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AP Statistics

AP Statistics. 9.1 Sampling Distribution. Learning Objectives. 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

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AP Statistics

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  1. AP Statistics 9.1 Sampling Distribution

  2. Learning Objectives • 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. Examples • 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. Sampling Variability • Sample proportion: (“p hat”) • 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/Reeses/ReesesPieces.html

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

  10. Describing Sampling Distributions • 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

  11. Unbiased Statistic • a statistic is unbiased if the mean of the sampling distribution is equal to the true value of the parameter being estimated

  12. Variability of a statistic • 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.

  13. High bias; Low bias; low variability high variability High bias; Low bias; high variability low variability

  14. Example • 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 (0.502,0.698)

  15. n = 2500 (0.5804 and 0.6196)

  16. Why does the size of the population have little influence on how statistics from a random sample behave? The larger the sample size, the smaller the standard deviation.

  17. Ch 9.2 Sample Proportions 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

  18. Sampling distribution of • 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.

  19. Assumption 1 The standard deviation p for __ can only be used when the population is at least 10 times as large as the sample • Assumption 2 We can say that the sampling distribution of is approximately normal when np>10 and n(1-p)>10. *(some books use np>5)

  20. Example • 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?

  21. Assumptions: -random sample -population is at least 10X the sample -(1500)(0.35)> 10 525>10 -(1500)(0.65)>10 975>10

  22. Complete 9.15 pg. 477 • Assumptions: -SRS -population is at least 10X the sample -(1540)(0.15)> 10 231>10 -(1540)(0.85)>10 1309>10

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