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**Chapter 8 Estimation Understandable Statistics Ninth Edition**

By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania

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**Estimating µ When σ is Known**

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Point Estimate An estimate of a population parameter given by a single number.

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Margin of Error Even if we take a very large sample size, will differ from µ.

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Confidence Levels A confidence level, c, is any value between 0 and 1 that corresponds to the area under the standard normal curve between –zc and +zc.

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Critical Values

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**Common Confidence Levels**

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**Recall From Sampling Distributions**

If we take samples of size n from our population, then the distribution of the sample mean has the following characteristics:

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**A Probability Statement**

In words, c is the probability that the sample mean will differ from the population mean by at most

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**Maximal Margin of Error**

Since µ is unknown, the margin of error | µ| is unknown. Using confidence level c, we can say that differs from µ by at most:

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Confidence Intervals

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**Critical Thinking Since is a random variable, so are the endpoints**

After the confidence interval is numerically fixed for a specific sample, it either does or does not contain µ.

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Critical Thinking If we repeated the confidence interval process by taking multiple random samples of equal size, some intervals would capture µ and some would not! Equation states that the proportion of all intervals containing µ will be c.

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**Multiple Confidence Intervals**

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**Estimating µ When σ is Unknown**

In most cases, researchers will have to estimate σ with s (the standard deviation of the sample). The sampling distribution for will follow a new distribution, the Student’s t distribution.

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The t Distribution

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The t Distribution

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The t Distribution Use Table 6 of Appendix II to find the critical values tc for a confidence level c. The figure to the right is a comparison of two t distributions and the standard normal distribution.

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**Using Table 6 to Find Critical Values**

Degrees of freedom, df, are the row headings. Confidence levels, c, are the column headings.

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**Maximal Margin of Error**

If we are using the t distribution:

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**What Distribution Should We Use?**

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**Estimating p in the Binomial Distribution**

We will use large-sample methods in which the sample size, n, is fixed. We assume the normal curve is a good approximation to the binomial distribution if both np > 5 and nq = n(1-p) > 5.

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**Point Estimates in the Binomial Case**

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Margin of Error The magnitude of the difference between the actual value of p and its estimate is the margin of error.

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The Distribution of The distribution is well approximated by a normal distribution.

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**A Probability Statement**

With confidence level c, as before.

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Public Opinion Polls

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Choosing Sample Sizes When designing statistical studies, it is good practice to decide in advance: The confidence level The maximal margin of error Then, we can calculate the required minimum sample size to meet these goals.

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**Sample Size for Estimating μ**

If σ is unknown, use σ from a previous study or conduct a pilot study to obtain s. Always round n up to the next integer!!

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**Sample Size for Estimating**

If we have no preliminary estimate for p, use the following modification:

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Independent Samples Two samples are independent if sample data drawn from one population is completely unrelated to the selection of a sample from the other population. Occurs when we draw two random samples

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Dependent Samples Two samples are dependent if each data value in one sample can be paired with a corresponding value in the other sample. Occur naturally when taking the same measurement twice on one observation Example: your weight before and after the holiday season.

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**Confidence Intervals for μ1 – μ2 when σ1, σ2 known**

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**Confidence Intervals for μ1 – μ2 when σ1, σ2 known**

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**Confidence Intervals for μ1 – μ2 when σ1, σ2 unknown**

If σ1, σ2 are unknown, we use the t distribution (just like the one-sample problem).

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What if σ1 = σ2 ? If the sample standard deviations s1 and s2 are sufficiently close, then it may be safe to assume that σ1 = σ2. Use a pooled standard deviation. See Section 8.4, problem 27.

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**Summarizing Intervals for Differences in Population Means**

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**Estimating the Difference in Proportions**

We consider two independent binomial distributions. For distribution 1 and distribution 2, respectively, we have: n1 p1 q1 r1 n2 p2 q2 r2 We assume that all the following are greater than 5:

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**Estimating the Difference in Proportions**

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Critical Thinking

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