1. Lecture 11 Dustin Lueker

2. 2  The larger the sample size, the smaller the sampling variability  Increasing the sample size to 25… 10 samples of size n=25 100 samples of size n=25 1000 samples of size n=25 STA 291 Summer 2010 Lecture 11

3. Population with mean  and standard deviation  If you repeatedly take random samples and calculate the sample mean each time, the distribution of the sample mean follows a pattern This pattern is the sampling distribution 3STA 291 Summer 2010 Lecture 11

4.  As n increases, the variability decreases and the normality (bell-shapedness) increases. 4STA 291 Summer 2010 Lecture 11

5. 5  The larger the sample size n, the smaller the standard deviation of the sampling distribution for the sample mean ◦ Larger sample size = better precision  As the sample size grows, the sampling distribution of the sample mean approaches a normal distribution ◦ Usually, for about n=30, the sampling distribution is close to normal ◦ This is called the “Central Limit Theorem” STA 291 Summer 2010 Lecture 11

6.  If X is a random variable from a normal population with a mean of 20, which of these would we expect to be greater? Why? ◦ P(15<X<25) ◦ P(15< <25)  What about these two? ◦ P(X<10) ◦ P( <10) STA 291 Summer 2010 Lecture 116

7. 7  When we calculate the sample mean,, we do not know how close it is to the population mean ◦ Because is unknown, in most cases.  On the other hand, if n is large, ought to be close to STA 291 Summer 2010 Lecture 11

8. 8  If we take random samples of size n from a population with population mean and population standard deviation, then the sampling distribution of ◦ has mean ◦ and standard error  The standard deviation of the sampling distribution of the mean is called “standard error” to distinguish it from the population standard deviation STA 291 Summer 2010 Lecture 11

9. 9  The example regarding students in STA 291  For a sample of size n=4, the standard error of is  For a sample of size n=25, STA 291 Summer 2010 Lecture 11

10. 10  For random sampling, as the sample size n grows, the sampling distribution of the sample mean,, approaches a normal distribution ◦ Amazing: This is the case even if the population distribution is discrete or highly skewed  Central Limit Theorem can be proved mathematically ◦ Usually, the sampling distribution of is approximately normal for n≥30 ◦ We know the parameters of the sampling distribution STA 291 Summer 2010 Lecture 11

11. 11  Household size in the United States (1995) has a mean of 2.6 and a standard deviation of 1.5  For a sample of 225 homes, find the probability that the sample mean household size falls within 0.1 of the population mean  Also find STA 291 Summer 2010 Lecture 11

12. Binomial Population with proportion p of successes If you repeatedly take random samples and calculate the sample proportion each time, the distribution of the sample proportion follows a pattern 12STA 291 Summer 2010 Lecture 11

13.  As n increases, the variability decreases and the normality (bell-shapedness) increases. 13STA 291 Summer 2010 Lecture 11

14.  For random sampling, as the sample size n grows, the sampling distribution of the sample proportion,, approaches a normal distribution ◦ Usually, the sampling distribution of is approximately normal for np≥5, nq≥5 ◦ We know the parameters of the sampling distribution 14STA 291 Summer 2010 Lecture 11

15.  Take a SRS with n=100 from a binomial population with p=.7, let X = number of successes in the sample  Find  Does this answer make sense?  Also Find  Does this answer make sense? 15STA 291 Summer 2010 Lecture 11

16. 16  Mean/center of the sampling distribution for sample mean/sample proportion is always the same for all n, and is equal to the population mean/proportion. STA 291 Summer 2010 Lecture 11

17. 17  The larger the sample size n, the smaller the variability of the sampling distribution  Standard Error ◦ Standard deviation of the sample mean or sample proportion ◦ Standard deviation of the population divided by STA 291 Summer 2010 Lecture 11