Fundamentals of Business Statistics chapter7 Sampling and Sampling Distributions 上海金融学院统计系 Statistics Dept., Shanghai Finance University.

Fundamentals of Business Statistics chapter7 Sampling and Sampling Distributions 上海金融学院统计系 Statistics Dept., Shanghai Finance University

上海金融学院上海金融学院 Chapter Goals After completing this chapter, you should be able to: Understand the importance of sampling and how results from samples can be used to provide estimates of population characteristics:  the population mean  the population standard deviation, and  the population proportion. Know what simple random sampling is and how simple random samples are selected. Understand the concept of a sampling distribution. Understand the central limit theorem and the important role it plays in sampling. Specifically know the characteristics of the sampling distribution of the sample mean and the sampling distribution of the sample proportion.

上海金融学院上海金融学院 Chapter Goals After completing this chapter, you should be able to: Know the definition of the following terms:  parameter  sampling distribution  sample statistic  finite population correction factor  simple random sampling  standard error  sampling without replacement  central limit theorem  sampling with replacement  unbiased  point estimator  relative efficiency  point estimate  consistency

上海金融学院上海金融学院 Section 7.1 The Electronics Associates Sampling Problem

上海金融学院上海金融学院

上海金融学院上海金融学院 Example: St. Andrew’s St. Andrew ’ s College receives 900 applications annually from prospective students. The application form contains a variety of information including the individual ’ s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus housing.

上海金融学院上海金融学院 The purpose of statistical inference is to obtain The purpose of statistical inference is to obtain information about a population from information information about a population from information contained in a sample. contained in a sample. The purpose of statistical inference is to obtain The purpose of statistical inference is to obtain information about a population from information information about a population from information contained in a sample. contained in a sample. Statistical Inference A population is the set of all the elements of interest. A sample is a subset of the population. The sample results provide only estimates of the The sample results provide only estimates of the values of the population characteristics. values of the population characteristics. The sample results provide only estimates of the The sample results provide only estimates of the values of the population characteristics. values of the population characteristics. A parameter is a numerical characteristic of a A parameter is a numerical characteristic of a population. population. A parameter is a numerical characteristic of a A parameter is a numerical characteristic of a population. population. With proper sampling methods, the sample results With proper sampling methods, the sample results can provide “good” estimates of the population can provide “good” estimates of the population characteristics. characteristics. With proper sampling methods, the sample results With proper sampling methods, the sample results can provide “good” estimates of the population can provide “good” estimates of the population characteristics. characteristics.

上海金融学院上海金融学院

上海金融学院上海金融学院 Example: St. Andrew’s The director of admissions would like to know the following information: the average SAT score for the 900 applicants, and the proportion of applicants that want to live on campus.

上海金融学院上海金融学院 Example: St. Andrew’s We will now look at three alternatives for obtaining the desired information.  Conducting a census of the entire 900 applicants  Selecting a sample of 30 applicants, using a random number table  Selecting a sample of 30 applicants, using Excel

上海金融学院上海金融学院 Conducting a Census n If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3. n We will assume for the moment that conducting a census is practical in this example.

上海金融学院上海金融学院 Conducting a Census  Population Mean SAT Score  Population Standard Deviation for SAT Score  Population Proportion Wanting On-Campus Housing

上海金融学院上海金融学院 Simple Random Sampling The applicants were numbered, from 1 to 900, as The applicants were numbered, from 1 to 900, as their applications arrived. their applications arrived. She decides a sample of 30 applicants will be used. She decides a sample of 30 applicants will be used. Furthermore, the Director of Admissions must obtain Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for estimates of the population parameters of interest for a meeting taking place in a few hours. a meeting taking place in a few hours. Now suppose that the necessary data on the Now suppose that the necessary data on the current year’s applicants were not yet entered in the current year’s applicants were not yet entered in the college’s database. college’s database.

