Econ 3790: Business and Economics Statistics. Instructor: Yogesh Uppal

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1 Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal

2 Chapter 8: Interval Estimation Population Mean: Known Population Mean: Unknown

3 Margin of Error and the Interval Estimate A point estimator cannot be expected to provide the exact value of the population parameter. An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. Point Estimate +/ Margin of Error The purpose of an interval estimate is is to provide information about how close the point estimate is is to the value of the parameter.

4 Margin of Error and the Interval Estimate The general form of an interval estimate of a population mean is is x Margin of Error In order to develop an interval estimate of a population mean, the margin of error must be computed using either: the population standard deviation, or the sample standard deviation s These are also Confidence Interval.

5 Interval Estimate of a Population Mean: Known Interval Estimate of x z / 2 n x where: is the sample mean 1 - is the confidence coefficient z /2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution is the population standard deviation n is the sample size

6 Interval Estimation of a Population Mean: Known There is a 1 probability that the value of a sample mean will provide a margin of error of z or less. Sampling distribution of x /2 1 - of all /2 x values /2 x z /2 x z /2 x x

7 Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter Parameter Value Point Estimator Point Estimate = Population mean 40.9 x = Sample mean = Population std. deviation 20.5 s = Sample std. deviation. p = Population pro- portion.62 p = Sample pro- portion

8 Example: Air Quality Consider our air quality example. Suppose the population is approximately normal with μ = 40.9 and σ = This is σ known case. If you guys remember, we picked a sample of size 5 (n =5). Given all this information, What is the margin of error at 95% confidence level?

9 Example: Air Quality What is the margin of error at 95% confidence level z n 5 We can say with 95% confidence that population mean (μ) is between ± 18 of the sample mean. x z n With 95% confidence, μ is between. and...

10 Interval Estimation of a Population Mean: Unknown If an estimate of the population standard deviation cannot be developed prior to sampling, we use the sample standard deviation s to estimate. This is the unknown case. In this case, the interval estimate for is based on the t distribution. (We ll assume for now that the population is normally distributed.)

11 t Distribution The t distribution is is a family of similar probability distributions. A specific t distribution depends on a parameter known as the degrees of freedom. Degrees of freedom refer to the number of independent pieces of information that go into the computation of s. s.

12 t Distribution A t distribution with more degrees of freedom has less dispersion. As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller.

13 t Distribution Standard normal distribution t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) 0 z, t

14 t Distribution For more than 100 degrees of freedom, the standard normal z value provides a good approximation to the t value. The standard normal z values can be found in the infinite degrees ( ) row of the t distribution table.

15 t Distribution Degrees Area in Upper Tail of Freedom Standard normal z values

16 Interval Estimation of a Population Mean: Unknown Interval Estimate s x t / 2 n where: 1 - = the confidence coefficient t /2 = the t value providing an area of /2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation

17 Example: Air quality when σ is unknown Now suppose that you did not know what σ is. You can estimate using the sample and then use t-distribution to find the margin of error. What is 95% confidence interval in this case? The sample size n =5. So, the degrees of freedom for the t-distribution is 4. The level of significance ( ) is s = x t / 2 s n

18 Summary of Interval Estimation Procedures for a Population Mean x Yes Use z z /2 Known Case n Can the population standard deviation be assumed known? Unknown Case No Use the sample standard deviation s to estimate σ Use s x t /2 n

19 Interval Estimation of a Population Proportion The general form of an interval estimate of a population proportion is is p Margin of Error

20 Interval Estimation of a Population Proportion Interval Estimate p z / 2 p ( 1 p ) n where: 1 - is the confidence coefficient z /2 is the z value providing an area of /2 in the upper tail of the standard normal probability distribution p is the sample proportion

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