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week10.pptx4-1 Statistical Inference
The field of statistical inference consists of those
methods used to make decisions or draw
conclusions about a population.
These methods utilize the information contained in
a sample from the population in drawing
conclusions.
4-1 Statistical Inference
4-2 Point Estimation
4-2 Point Estimation
4-2 Point Estimation
4-3 Hypothesis Testing
We like to think of statistical hypothesis testing as the data analysis stage of a comparative experiment, in which the engineer is interested, for example, in comparing the mean of a population to a specified value (e.g. mean pull strength).
4-3.1 Statistical Hypotheses
4-3 Hypothesis Testing
For example, suppose that we are interested in the burning rate of a solid propellant used to power aircrew escape systems.
Now burning rate is a random variable that can be described by a probability distribution.
Suppose that our interest focuses on the mean burning rate (a parameter of this distribution).
Specifically, we are interested in deciding whether or not the mean burning rate is 50 centimeters per second.
4-3.1 Statistical Hypotheses
4-3 Hypothesis Testing
4-3.1 Statistical Hypotheses
Two-sided Alternative Hypothesis
One-sided Alternative Hypotheses
4-3 Hypothesis Testing
4-3.1 Statistical Hypotheses
Test of a Hypothesis
A procedure leading to a decision about a particular hypothesis
Hypothesis-testing procedures rely on using the information in a random sample from the population of interest.
If this information is consistent with the hypothesis, then we will conclude that the hypothesis is true; if this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false.
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
Sometimes the type I error probability is called the significance level, or the -error, or the size of the test.
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
4-3 Hypothesis Testing
4-3.2 Testing Statistical Hypotheses
The power is computed as 1 - b, and power can be interpreted as the probability of correctly rejecting a false null hypothesis. We often compare statistical tests by comparing their power properties.
For example, consider the propellant burning rate problem when
we are testing H 0 : m = 50 centimeters per second against H 1 : m not equal 50 centimeters per second . Suppose that the true value of the mean is m = 52. When n = 10, we found that b = 0.2643, so the power of this test is 1 - b = 1 - 0.2643 = 0.7357 when m = 52.
4-3 Hypothesis Testing
4-3.3 One-Sided and Two-Sided Hypotheses
Two-Sided Test:
One-Sided Tests:
4-3 Hypothesis Testing
4-3.3 P-Values in Hypothesis Testing
4-3 Hypothesis Testing
4-3.5 General Procedure for Hypothesis Testing
4-4 Inference on the Mean of a Population,
Variance Known
Assumptions
4-4 Inference on the Mean of a Population,
Variance Known
4-4.1 Hypothesis Testing on the Mean
We wish to test:
The test statistic is:
4-4 Inference on the Mean of a Population,
Variance Known
4-4.1 Hypothesis Testing on the Mean
Reject H0 if the observed value of the test statistic z0 is either:
or
Fail to reject H0 if
4-4 Inference on the Mean of a Population,
Variance Known
4-4.1 Hypothesis Testing on the Mean
4-4 Inference on the Mean of a Population,
Variance Known
4-4.1 Hypothesis Testing on the Mean
4-4 Inference on the Mean of a Population,
Variance Known
4-4.2 Type II Error and Choice of Sample Size
Finding The Probability of Type II Error
4-4 Inference on the Mean of a Population,
Variance Known
4-4.2 Type II Error and Choice of Sample Size
Sample Size Formulas
4-4 Inference on the Mean of a Population,
Variance Known
4-4.2 Type II Error and Choice of Sample Size
Sample Size Formulas
4-4 Inference on the Mean of a Population,
Variance Known
4-4.2 Type II Error and Choice of Sample Size
4-4 Inference on the Mean of a Population,
Variance Known
4-4.2 Type II Error and Choice of Sample Size
4-4 Inference on the Mean of a Population,
Variance Known
4-4.3 Large Sample Test
In general, if n 30, the sample variance s2 will be close to σ2 for most samples, and so s can be substituted for σ in the test procedures with little harmful effect.
4-4 Inference on the Mean of a Population,
Variance Known
4-4.4 Some Practical Comments on Hypothesis
Testing
The Seven-Step Procedure
Only three steps are really required:
4-4 Inference on the Mean of a Population,
Variance Known
4-4.4 Some Practical Comments on Hypothesis
Testing
Statistical versus Practical Significance
4-4 Inference on the Mean of a Population,
Variance Known
4-4.4 Some Practical Comments on Hypothesis
Testing
Statistical versus Practical Significance
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
Two-sided confidence interval:
One-sided confidence intervals:
Confidence coefficient:
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
4-4 Inference on the Mean of a Population,
Variance Known
4-4.6 Confidence Interval on the Mean
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
Relationship between Tests of Hypotheses and Confidence Intervals
If [l,u] is a 100(1 - a) percent confidence interval for the parameter, then the test of significance level a of the hypothesis
will lead to rejection of H0 if and only if the hypothesized value is not in the 100(1 - a) percent confidence interval [l, u].
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
Confidence Level and Precision of Estimation
The length of the two-sided 95% confidence interval is
whereas the length of the two-sided 99% confidence
interval is
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
Choice of Sample Size
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
Choice of Sample Size
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
Choice of Sample Size
4-4 Inference on the Mean of a Population,
Variance Known
4-4.5 Confidence Interval on the Mean
One-Sided Confidence Bounds
4-4 Inference on the Mean of a Population,
Variance Known
4-4.6 General Method for Deriving a Confidence Interval
