Math 140 Introductory Statistics
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1 Math 140 Introductory Statistics
2 Let s make our own sampling! If we use a random sample (a survey) or if we randomly assign treatments to subjects (an experiment) we can come up with proper, unbiased conclusions We should work with randomized data to avoid bias But HOW to produce, collect and analyze data?
3 7.1 Generating sampling distributions Generate sampling distributions and study: The sample mean The sample shape, the center, the spread How to draw proper conclusions about what is likely and what is rare. How to relate this to the entire population? How to do this in the easiest way?
4 Sample vs. Population Sampling size n Population Samples
5 Our friends at Westvaco Recall, the people laid off were 55, 55 and 64 is this discrimination or not? We need to compare with RANDOM layoffs of three people
6 Perform a random simulation All possible sets of 3 people chosen from 10 " 10% $ ' =120 # 3 & For each of these groups calculate average age and create a dot plot - this is your sampling distribution
7 Conclusions from Westvaco The average age of the people that were actually laid off was 58 Common sense: It is rather hard for this to happen by chance - Westvaco has some explaining to do
8 Generate a sample distribution Simulated sampling distribution: distribution of summary statistics obtained from taking repeated random samples. I. Take a random sample of a fixed size n from a population. II. Compute a summary statistic for this sample. III. Repeat steps I and II many times. IV. Display the distribution of the summary statistic.
9 We will often have access to samples, but not necessarily to the entire population (too big or inaccessible)
10 Our friends at the NBA These are the salaries of NBA players. The mean is $4.6 million and the SD is $4.7 million. Highly skewed THESE ARE POPULATION STATISTICS (EVERYBODY)
11 Our friends at the NBA Suppose this data was not public and I am an NBA player who wants to know the average salary of my colleagues. I can only access 10 people at random. How is the average I find different from the true average? Since the distribution is skewed, should I be concerned?
12 Lets simulate a sampling distribution Select random samples of 10 from our distribution Calculate average salary Repeat many times (200?) Place them in a chart THESE ARE SAMPLE STATISTICS
13 Average simulated salaries 200 simulations The distribution is approximately Normal Centered at about $4.6 million Equivalent of what we did for Westvaco! SD is about $1.5 million
14 Average simulated salaries From our 200 simulations The distribution is approximately normal and centered at about $4.6 million, the SD is about $1.5 million The mean of the entire population was $4.6 million and the SD was $4.7 million.
15 Recall properties of the normal distribution For us the mean is $4.6 million and the SD is $1.5 million We can be 95% sure that our sample mean is within 3 million from the population mean
16 Normal distribution We can be 95% sure ANY mean of 10 people we pick falls between $1.6 and $7.6 million and centered about $4.6 million $1.6 mil $7.6 mil $4.6 mil
17 Average simulated salaries We can be pretty confident that the selection of 10 people will give us a good idea about the average salary of NBA players We did not need to sample the entire population! The SD from our SAMPLING DISTRIBUTION is $1.5 million. The SD from our POPULATION DISTRIBUTION Is $4.7 million
18 Average simulated salaries The SD from our SAMPLING DISTRIBUTION is $1.5 million. This is called the STANDARD ERROR The SD from our POPULATION DISTRIBUTION Is $4.7 million This is called the POPULATION STANDARD DEVIATION
19 Definitions Values that lie in the middle 95% of a sampling distribution are called REASONABLY LIKELY EVENTS Values that lie in the left 2.5% and in the right 2.5% Sides of a sampling distribution are called RARE EVENTS
20 Let s compare population sample Would we be surprised to draw a player with an $3 million salary? What about $8 million salary? Would be surprised to draw 10 players with an average salary of $8 million?
21 Utah s national parks Create the sampling distribution for the total number of square miles in any 2 parks. Use all possible samples of 2 parks.
22 Utah s national parks
23 Utah s national parks How many possible ways of selecting 2 parks? We can only survey 600 square miles a year. What is the probability that we DO NOT finish the survey within the first year?
24 Utah s national parks We can use all possible combinations
25 Utah s national parks Probability we don t finish survey is 4/10
26 Sample and population means Any sample mean x Population mean µ Usually they are different, but OVER MANY SAMPLES they tend to be the same Also, THE LARGER THE SAMPLE SIZE the closer they will be
27 Estimator points Any sample mean x Population mean µ When we use a summary statistic derived from the sample, (such as the sample mean) as an estimate of the population statistic (such as the population mean) we call it an estimator point.
28 Desired estimator points The mean of the sampling distribution should be the same if you calculated the mean of the entire population unbiased Also it is desirable that as the sample size increases, The SD should decrease So that we have the most precision possible And the least standard error
29 Back to Utah Calculate mean and SD for all parks Then do the same for all 10 samples of 2 parks
30 Back to Utah At the end calculate the mean area for all your samples. Is this mean the same as for the initial distribution? If so, our sample mean an unbiased estimator. Now calculate the SD of the sampling distribution. Compare with the previous SD.
31 Back to Utah The SD should be smaller here (105.23) than for the entire population (171.85)
32 Back to Utah Sample size 1 Sample size 2 This means that the spread we have is less if we use Sample sizes of 2 than if we use sample sizes of 1. The mean is the same, no bias The spread is different
33 Concepts A simulated sample distribution is the distribution of a sample statistic (the mean) for a large number of repeated samples The sample distributions are best described by shape, center and spread Sampling distributions DO NOT necessarily have the same shape as the population from which they were taken
34 Concepts The SD of the sampling distribution is called the standard error If the sampling distribution is normal, reasonably likely outcomes are those that lie within 2 SD of the mean (95% of data)
35 P5 page 319 Estimate the range of Utah s national parks Range = Largest Area - Smallest Area Select 3 parks at random and calculate the range 1) What is the range of the entire POPULATION? 2) Make a table for the range of groups of 3 3) Place your values on a dot plot 4) What is the mean of the sample? 5) Is the sample range biased or unbiased?
36 Practice Page 321 P3, P4, P5, E1, E2, E3, E5, E6, E7, E10,
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