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# Note 10 of 5E Statistics with Economics and Business Applications Chapter 7 Estimation of Means and Proportions Large-Sample Estimation.

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Note 10 of 5E Statistics with Economics and Business Applications Chapter 7 Estimation of Means and Proportions Large-Sample Estimation

Note 10 of 5E Review Review I. What’s in last lecture? Point Estimation Interval Estimation/Confidence Interval Chapter 7 II. What's in this lecture? Large-Sample Estimation Read Chapter 7

Note 10 of 5E The Margin of Error In this note we assume that the sample sizes are large approximately normal under certain assumptionsFrom the Central Limit Theorem, the sampling distributions of and will be approximately normal under certain assumptions For unbiasedFor unbiased estimators with normal sampling distributions, 95% of all point estimates will lie within 1.96 standard deviations of the parameter of interest. Margin of error: provides a upper bound to the difference between a particular estimate and the parameter that it estimates. It isMargin of error: provides a upper bound to the difference between a particular estimate and the parameter that it estimates. It is calculated as

Note 10 of 5E Estimating Means and Proportions For a quantitative population, For a binomial population,

Note 10 of 5E Example A homeowner randomly samples 64 homes similar to her own and finds that the average selling price is \$250,000 with a standard deviation of \$15,000. Estimate the average selling price for all similar homes in the city.

Note 10 of 5EExample A quality control technician wants to estimate the proportion of soda cans that are underfilled. He randomly samples 200 cans of soda and finds 10 underfilled cans.

Note 10 of 5E Interval Estimation/Confidence Interval Create an interval (a, b) so that you are fairly sure that the parameter lies between these two values. confidence coefficient, 1 “Fairly sure” means “with high probability”, measured using the confidence coefficient, 1  Usually, 1-  100(1-  )% Confidence Interval: Point Estimator  z  SE 100(1-  )% Confidence Interval: Point Estimator  z  SE For large-Sample size,

Note 10 of 5E To Change the Confidence Level To change to a general confidence level, 1- , pick a value of z that puts area 1-  in the center of the z distribution. Tail area  z  /2.051.645.0251.96.0052.58 95% of the intervals constructed in this manner will enclose the population mean Suppose 1-  =.95,

Note 10 of 5E Interval Estimation/Confidence Interval Since we don’t know the value of the parameter, consider which has a variable center. Only if the estimator falls in the tail areas will the interval fail to enclose the parameter. This happens only 5% of the time. Point Estimator  1.96SE Worked Failed

Note 10 of 5E Interpretation of A Confidence Interval A confidence interval is calculated from one given sample. It either covers or misses the true parameter. Since the true parameter is unknown, you'll never know which one is true. If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter. The confidence level associated with a confidence interval is the success rate of the confidence interval. SticiGui

Note 10 of 5E Confidence Intervals for Means and Proportions For a quantitative population, For a binomial population,

Note 10 of 5E Example A random sample of n = 50 males showed a mean average daily intake of dairy products equal to 756 grams with a standard deviation of 35 grams. Find a 95% confidence interval for the population average 

Note 10 of 5E Example Find a 99% confidence interval for  the population average daily intake of dairy products for men. The interval must be wider to provide for the increased confidence that is does indeed enclose the true value of .

Note 10 of 5E Example Of a random sample of n = 150 college students, 104 of the students said that they had played on a soccer team during their K-12 years. Estimate the proportion of college students who played soccer in their youth with a 90% confidence interval.

Note 10 of 5E Estimating the Difference between Two Means Sometimes we are interested in comparing the means of two populations. The average growth of plants fed using two different nutrients. The average scores for students taught with two different teaching methods. To make this comparison,

Note 10 of 5E Notations - Comparing Two Means MeanVarianceStandard Deviation Population 1 µ 1 σ 1 2 σ 1 Population 2 µ 2 σ 2 2 σ 2 Sample size MeanVarianceStandard Deviation Sample from Population 1 n 1 s 1 2 s 1 Sample from Population 2 n 2 s 2 2 s 2

Note 10 of 5E Estimating the Difference between Two Means We compare the two averages by making inferences about   -  , the difference in the two population averages. If the two population averages are the same, then  1 -   = 0. The best estimate of  1 -    is the difference in the two sample means,

Note 10 of 5E The Sampling Distribution of

Note 10 of 5E Estimating  1 -   For large samples, point estimates and their margin of error as well as confidence intervals are based on the standard normal (z) distribution.

Note 10 of 5EExample Compare the average daily intake of dairy products of men and women using a 95% confidence interval. Avg Daily IntakesMenWomen Sample size50 Sample mean756762 Sample Std Dev3530

Note 10 of 5E Example, continued Could you conclude, based on this confidence interval, that there is a difference in the average daily intake of dairy products for men and women?  1 -   = 0.  1 =  The confidence interval contains the value  1 -   = 0. Therefore, it is possible that  1 =   You would not want to conclude that there is a difference in average daily intake of dairy products for men and women.

Note 10 of 5E Estimating the Difference between Two Proportions Sometimes we are interested in comparing the proportion of “successes” in two binomial populations. The germination rates of untreated seeds and seeds treated with a fungicide. The proportion of male and female voters who favor a particular candidate for governor. To make this comparison,

Note 10 of 5E Notations - Comparing Two Proportions Sample size Sample Proportion Sample Variance Standard Deviation Sample from Population 1 n 1 Sample from Population 2 n 2

Note 10 of 5E Estimating the Difference between Two Means We compare the two proportions by making inferences about p  -p , the difference in the two population proportions. If the two population proportions are the same, then p 1 -p  = 0. The best estimate of p 1 -p   is the difference in the two sample proportions,

Note 10 of 5E The Sampling Distribution of

Note 10 of 5E Estimating p 1 -p  For large samples, point estimates and their margin of error as well as confidence intervals are based on the standard normal (z) distribution.

Note 10 of 5E Example Compare the proportion of male and female college students who said that they had played on a soccer team during their K-12 years using a 99% confidence interval. Youth SoccerMaleFemale Sample size8070 Played soccer6539

Note 10 of 5E Example, continued Could you conclude, based on this confidence interval, that there is a difference in the proportion of male and female college students who said that they had played on a soccer team during their K-12 years? p 1 -p  = 0.p 1 = p The confidence interval does not contains the value p 1 -p  = 0. Therefore, it is not likely that p 1 = p  You would conclude that there is a difference in the proportions for males and females. A higher proportion of males than females played soccer in their youth.

Note 10 of 5E Key Concepts I. Large-Sample Point Estimators To estimate one of four population parameters when the sample sizes are large, use the following point estimators with the appropriate margins of error.

Note 10 of 5E Key Concepts II. Large-Sample Interval Estimators To estimate one of four population parameters when the sample sizes are large, use the following interval estimators.

Note 10 of 5E Key Concepts 1.All values in the interval are possible values for the unknown population parameter. 2.Any values outside the interval are unlikely to be the value of the unknown parameter. 3.To compare two population means or proportions, look for the value 0 in the confidence interval. If 0 is in the interval, it is possible that the two population means or proportions are equal, and you should not declare a difference. If 0 is not in the interval, it is unlikely that the two means or proportions are equal, and you can confidently declare a difference.

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