上海金融学院上海金融学院 Section 7.2 Simple Random Sampling  Sampling from a Finite Population  Sampling from an Infinite Population

上海金融学院上海金融学院  Taking a Sample of 30 Applicants Simple Random Sampling: Using a Random Number Table We will use the last three digits of the 5-digit We will use the last three digits of the 5-digit random numbers in the third column of the random numbers in the third column of the textbook’s random number table, and continue textbook’s random number table, and continue into the fourth column as needed. into the fourth column as needed. Because the finite population has 900 elements, we Because the finite population has 900 elements, we will need 3-digit random numbers to randomly will need 3-digit random numbers to randomly select applicants numbered from 1 to 900. select applicants numbered from 1 to 900.

上海金融学院上海金融学院  Taking a Sample of 30 Applicants Simple Random Sampling: Using a Random Number Table We will go through all of column 3 and part of We will go through all of column 3 and part of column 4 of the random number table, encountering in the process five numbers greater than 900 and one duplicate, 835. We will continue to draw random numbers until We will continue to draw random numbers until we have selected 30 applicants for our sample. we have selected 30 applicants for our sample. The numbers we draw will be the numbers of the applicants we will sample unless The numbers we draw will be the numbers of the applicants we will sample unless the random number is greater than 900 or the random number is greater than 900 or the random number has already been used. the random number has already been used.

上海金融学院上海金融学院  Use of Random Numbers for Sampling Simple Random Sampling: Using a Random Number Table 744 436 865 790 835 902 190 836... and so on 3-Digit 3-Digit Random Number Applicant Included in Sample No. 436 No. 865 No. 790 No. 835 Number exceeds 900 No. 190 No. 836 No. 744

上海金融学院上海金融学院  Sample Data Simple Random Sampling: Using a Random Number Table 1744 Conrad Harris1025 Yes 2436 Enrique Romero 950 Yes 3865 Fabian Avante1090 No 4790 Lucila Cruz1120 Yes 5835 Chan Chiang 930 No..... 30 498 Emily Morse 1010 No No. RandomNumber Applicant SAT Score Score Live On- Campus.....

上海金融学院上海金融学院  Taking a Sample of 30 Applicants Then we choose the 30 applicants as our sample. Then we choose the 30 applicants as our sample. For example, Excel’s function For example, Excel’s function = RANDBETWEEN(1,900) = RANDBETWEEN(1,900) can be used to generate random numbers between can be used to generate random numbers between 1 and 900. 1 and 900. Computers can be used to generate random Computers can be used to generate random numbers for selecting random samples. numbers for selecting random samples. Simple Random Sampling: Using a Computer

上海金融学院上海金融学院  as Point Estimator of   as Point Estimator of p Point Estimation Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.  s as Point Estimator of 

上海金融学院上海金融学院 PopulationParameter PointEstimator PointEstimateParameterValue  = Population mean SAT score SAT score 990997  = Population std. deviation for deviation for SAT score SAT score 80 s = Sample std. s = Sample std. deviation for deviation for SAT score SAT score75.2 p = Population proportion wanting portion wanting campus housing campus housing.72.68 Summary of Point Estimates Obtained from a Simple Random Sample = Sample pro- = Sample proportion wanting portion wanting campus housing campus housing = Sample mean = Sample mean SAT score SAT score

上海金融学院上海金融学院 Simple Random Sampling: Finite Population Finite populations are often defined by lists such as: Organization membership roster Credit card account numbers Inventory product numbers n A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

上海金融学院上海金融学院 Simple Random Sampling: Finite Population In large sampling projects, computer-generated In large sampling projects, computer-generated random numbers are often used to automate the random numbers are often used to automate the sample selection process. sample selection process. Sampling without replacement is the procedure Sampling without replacement is the procedure used most often. used most often. Replacing each sampled element before selecting Replacing each sampled element before selecting subsequent elements is called sampling with subsequent elements is called sampling with replacement. replacement.

