1 Sampling Distributions Lecture 9

2 Background  We want to learn about the feature of a population (parameter)  In many situations, it is impossible to examine all elements of a population because elements are physically inaccessible, too costly to do so, or the examination involved may destroy the item.  Sample is a relatively small subset of the total population.  We study a random sample to draw conclusions about a population, this is where statistics come into the picture.  Statistics, such as the sample mean and sample variance, computed from sample measurements, vary from sample to sample. Therefore, they are random variables.  The probability distribution of a statistic is called a sampling distribution.

3 Sampling Distributions Sampling Distribution of the Mean Sampling Distribution of the Proportion A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population

4 Developing a Sampling Distribution  Assume there is 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

A B C D P(x) x (continued) Summary Measures for the Population Distribution: Developing a Sampling Distribution

6 Sampling with replacement SamplesAgeSample means A, A18, 1818 A, B18, 2019 A, C18, 2220 A, D18, 2421 B, A20, 1819 B, B20, 2020 B, C20, 2221 B, D20, 2422 C, A22, 1820 C, B22, 2021 C, C22, 2222 C, D22, 2423 D, A24, 1821 D, D24, 2022 D, C24, 2223 D, D24, 2424

7 Sampling Distribution of All Sample Means P(X) X Sample Means Distribution 16 Sample Means _ Developing a Sampling Distribution (continued) _

8 Summary Measures of this Sampling Distribution (note that N=16 for the population of sample means): Developing a Sampling Distribution (continued)

9 Comparing the Population with its Sampling Distribution (with replacement) P(X) X A B C D Population N = 4 P(X) X _ Sample Means Distribution n = 2 _

10 Mean and standard error of the sample Mean (sample with replacement)  The mean of the distribution of sample mean:  A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: ( This assumes that sampling is with replacement or sampling is without replacement from an infinite population)  Note that the standard error of the mean decreases as the sample size increases

11 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  Or, equivalently, the sampling distribution of is normally distributed with and

12 Sampling Distribution Properties As n increases, decreases Larger sample size Smaller sample size (continued)

13 If the Population is not normal  The central limit theorem states that when the number of observations in each sample (called sample size) gets large enough  The sampling distribution of is approximately normally distributed with and  Or, equivalently, the sampling distribution of is also approximately normally distributed with and

14 Z value for means Standardize the sample mean:

15 Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation Larger sample size Smaller sample size Visualizing the Central Limit Theorem Sampling distribution properties:

16 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  Recall that, for normal population distributions, the sampling distribution of the mean is always normally distributed regardless of sample size n

17 Calculating probabilities  Suppose we want to find out  If the population is normal, then regardless of the value of n:  If the population is not normal, then, when n is large enough (n > 30)

18 Example  Suppose a population has mean μ = 10 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 9.7 and 10.3?

19 Example Solution:  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 = 10  …and standard deviation (continued)

20 Example Solution (continued): (continued) Sampling Distribution Population Distribution ? ? ? ? ? ? ?? ? ? ? ? Sample X

21 One more example  Time spent using per session is normally distributed with =8 minutes and =2 minutes. 1.If a random sample of 25 sessions were selected, what proportion of the sample mean would be between 7.8 and 8.2 minutes?

22 Example (Cont’d) 2.If a random sample of 100 sessions were selected, what proportion of the sample mean would be between 7.8 and 8.2 minutes? 3.What sample size would you suggest if it is desired to have at least 0.90 probability that the sample mean is within 0.2 of the population mean?

23 Sampling Distribution of the Proportion Sampling Distributions Sampling Distribution of the Mean Sampling Distribution of the Proportion

24 Population Proportions In Bernoulli trials, let π = the proportion of successes  Recall that Y = the number of successes in n Bernoulli trials follows Bin(n, π)  For the ith Bernoulli trial, Define  Then, obviously

25 Population proportions (Cont’d)  For large n, apply the CLT to sample mean and sum  How large is large? Or

26 Z-Value for Proportions Standardize p to a Z value with the formula:

27 Example  If the true proportion of voters who support Proposition A is π = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45?  i.e.: if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ?

28 Example (continued) Find : Convert to standard normal:

29 Review example  The number of claims received by an automobile insurance company on collision insurance on one day follows the following probability distribution: With  Suppose the number of claims received are independent from day to day. x01234 p(x)

30 Review example (cont’d)  For a 50-day period, Find the probability of the following events: 1)The total number of claims exceeds 20 2)On more than 20 days, at least one claim is received

31 Sampling distribution of difference of two independent populations  An important estimation problem involves the comparison of means of the two populations. For example, you may want to make comparisons like these:  The average scores on GRE for students who majored in mathematics versus chemistry  The average income for male and female college graduates  The proportion of patients receiving different medications who recovered from a certain disease

32 Sample distributions of difference of two independent sample means  Suppose there are two populations  Independent random samples of size n ₁ and n ₂ observations have been selected from the two populations with sample means and respectively  Recall that when n ₁ and n ₂ are large, and are approximately normally distributed with PopulationMeanS.d. I II

33  Since the two samples are independent  Standardize:

34 Example  A light bulb factory operates two different types of machines. The mean life expectancy is 385 hours from machine I and 365 hours from machine II. The process standard deviation of life expectancy of machine I is 110 hours and of machine II is 120 hours.  What is the probability that the average life expectancy of a random sample of 100 light bulbs from Machine I is shorter than the average life expectancy of 100 light bulbs from Machine II?

35 Example (Cont’d)  Note that  Therefore

36 Sampling distribution of difference of two independent sample proportions  Assume that independent random samples of n ₁ and n ₂ observations have been selected from binomial populations with parameters and, respectively.  The sampling distribution of the difference in sample proportions (p ₁ -p ₂ ) can be approximated by a normal distribution with mean and standard deviation  The Z statistic is

37 Example  From a study by the Charles Schwab Corporation, 74% of African Americans and 84% of Whites with an annual income above $50,000 owned stocks.  For a random sample of 500 African American and a random sample of 500 Whites with income above $50,000, what is the probability that more whites own stocks?

38 Example (Cont’d)  Summary data:  It follows that

39 Important Summary of sampling distributions Param.Point estimate Sampling distribution Standardized Z μ

40 Sampling methods  Simple random samples  Stratified samples

41 Simple Random Samples  Every individual or item from the frame has an equal chance of being selected  Selection may be with replacement or without replacement  Samples obtained from table of random numbers or computer random number generators  Simple to use  May not be a good representation of the population’s underlying characteristics

42 Stratified Samples  Divide population into two or more subgroups (called strata) according to some common characteristic  A simple random sample is selected from each subgroup, with sample sizes proportional to strata sizes  Samples from subgroups are combined into one  Ensures representation of individuals across the entire population Population Divided into 4 strata Sample

43 Types of Survey Errors  Coverage error  Non response error  Sampling error  Measurement error Excluded from frame Follow up on nonresponses Random differences from sample to sample Bad or leading question (continued)