known Statistical Test: z -test

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Chapter 9 Inferences Involving One Population
Today 1. Confidence Interval for μ when σ is NOT known 2. Hypothesis test: μ = ‘some value’ when σ is NOT known 3. Confidence Interval for proportion = p 4. Hypothesis test: Comparing ( p ) to ‘some value’ 5. Hypothesis test: σ = ‘some value’ Why are these all considered 1 population tests?

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Finally time to consider: σ NOT known We used the z table when the population standard deviation ( σ ) is known: If σ not known, we have to use the sample standard deviation ( s ) to calculate the test statistic t* is the Student’s t -statistic * - = n x z σ μ n s x t μ - = *
Question about μ σ Known σ Not Known If Normality can’t be assumed, must use a NONPARAMETRIC Test (not covered in this class)

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0 t Student’s t , Student’s t , Normal distribution Student’s t -distribution matches normal distribution ( z -distribution) when n is large . When n is smaller -- the t -distribution is less peaked and more spread out
To test hypotheses, we need to find and critical t values just like for the z- distribution Notation: t (df, α ) Read as: t of df, α t 0 ) df, α ( t α One-tailed test 2 tailed test α 2 α 2 -t (df, α/2) +t (df, α/2)

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Table 6 deals with both 1 and 2 tailed alternative hypotheses A -t value means you are in the left tail but since the t -distribution is symmetric about mean, table only gives + t values What is the 2-tailed t value for t (26,0.01) ?
Confidence Interval Procedure : Use t in place of z, and use s in place of σ The formula for the 1 -α confidence interval for μ is: 1 df where t (df, α/229 to - = + - n n s x n s x t (df, α/229 Still assumes the sampled population is normally distributed OR that the sample size is large (i.e., SDSM is normally distributed) So…now we can calculate confidence intervals for μ using sample data

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Example: You conduct a study measuring the size of milkweed beetles in Maryland. From your sample data, you calculate a sample mean of 7.8 cm and a sample standard deviation of 2.3 cm based on 17 beetles. Find a 95% confidence interval for the population mean milkweed beetle size. Assume the size of milkweed beetles is normally distributed. Example
3 . 2 and , 8 . 7 , 17 = = = s x n The Sample Evidence : The Statement : Remember: df = n -1

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1. The t -statistic is used to complete a hypothesis test about a population mean when σ is not known 2. Hypothesis-Testing when σ NOT known 1 df with * - = - = n n s x t μ 2. The test statistic: x 3. The calculated t is the number of estimated standard errors is from the hypothesized mean μ
Example: You are interested in the size of the milkweed leaves on which you find your beetles in Western Maryland. For a random sample of 25 leaves you find the mean length is 22.6 cm and the standard deviation was 8.0 cm. Is there any evidence to support the your advisor’s claim that the average leaf length for beetle-containing leaves is more than 20 cm? Use α = 0.05 and assume leaf length is approximately normal distributed.

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Chapter 9 - Hypothesis Testing So far Ho = some value...

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