9813562496218316 9512301622865263 155854125180591 2499424629202 197479751577433 Random numbers between 1 to 100 (Excel) Using Excel function: =RANDBETWEEN(bottom,top) Such as: =RANDBETWEEN(1,100)

上海金融学院上海金融学院 Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely. Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely. Simple Random Sampling: Infinite Population n A simple random sample from an infinite population is a sample selected such that the following conditions is a sample selected such that the following conditions are satisfied. are satisfied. Each element selected comes from the same population. Each element selected comes from the same population. Each element is selected independently. Each element is selected independently.

上海金融学院上海金融学院 Simple Random Sampling: Infinite Population The random number selection procedure cannot be The random number selection procedure cannot be used for infinite populations. used for infinite populations. In the case of infinite populations, it is impossible to In the case of infinite populations, it is impossible to obtain a list of all elements in the population. obtain a list of all elements in the population.

上海金融学院上海金融学院 Section 7.3 Point Estimation

上海金融学院上海金融学院 In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. Point Estimation is the point estimator of the population proportion p. is the point estimator of the population proportion p. s is the point estimator of the population standard deviation. s is the point estimator of the population standard deviation. We refer to as the point estimator of the population mean. We refer to as the point estimator of the population mean.   

上海金融学院上海金融学院 Sampling Error  Sample Statistics are used to estimate Population Parameters ex: X is an estimate of the population mean μ  Problems: Different samples provide different estimates of the population parameter. Sample results have potential variability, thus sampling error exits.

上海金融学院上海金融学院 Sampling Error Statistical methods can be used to make probability statements about the size of the sampling error. Statistical methods can be used to make probability statements about the size of the sampling error. Sampling error is the result of using a subset of the population (the sample), and not the entire population. Sampling error is the result of using a subset of the population (the sample), and not the entire population. The absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. The absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased. When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased.

上海金融学院上海金融学院 Example: (for the mean) Sampling Error  Sampling Error The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population where

上海金融学院上海金融学院 If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is n The sampling errors are: for sample proportion for sample standard deviation for sample mean Sampling Error

上海金融学院上海金融学院  Different samples will yield different sampling errors. Sampling Error  The sampling error may be positive or negative. ( may be greater than or less than μ)  The expected sampling error decreases as the sample size increases.

上海金融学院上海金融学院 Section 7.4 Introduction to Sampling Distributions

上海金融学院上海金融学院 Sampling Distribution  A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population. Population size N=4 Random variable, x, is age of individuals Values of x: 18, 20, 22, 24 (years) A B C D Assume there is a population :

上海金融学院上海金融学院.3.2.1 0 18 20 22 24 A B C D Uniform Distribution P(x) x (continued) Summary Measures for the Population Distribution: Sampling Distribution

16 possible samples (sampling with replacement) Now consider all possible samples of size: n=2 16 Sample Means Sampling Distribution

上海金融学院上海金融学院 Sampling Distribution of All Sample Means 18 19 20 21 22 23 24 0.1.2.3 P(x) x Sample Means Distribution 16 Sample Means _ (continued) (no longer uniform) Sampling Distribution

上海金融学院上海金融学院 Summary Measures of this Sampling Distribution: (continued)

上海金融学院上海金融学院 Comparing the Population with its Sampling Distribution 18 19 20 21 22 23 24 0.1.2.3 P(x) x 18 20 22 24 A B C D 0.1.2.3 Population N = 4 P(x) x _ Sample Means Distribution n = 2

上海金融学院上海金融学院 Section 7.5 Sampling Distribution of  Expected Value of  Standard Deviation of  Form of the Sampling Distribution of  Sampling Distribution of for the EAI Problem  Practical Value of the Sampling Distribution of  Relationship Between the Sampling Size and the Sampling Distribution of

上海金融学院上海金融学院  Process of Statistical Inference The value of is used to make inferences about the value of . The sample data provide a value for the sample mean. A simple random sample of n elements is selected from the population. Population with mean  = ? Sampling Distribution of

