1. 9.1 Sampling Distribution

2. ◦ Know the difference between a statistic and a parameter ◦ Understand that the value of a statistic varies between samples ◦ Be able to describe the shape, center and spread of a given sampling distribution ◦ Understand how bias and variability of a statistic affects the sampling distribution

3.  Parameter - a number that describes the population (usually it’s unknown)  Statistic - a number computed from the sample data

4.  Is the boldfaced number a parameter or a statistic? 1. 60,000 members of the labor force were interviewed of whom 7.2% were unemployed statistic

5. 2. A lot of ball bearings has a mean diameter of 2.5003 cm. A 100 bearings are selected from the lot and have a mean diameter of 2.5009 cm. 2.5003- parameter 2.5009- statistic

6.  3. A telemarketing firm in Los Angeles randomly dials telephone numbers. Of the first 100 numbers dialed 48% are unlisted. This is not surprising because 52% of all Los Angeles residential phones are unlisted. 48%- statistic 52%- parameter

7.  Sample proportion: (“p hat”) (Our true population proportion is “p”, so our sample proportion needs to be denoted differently)  Example: A poll found that 1650 out of 2500 randomly selected adults agreed with the statement that shopping is frustrating. What is the proportion of the sample who agreed? =1650/2500

8.  Sampling variability – the value of a statistic varies with repeated sampling  Applet: http://www.rossmanchance.com/applets/Ree ses/ReesesPieces.html http://www.rossmanchance.com/applets/Ree ses/ReesesPieces.html **Unclick the animate button. The true proportion of orange reese’s pieces is 0.5. Let’s take sample and see how they compare!!!

11.  So with a sample size of 25, our sampling distribution had a mean of 0.501 and a spread of 0.31-0.65  Our sample size of 100, the sampling distribution had a mean of 0.494 and a spread of 0.39-0.62.  So what do we notice: 1-The larger the sample size, the closer your sample proportion gets to the true proportion. 2- The larger the sample size, the less the variability of your sampling distribution.

12. the distribution of values taken by the statistic in all possible samples of the same size from the same population.

13.  1. the overall shape is symmetric (normal)  2. there are no outliers or other important deviations from the overall pattern  3. the center of the distribution is the true value p  4. the values of have a large spread

14.  a statistic is unbiased if the mean of the sampling distribution is equal to the true value of the parameter being estimated

15.  1. Is described by the spread of its sampling distribution.  2. This spread is determined by the sampling design and the size of the sample.  3. Larger samples give smaller spread.

16. High bias; Low bias; low variabilityhigh variability High bias; Low bias; high variabilitylow variability

17. To calculate the standard deviation for proportion:

18. Ex: 60% of people find clothes shopping frustrating.  Find the proportion of people that fall within 2 standard deviations of the mean for samples of size n =100. Step 1: Calculate σ Step 2: Draw your curve (0.502,0.698)

19.  n = 2500 (0.5804 and 0.6196)

20.  Why does the size of the population have little influence on how statistics from a random sample behave?  The spread of your curve is computed from n (sample size) not the size of the population. If you have a large sample size (n>30), the closer your sampling distribution will get to a normal distribution. (even if the population was not normal)

21. Learning Objectives: ◦ Know the characteristics of the sampling distribution of ◦ Know when to use the normal approximation for ◦ Be able to solve problems using the normal approximation for

22.  Choose an SRS of size n from a large population, then:  1. the sampling distribution of is approximately normal. (closer to a normal dist. when n is large)  2. the mean of the sampling dist. is exactly p  3.

23. In order to analyze data,we have to make some assumptions before we begin our calculations!  Assumption 1 The data was taken from a random sample  Assumption 2 The standard deviation p for __ can only be used when the population is at least 10 times as large as the sample  Assumption 3 We can say that the sampling distribution of is approximately normal when np>10 and n(1- p)>10. *(some books use np>5)

24.  There are 1.7 million first-year college students of those, 1500 first-year college students are asked whether they applied for admission to any other college. In fact 35% of all first-year students applied to a college other than the one they are attending. What is the probability that your sample will give a result within 2 percentage points of this true value?

25.  Step 1: Assumptions: -random sample -population is at least 10X the sample -(1500)(0.35)> 10 525>10 -(1500)(0.65)>10 975>10 Step 2: Calculate σ, then write out in terms of the problem and convert to a z-scores.

26.  Assumptions: -Random Sample -population is at least 10X the sample -(1540)(0.15)> 10 231>10 -(1540)(0.85)>10 1309>10