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STAT 312 Chapter 7 - Statistical Intervals Based on a Single Sample
7.1 - Basic Properties of Confidence Intervals 7.2 - Large-Sample Confidence Intervals for a Population Mean and 7.3 - Intervals Based on a Normal Population Distribution 7.4 - Confidence Intervals for the Variance and Standard Deviation of a Normal Pop Proportion Chapter 8 - Tests of Hypotheses Based on a Single Sample 8.1 - Hypotheses and Test Procedures 8.2 - Z-Tests for Hypotheses about a Population Mean 8.3 - The One-Sample T-Test 8.4 - Tests Concerning a Population Proportion 8.5 - Further Aspects of Hypothesis Testing
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Sampling Distribution
POPULATION X = random variable, numerical (discrete or continuous) X ~ Dist(, ) = mean 2 = variance Parameter Estimation Parameters RANDOM SAMPLE size n Estimator Sampling Distribution Mean Variance
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Discrete random variable
POPULATION POPULATION Success Failure For any randomly selected individual, first define a binary random variable: Parameter Estimator = ? Parameter RANDOM SAMPLE size n Discrete random variable X = # Successes in sequence of n Bernoulli trials (0, …, n) Sampling Distribution Sampling Distribution
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Discrete random variable
POPULATION POPULATION Success Failure For any randomly selected individual, first define a binary random variable: Parameter Estimator = ? RANDOM SAMPLE size n Discrete random variable X = # Successes in sequence of n Bernoulli trials (0, …, n) If n 15 and n (1 – ) 15, then via the Normal Approximation to the Binomial… Sampling Distribution Sampling Distribution
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Discrete random variable
POPULATION POPULATION Success Failure For any randomly selected individual, first define a binary random variable: Parameter Estimator = ? RANDOM SAMPLE size n Discrete random variable X = # Successes in sequence of n Bernoulli trials (0, …, n) If n 15 and n (1 – ) 15, then via the Normal Approximation to the Binomial… s.e. DOES depend on s.e. does not depend on Sampling Distribution Sampling Distribution
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Sampling Distribution
Example Null Distribution Sampling Distribution
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Example Null Distribution
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Example Null Hypothesis Alternative Hypothesis Sample n = 100
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point estimate of true
Sample n = 100 X = 10 Example Null Hypothesis Alternative Hypothesis point estimate of true 95% Margin of Error 95% Confidence Interval (for ) = does not contain null value = 0.2 Reject at = .05 Statistical significance at = .05… Evidence that < 0.2, based on study. .04 .16
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point estimate of true
Sample n = 100 X = 10 Note that the textbook gives and justifies a different formula that is more precise. However, the two give approximately equal results when n is large; see pages (You are not responsible for this.) Example Null Hypothesis Alternative Hypothesis point estimate of true 95% Margin of Error 95% Confidence Interval (for ) = does not contain null value = 0.2 Reject at = .05 Statistical significance at = .05… Evidence that < 0.2, based on study. .04 .16
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point estimate of true
Example Null Hypothesis Alternative Hypothesis Sample n = 100 X = 10 Example Null Hypothesis Alternative Hypothesis point estimate of true 95% Margin of Error 95% Acceptance Region (for H0) = 95% Confidence Interval (for ) = does not contain null value = 0.2 Reject at = .05 Statistical significance at = .05… Evidence that < 0.2, based on study. .04 .16
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point estimate of true
Example Null Hypothesis Alternative Hypothesis Sample n = 100 X = 10 Example Null Hypothesis Alternative Hypothesis point estimate of true 95% Margin of Error 95% Acceptance Region (for H0) = does not contain point estimate = 0.1 Reject at = .05 does not contain null value = 0.2 Reject at = .05 Statistical significance at = .05… Evidence that < 0.2, based on study. .12 .28
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point estimate of true
Example Null Hypothesis Alternative Hypothesis Sample n = 100 X = 10 Example Null Hypothesis Alternative Hypothesis point estimate of true p-value = Reject at = .05, etc. .12 .28
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