上海金融学院上海金融学院 The sampling distribution of is the probability distribution of all possible values of the sample mean. where:  = the population mean  = the population mean E ( ) =  Expected Value of Sampling Distribution of

上海金融学院上海金融学院 Finite Population Infinite Population is referred to as the standard error of the is referred to as the standard error of the mean. mean. A finite population is treated as being A finite population is treated as being infinite if n / N <.05. infinite if n / N <.05. is the finite correction factor. is the finite correction factor. Sampling Distribution of Standard Deviation of

上海金融学院上海金融学院 z-value for Sampling Distribution of x  Z-value for the sampling distribution of : where: = sample mean = population mean = population standard deviation n = sample size

上海金融学院上海金融学院 Finite Population Correction  Apply the Finite Population Correction if: the sample is large relative to the population (n is greater than 5% of N) and … Sampling is without replacement Then

上海金融学院上海金融学院 Form of the Sampling Distribution of If we use a large ( n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of can be approximated by a normal distribution. When the simple random sample is small ( n < 30), the sampling distribution of can be considered normal only if we assume the population has a normal distribution. If the Population is Normal If the Population is not Normal

上海金融学院上海金融学院 If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and

上海金融学院上海金融学院 Normal Population Distribution Normal Sampling Distribution (has the same mean) (i.e. is unbiased ) Sampling Distribution Properties

上海金融学院上海金融学院 For sampling with replacement: As n increases, decreases Larger sample size Smaller sample size (continued)

上海金融学院上海金融学院 If the Population is not Normal  We can apply the Central Limit Theorem: Even if the population is not normal, … sample means from the population will be approximately normal as long as the sample size is large enough … and the sampling distribution will have and

上海金融学院上海金融学院 n↑n↑ ~~ Central Limit Theorem ~~ As the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population If the Population is not Normal

上海金融学院上海金融学院 Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation (Sampling with replacement) Larger sample size Smaller sample size (continued) Sampling distribution properties: If the Population is not Normal ~~ Central Limit Theorem ~~

上海金融学院上海金融学院 Population Distribution Sampling Distribution of (n=2) Samling Distribution of (n=5) Sampling Distribution of (n=30)

上海金融学院上海金融学院 How Large is Large Enough?  For most distributions, n > 30 will give a sampling distribution that is nearly normal  For fairly symmetric distributions, n > 15  For normal population distributions, the sampling distribution of the mean is always normally distributed  Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.  What is the probability that the sample mean is between 7.8 and 8.2? Example

上海金融学院上海金融学院  Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.  What is the probability that the sample mean is between 7.8 and 8.2? Example Even if the population is not normally distributed, the central limit theorem can be used (n > 30) … so the sampling distribution of is approximately normal … with mean = 8 … and standard deviation Solution:

上海金融学院上海金融学院 Solution Solution(continued): z 7.8 8.2 -0.4 0.4 Sampling Distribution Standard Normal Distribution.1554 +.1554 Population Distribution ? ? ? ? ? ? ?? ? ? ? ? Sample Standardize  Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.  What is the probability that the sample mean is between 7.8 and 8.2? Example

上海金融学院上海金融学院  Sample Data Simple Random Sampling: SAT Scores 1744 Conrad Harris1025 Yes 2436 Enrique Romero 950 Yes 3865 Fabian Avante1090 No 4790 Lucila Cruz1120 Yes 5835 Chan Chiang 930 No..... 30 498 Emily Morse 1010 No No. RandomNumber Applicant SAT Score Score Live On- Campus.....

