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Generating Sampling Distributions Section 7.1 Generating Sampling Distributions
Why Statistics?
Why Statistics? Statistics allow us to make inferences (or draw conclusions) about a population.
Population How do we describe populations?
Population How do we describe populations? If we can graph data about the population, we can use shape, center, and spread.
Population How do we describe populations? If we can graph data about the population, we can use shape, center, and spread. We may use summary numbers such as the mean ( ), standard deviation ( ), median, quartiles, etc.
Population What do we call a summary number that describes a population or a probability distribution?
Population What do we call a summary number that describes a population or a probability distribution? parameter
Summary Statistic What is a summary statistic?
Summary Statistic Summary statistic: a summary number calculated from a sample taken from a population
Summary Statistic Examples: sample mean x, standard deviation , 5-number summary
Sample statistic Population parameter
Sampling Distribution Suppose we take a random sample of a fixed size n from our population and compute a summary statistic.
Sampling Distribution Suppose we take a random sample of a fixed size n from our population and compute a summary statistic. Then, suppose we repeat this process many times.
Sampling Distribution Sampling distribution: distribution of summary statistics you get from taking repeated random samples
Sampling Distribution Sampling distribution: distribution of summary statistics you get from taking repeated random samples Answers the question “How does my summary statistic behave when I repeat the process many times?” (Think Law of Large Numbers).
Sampling Distribution Two types of sampling distributions:
Sampling Distribution Two types of sampling distributions: Exact Sampling Distribution
Sampling Distribution Two types of sampling distributions: Exact Sampling Distribution Approximate Sampling Distribution
Exact Sampling Distributions When the population size is very small,
Exact Sampling Distributions When the population size is very small, you can construct sampling distributions exactly by listing distribution of summary statistic for all possible samples for a fixed size, n.
Exact Sampling Distributions Utah has five national parks. Your company has been hired to make maps of two of these parks, which will be selected at random. 1. Construct the sampling distribution for the total number of square miles you would map. 2. Find P(map > 600 mi2)
Exact Sampling Distributions Utah has five national parks. Your company has been hired to make maps of two of these parks, which will be selected at random. How many possible samples of size n = 2 are there?
Exact Sampling Distributions Utah has five national parks. Your company has been hired to make maps of two of these parks, which will be selected at random. How many possible samples of size n = 2 are there? 5C2 = 10 samples
Exact Sampling Distributions Construct the sampling distribution for the total number of square miles you would map. Find P(map > 600 mi2).
Exact Sampling Distributions
Find P(map > 600 mi2).
Find P(map > 600 mi2) = 4/10
Exact Sampling Distributions
Should we always use exact sampling distributions?
Always use Exact Sampling Distribution? Suppose you have a population of 100 rectangles with varying dimensions and you had to construct a sampling distribution for a sample of size 5. How many ways can you choose 5 rectangles at a time from the population of 100?
Always use Exact Sampling Distribution? Suppose you have a population of 100 rectangles with varying dimensions and you had to construct a sampling distribution for a sample of size 5. How many ways can you choose 5 rectangles at a time from the population of 100? 100C5 = 75,287,520 Would you construct an exact sampling distribution here?
Approximate Sampling Distribution Approximate sampling distribution is AKA simulated sampling distribution.
Approximate Sampling Distribution Approximate sampling distribution is AKA simulated sampling distribution. 4-step process:
Approximate Sampling Distribution Approximate sampling distribution is AKA simulated sampling distribution. 4-step process: 1. Take random sample of fixed size n from population
Approximate Sampling Distribution Approximate sampling distribution is AKA simulated sampling distribution. 4-step process: 1. Take random sample of fixed size n from population 2. Compute summary statistic of interest For example: mean, median, min, or max
Approximate Sampling Distribution Approximate sampling distribution is AKA simulated sampling distribution. 4-step process: 1. Take random sample of fixed size n from population 2. Compute summary statistic of interest 3. Repeat steps 1 and 2 many times
Approximate Sampling Distribution Approximate sampling distribution is AKA simulated sampling distribution. 4-step process: 1. Take random sample of fixed size n from population 2. Compute summary statistic of interest 3. Repeat steps 1 and 2 many times 4. Display distribution of the summary statistic
Each dot represents 1 rectangle. Display 7.2, p. 411 Each dot represents 1 rectangle.
Display 7.2, p. 411 Shape: skewed right Center: μx = 7.4 Spread: σx = 5.2
Now, 5 rectangles were selected at random
Now, 5 rectangles were selected at random Mean area of these five was calculated
Now, 5 rectangles were selected at random Mean area of these five was calculated This was repeated 1000 times
Display 7.3, p. 411
Display 7.3, p. 411 Shape: approx. normal Center: μx = 7.4 Spread: σx = 2.3
Population Sampling Dist. Shape: skewed right approx. normal Center: μx = 7.4 μx = 7.4 Spread: σx = 5.2 σx = 2.3
Reasonably Likely vs Rare Events What are these?
Reasonably Likely vs Rare Events Reasonably likely events: values that lie in the middle 95% of sampling distribution
Reasonably Likely vs Rare Events Reasonably likely events: values that lie in the middle 95% of sampling distribution Rare events: values that lie in outer 5% of sampling distribution
Display 7.3, p. 411
Reasonably Likely vs Rare Events In normal distribution, rare events lie more than approximately 2 standard deviations from the mean
Vocabulary Population standard deviation is the standard deviation of the population,
Vocabulary
Point Estimators Point estimator: a statistic from a sample that provides a single point (number) as a plausible value of a population parameter
Sampling Bias Recall, when we discussed sampling bias. What happens if you have biased results?
Sampling Bias Recall, when we discussed sampling bias. What happens if you have biased results? The estimate from the sample is larger or smaller, on average, than the population parameter being estimated
Point Estimators A summary statistic is a biased estimator of a population parameter if it gives results that are too large or too small on average
Biased or Unbiased Estimators? For a sampling distribution, are these biased or unbiased estimators? 1) Sample mean Sample median Sample maximum Sample minimum Sample range 6) Sample standard deviation
Estimators Sample mean is unbiased estimator of the population mean because the mean of the sampling distribution of the sample mean is equal to the population mean Sample median?
Estimators Sample median is nearly unbiased estimator of population median for large samples Sample maximum?
Estimators Sample maximum is biased estimator of population maximum and is biased in direction of being too small Sample minimum?
Estimators Sample minimum is biased estimator of population minimum and tends to be too large Sample range?
Estimators Sample range is biased estimator of population range and tends to be too small Sample standard deviation?
Estimators Sample standard deviation is biased estimator of population standard deviation and tends to be too small
Biased or Unbiased Estimators? For a sampling distribution, are these biased or unbiased estimators? 1) Sample mean : unbiased Sample median: nearly unbiased Sample maximum Sample minimum Sample range 6) Sample standard deviation
Questions?