# VCU STAT 210 - Lecture32 (57 pages)

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## Lecture32

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- Pages:
- 57
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 32 Confidence Intervals for m When s is Known November 10 2017 Practice Problems Sailboat Pages 220 through 230 Relevant problems VIII 7 VIII 10 and VIII 15 a and b Recommended problems VIII 8 and VIII 10 Hummingbird Pages 250 through 260 Relevant problems IX 7 IX 10 and IX 15 a and b Recommended problems IX 8 and IX 10 Additional Reading and Examples Sailboat Read pages 213 218 Pay special attention to pages 214 215 Hummingbird Read page 243 248 Pay special attention to pages 244 245 Top Hat Inference Statistical inference involves using statistics computed from data collected in a sample to make statements inferences about unknown population parameters In this chapter we will discuss statistical inferences confidence intervals and statistical tests for the population mean m Top Hat Confidence Intervals To estimate m we will select a simple random sample from the population and compute the sample mean X for the data in the sample This sample mean X will be the point estimate of m To this point estimate we will subtract and add a margin of error creating an interval of values that we hope the unknown population mean m is between This interval is referred to as a confidence interval Confidence Intervals The term confidence refers to the amount of confidence that we have that our interval will contain m Since m is unknown we will never know for sure whether the interval contains it or not but we typically choose a confidence level that is relatively high such as 90 95 98 and 99 so that our confidence of success is high Note the only way to have 100 confidence is to actually know the value of m Confidence Intervals Both the confidence interval application discussed today and the confidence interval application discussed in the next lecture will be based on the sampling distribution theory of X derived in the last lecture and hence both will require the following two assumptions Confidence Intervals Both the confidence interval application discussed today and the confidence interval application discussed in the next lecture will be based on the sampling distribution theory of X derived in the last lecture and hence both will require the following two assumptions 1 We must have a simple random sample from the population 2 The population must be normal or the sample size must be large enough for the central limit theorem to apply Confidence Interval for m s Known Assumptions 1 We have a simple random sample from the population 2 Either i the population is normal or ii the sample size is large enough for the central limit theorem to apply 3 The population standard deviation s is known Confidence Interval for m s Known 1 We have a simple random sample from the population 2 Either i the population is normal or ii the sample size is large enough for the central limit theorem to apply 3 The population standard deviation s is known With these assumptions the sampling distribution of X is X N m s n Confidence Interval for m s Known So X N m s n We can then apply the Z score transformation to create the statistic X m Z s n That has a standard normal distribution Z N 0 1 Confidence Interval for m s Known So X N m s n We can then apply the Z score transformation X m Z s n Solving this for the unknown population mean m yields the following confidence interval formula X Z s n Confidence Interval for m s Known Solving this for the unknown population mean m yields the following confidence interval formula X Z s n In this expression the symbol means subtract and add We can get a lower limit L X Z s n and an upper limit U X Z s n The quantity Z s n is the margin of error Confidence Interval for m s Known X Z s n The value of Z depends on the amount of confidence stated and is determined from the t table on page 340 One looks up the confidence level across the bottom row and then reads the Z value from the row directly above Z Confidence level Confidence Interval for m s Known X Z s n The value of Z depends on the amount of confidence stated and is determined from the t table on page 340 The most common values are as follows 90 CI Z 1 645 98 CI Z 2 326 95 CI Z 1 960 99 CI Z 2 576 Confidence Interval for m s Known X Z s n The interpretation of the confidence interval is the statistical inference and should be stated as follows We have 100 C confidence that the population mean m falls between the lower limit X Z s n and the upper limit X Z s n Confidence Interval for m s Known X Z s n For example if estimating m the mean age of all students at the university with a 95 confidence interval and if the lower limit is 20 5 and the upper limit is 28 5 then we write We have 95 confidence that the mean age of all students at this university falls between 20 5 and 28 5 Confidence Interval for m s Known X Z s n We have 95 confidence that the mean age of all students at this university falls between 20 5 and 28 5 In writing an interpretation we do not use the term probability confidence instead we do not talk about the sample mean population mean instead and we do not talk about individual values TI 83 84 Calculator See page 220 for instructors for using a calculator to construct confidence intervals based on a Zdistribution Select STAT Choose TESTS Choose option 7 ZInterval Next to Inpt choose Stats Enter values and hit Calculate Example 71 84 Population of interest Parameter of interest Example 71 84 Population of interest All VCU employees Parameter of interest m mean salary of all VCU employees Top Hat Example 71 84 m mean salary of all VCU employees n X s Example 71 84 m mean salary of all VCU employees n 49 X 42000 s standard deviation of all employees 6000 Example 71 84 m mean salary of all VCU employees n 49 X 42000 s standard deviation of all employees 6000 Assumptions 1 Simple random sample Example 71 84 m mean salary of all VCU employees n 49 X 42000 s standard deviation of all employees 6000 Assumptions 1 Simple random sample YES Example 71 84 m mean salary of all VCU employees n 49 X 42000 s standard deviation of all employees 6000 Assumptions 1 Simple random sample 2 Normal population Example 71 84 m mean salary of all VCU employees n 49 X 42000 s standard deviation of all employees 6000 Assumptions 1 Simple random sample 2 Normal population NO Example 71 84 m mean salary of all VCU employees n 49 X 42000 s standard deviation of all employees 6000 Assumptions 1 Simple random sample 2 Large …

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