上海金融学院上海金融学院 Sampling Distribution of for SAT Scores SamplingDistributionof

上海金融学院上海金融学院 What is the probability that a simple random sample What is the probability that a simple random sample of 30 applicants will provide an estimate of the population mean SAT score that is within +/  10 of the actual population mean  ? In other words, what is the probability that will be In other words, what is the probability that will be between 980 and 1000? Sampling Distribution of for SAT Scores Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (1000 - 990)/14.6=.68 P ( z <.68) =.2517+0.5=0.7517 Step 2: Find the area under the curve to the left of the upper endpoint. upper endpoint. Solution:

上海金融学院上海金融学院 Standard Normal Distribution Using the table of standard normal distribution

上海金融学院上海金融学院 9901000 Area =.7517 SamplingDistributionof Sampling Distribution of for SAT Scores

上海金融学院上海金融学院 Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve to the left of the lower endpoint. lower endpoint. z = (980 - 990)/14.6= -.68 P ( z.68) =.2483 = 1 -. 7517 = 1 - P ( z <.68) Sampling Distribution of for SAT Scores

上海金融学院上海金融学院 Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. the lower and upper endpoints of the interval. P (-.68 < z <.68) = P ( z <.68) - P ( z < -.68) =.7517 -.2483 =.5034 The probability that the sample mean SAT score will be between 980 and 1000 is: P (980 < < 1000) =.5034 Sampling Distribution of for SAT Scores

上海金融学院上海金融学院 Relationship Between the Sample Size and the Sampling Distribution of Suppose we select a simple random sample of 100 Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. applicants instead of the 30 originally considered. E ( ) =  regardless of the sample size. In our E ( ) =  regardless of the sample size. In our example, E ( ) remains at 990. example, E ( ) remains at 990. Whenever the sample size is increased, the standard Whenever the sample size is increased, the standard error of the mean is decreased. With the increase error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the in the sample size to n = 100, the standard error of the mean is decreased to: mean is decreased to:

上海金融学院上海金融学院 With n = 30, With n = 100, Relationship Between the Sample Size and the Sampling Distribution of

上海金融学院上海金融学院 Recall that when n = 30, P (980 < < 1000) =.5034. Recall that when n = 30, P (980 < < 1000) =.5034. We follow the same steps to solve for P (980 < < 1000) We follow the same steps to solve for P (980 < < 1000) when n = 100 as we showed earlier when n = 30. when n = 100 as we showed earlier when n = 30. Now, with n = 100, P (980 < < 1000) =.7888. Now, with n = 100, P (980 < < 1000) =.7888. Because the sampling distribution with n = 100 has a Because the sampling distribution with n = 100 has a smaller standard error, the values of have less smaller standard error, the values of have less variability and tend to be closer to the population variability and tend to be closer to the population mean than the values of with n = 30. mean than the values of with n = 30. Relationship Between the Sample Size and the Sampling Distribution of

上海金融学院上海金融学院 1000980990 Area =.7888 Relationship Between the Sample Size and the Sampling Distribution of SamplingDistributionof

上海金融学院上海金融学院 Section 7.6 Sampling Distribution of  Expected Value of  Standard Deviation of  Form of the Sampling Distribution of  Practical Value of the Sampling Distribution of

上海金融学院上海金融学院 Population Proportions, p p = the proportion of population having some characteristic  Sample proportion ( p ) provides an estimate of p:  If two outcomes, p has a binomial distribution

上海金融学院上海金融学院 A simple random sample of n elements is selected from the population. Population with proportion p = ? n Making Inferences about a Population Proportion The sample data provide a value for the sample proportion. The value of is used to make inferences about the value of p. Sampling Distribution of

上海金融学院上海金融学院 where: p = the population proportion The sampling distribution of is the probability distribution of all possible values of the sample proportion. Expected Value of Sampling Distribution of

上海金融学院上海金融学院 is referred to as the standard error of the proportion. is referred to as the standard error of the proportion. Finite Population Infinite Population Standard Deviation of Sampling Distribution of

上海金融学院上海金融学院 The sampling distribution of can be approximated by a normal distribution whenever the sample size is large. The sampling distribution of can be approximated by a normal distribution whenever the sample size is large. The sample size is considered large whenever these conditions are satisfied: The sample size is considered large whenever these conditions are satisfied: np > 5 n (1 – p ) > 5 and Form of the Sampling Distribution of For values of p near.50, sample sizes as small as 10 permit a normal approximation. With very small (approaching 0) or very large (approaching 1) values of p, much larger samples are needed.

