Probability and Statistics Test tomorrow In Class 2-2:50 Eastern time

Prep for Exam 4 – MATH224

Please review the textbook and lecture notes and also practice old exams and lecture note problems along with the following topics:

Chapter 7 Variation in Repeated Samples – Sampling Distributions

The distribution of a statistic is called the sampling distribution. In Chapter 7, we learned the distribution of .

If , then the sampling distribution of regardless of the sample size .

If , then when is large () by the Central Limit Theorem (CLT). If is unknown, then the sample standard deviation (SD), , will be used when . The standard deviation of depends on the sample size, .

Ex. For , evaluate

Chapter 8 Drawing Inferences from Large Samples

Statistical inference: drawing conclusion about population parameters (using the statistic derived) from an analysis of the sample data. Statistical inference is carried out via point estimation, interval estimation, and testing hypothesis Population parameters: population mean: and population proportion: --- In general both parameters are unknown Point estimation: a point estimator of is (sample mean) and a point estimator of p is (sample proportion) The point estimator bears error margin () because it is obtained from random samples instead of the entire population where is called a confidence level. For . For

	80%	90%	95%	98%	99%
	.80	.90	.95	.98	.99
	.20	.10	.05	.02	.01
	1.28	1.645	1.96	2.33	2.58

Interval estimation: confidence interval (CI) for an unknown population parameter,, is ( – , + ). CI for an unknown population parameter, , is ( , ).

Interpretation. We are confident that the unknown population parameter is between the lower bound and the upper bound. Meaning. For a 95% CI, if we have 100 random samples, then about 95% CIs among 100 CIs contain the true population parameter.

The higher the standard deviation is, the wider the confidence interval is.

The larger the sample size is, the narrower the confidence interval is.

The higher the confidence level is, the wider the confidence interval is.

Testing hypothesis:

1. Set up the null and alternative hypotheses: or or

or or

2. Set the level of significance,

3. Calculate the test statistic: -statistic for , for

4. Calculate the P-value using the test statistic. Note that the P-value is a probability so that .

Left-sided:

Right-sided:

Two-sided:

5. Statistical decision: If the P-value , then we reject in favor of Otherwise (P-value ) , we do not reject The smaller the P-value, the stronger the evidence against

6. Think about the research conclusion based on your statistical conclusion. If you reject , then there is a statistical evidence to support your research claim (expectation). If you do not reject then there is no statistical evidence to support your claim.