1. Sampling Distributions

2. Review Random phenomenon Individual outcomes unpredictable Sample space all possible outcomes Probability of an outcome long-run proportion for outcome Probability distribution probabilities for outcomes in sample space

3. Review parameter numerical fact about the population (e.g.  ) – the thing we want to know, but can’t statistic corresponding numerical fact in the sample (e.g. ) – the thing we can know

5. Fact about (when X has mean μ and s.d. σ) Law of Large Numbers – As n gets larger, “gets closer and closer” to the mean μ – More precisely, the chance of getting a “bad” gets smaller as n gets larger

6. Sampling Distribution A mental picture: imagine trying to estimate average height in this class (μ) – A sample of 5 persons is obtained and is calculated – Imagine all possible samples of size 5, with an for each sample – Collect all the ’s: This is the “sampling distribution of ” A definition: The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of size n

7. What we did….

8. Application to Statistics if you have a statistic calculated from a random sample or randomized experiment sample space = all possible values of sample statistic The probability distribution of the sample statistic is called the sampling distribution

9. iClicker Consider a simple random sample of 100 BYU students, asking them how many movies they watched last week (x), and then calculating. What is the sampling distribution? a.dist. of x for all BYU stud. b.dist. of x for 100 BYU stud. c.prob. of getting the particular of the sample d.prob. dist. of for samples of 100 BYU students

10. Why sampling distribution? sampling distribution allows us to assess uncertainty of sample results (i.e., “how reliable is ?”) if we knew the spread of the sampling distribution, we would know how far our might be from the true 

11. Height Data for Our Class μ = 68.18 inches (~ 5’ 8”) and σ = 4.49 inches What is sampling distribution for if n=5? – We can’t see the (theoretical) sampling distribution because we don’t have time to look at all possible samples of size 5 – We CAN approximate it with simulation How does the sampling distribution of compare with distribution of heights (x)?

12. What we did….

13. If we had truly random samples….

14. More facts about (when X has mean μ and s.d. σ) Sampling Distribution (aka “Theoretical Sampling Distribution”) for – Has a mean of exactly μ – Has a standard deviation of exactly

15. Height Data for Our Class μ = 68.18 inches (~ 5’ 8”) and σ = 4.49 inches Someone says BYU’s incoming class for Fall 2014 will have a mean height larger than 68.18, based on a random sample of n=5 incoming freshman with = 69.5. What do you think? – What if the came from a sample with n=16? n=100?

16. Height Data for Our Class μ = 68.18 inches (~ 5’ 8”) and σ = 4.49 inches QUIZ: What is the mean of the sampling distribution for if n=4? – A: impossible to know – B: exactly 68.18 inches – C: approximately 68.18 inches, give or take a little bit of room for error – D: a value that gets closer and closer to 68.18 inches as n gets larger and larger

17. Height Data for Our Class μ = 68.18 inches (~ 5’ 8”) and σ = 4.49 inches QUIZ: What is the standard deviation of the sampling distribution for if n=4? – A: impossible to know – B: 4.49 inches – C: 4.49/2 = 2.245 inches – D: 4.49/4 = 1.1225 inches

18. Investigating properties of sampling distributions use simulation to investigate nature of sampling distribution of

20. sampling distribution applet

21. T F 1. always estimates μ well (i.e., it’s always close to μ). 2.Using the sampling dist. we can compute probabilities on. 3. does not vary from sample to sample. 4.The mean of the sampling dist. of is µ. iClicker

22. Next… What if we don’t have the whole population to simulate from? What if we don’t have 600 Stat 121 students willing to calculate values based on 600 different samples? – What if we only have time for one sample of size n=35 (BYU students), and we get 6.9 hours as an average number of TV hours per week? Can we say that BYU students’ mean viewing time is significantly less than the national average of 10.6 hours for college students? (σ=8.0) What if knew somehow that the sampling distribution for is normal?

23. iClicker Our is 1.3, and the sampling distribution is somehow known to be normal with st. dev. = 0.4. The true mean is not likely higher than__. (Think of “likely” cases as those in the “middle 95%” of cases.) a.1.3 b.1.7 c.2.1 d.2.5

24. Vocabulary Statistic Parameter Probability Probability distribution Sampling distribution of statistic