上海金融学院上海金融学院  Approximated by a normal distribution if: where and (where p = population proportion) Sampling Distribution P( p ).3.2.1 0 0. 2.4.6 8 1 p Sampling Distribution of

上海金融学院上海金融学院 z-Value for Proportions If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor: Standardize p to a z value with the formula:

上海金融学院上海金融学院 Sampling Distribution of Example  If the true proportion of voters who support Proposition A is p =.4, what is the probability that a sample of size 200 yields a sample proportion between.40 and.45?  i.e.: if p =.4 and n = 200, what is P(.40 ≤ p ≤.45) ?

上海金融学院上海金融学院 Example (continued) Sampling Distribution of  if p =.4 and n = 200, what is P(.40 ≤ p ≤.45) ? Find : Convert to standard normal:

上海金融学院上海金融学院 Example (continued) Sampling Distribution of z.451.44.4251 Standardize Sampling Distribution Standardized Normal Distribution Use standard normal table: P(0 ≤ z ≤ 1.44) =.4251.400 p  if p =.4 and n = 200, what is P(.40 ≤ p ≤.45) ?

上海金融学院上海金融学院 Recall that 72% of the prospective students applying to St. Andrew ’ s College desire on-campus housing. n Example: St. Andrew’s College What is the probability that What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicant desiring on-campus housing that is within plus or minus.05 of the actual population proportion? Sampling Distribution of

上海金融学院上海金融学院 For our example, with n = 30 and p =.72, the normal distribution is an acceptable approximation because: n (1 - p ) = 30(.28) = 8.4 > 5 and np = 30(.72) = 21.6 > 5 Sampling Distribution of

上海金融学院上海金融学院 SamplingDistributionof

上海金融学院上海金融学院 Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (0.77 -0.72)/0.082 = 0.61 P ( z <.61) =0.2291+0.5=0.7291 Step 2: Find the area under the curve to the left of the upper endpoint. upper endpoint. Sampling Distribution of

上海金融学院上海金融学院 Standard Normal Distribution Using the table of standard normal distribution

上海金融学院上海金融学院.77.77.72 Area =.7291 Sampling Distribution of SamplingDistributionof

上海金融学院上海金融学院 Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve to the left of the lower endpoint. lower endpoint. z = (.67 -.72)/.082 = -.61 P ( z.61) =.2709 = 1 -. 7291 = 1 - P ( z <.61) Sampling Distribution of

上海金融学院上海金融学院.67.72 Area =.2709 Sampling Distribution of SamplingDistributionof

上海金融学院上海金融学院 P (.67 < <.77) =.4582 Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. the lower and upper endpoints of the interval. P (-.61 < z <.61) = P ( z <.61) - P ( z < -.61) =.7291 -.2709 =.4582 The probability that the sample proportion of applicants wanting on-campus housing will be within +/-.05 of the actual population proportion : Sampling Distribution of

上海金融学院上海金融学院.77.67.72 Area =.4582 Sampling Distribution of SamplingDistributionof

上海金融学院上海金融学院 Chapter Summary  Discussed sampling error  Introduced sampling distributions  Described the sampling distribution of the mean For normal populations Using the Central Limit Theorem  Described the sampling distribution of a proportion  Calculated probabilities using sampling distributions  Discussed sampling from finite populations

上海金融学院上海金融学院 Homework Chapter 7 Exercises Section 7.5 Page278 24 Section 7.6 Page284 40

Fundamentals of Business Statistics chapter7 Sampling and Sampling Distributions 上海金融学院统计系 Statistics Dept., Shanghai Finance University.

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Fundamentals of Business Statistics chapter7 Sampling and Sampling Distributions 上海金融学院统计系 Statistics Dept., Shanghai Finance University.

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Presentation on theme: "Fundamentals of Business Statistics chapter7 Sampling and Sampling Distributions 上海金融学院统计系 Statistics Dept., Shanghai Finance University."— Presentation transcript:

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