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Lesson #9: What Do Samples Tell Us?

I’m used to estimate for the population. Watch out for bias…. Lesson #9: What Do Samples Tell Us?. Parameters . A parameter is a number which describes a population. The parameter is a fixed number, but usually the value is not known. Statistics.

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Lesson #9: What Do Samples Tell Us?

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  1. I’m used to estimate for the population Watch out for bias… Lesson #9:What Do Samples Tell Us?

  2. Parameters • A parameter is a number which describes a population. • The parameter is a fixed number, but usually the value is not known.

  3. Statistics • A statistic is a number which describes a sample. • The statistic is known when a sample is taken. • The statistic can change from sample to sample. • The statistic can be used to estimate the parameter (that is, make generalized statements about the population).

  4. P and P-Hat • The proportion of a population is a parameter = p • The proportion of a sample is a statistic = p-hat. Can be used as an estimate when p is unknown. • P-hat = # in sample that met need / # in sample

  5. P and P-Hat • The value of p-hat will vary from sample to sample. • Too much variability will produce untrustworthy results. • Random sampling attacks bias (that is, by choosing randomly, you reduce “favoritism”). • By taking lots of random samples of the same size from the same population, the variation from sample to sample follows a predictable pattern.

  6. Gallup Poll Ex. • According to Gallup, “57% of Americans have bought a lottery ticket in the last 12 months”. This claim was made about the population of 200 million adults. 1523 people were chosen at random to be surveyed. • Gallup turned up the fact that 57% of the sample bought lottery tickets into an estimate that about 57% of alladults bought tickets. • Gallup used a fact about a sample to estimate the truth about the whole population.

  7. Two types of error in estimation • Bias: consistent, repeated deviation of the sample statistic from the population parameter in the same direction when we take many samples.

  8. Two types of error in estimation • Variability: describes how spread out the values of the sample statistic are when we take many samples. Large variability means that the result of sampling is not repeatable. • A good sampling method has both small bias and small variability.

  9. Example

  10. Example

  11. Example

  12. Example

  13. Managing bias and variability • To reduce bias: Use random sampling. Using a list of the entire population for an SRS produces unbiased estimates. • To reduce variability: Use a large sample. Population must be 10 times as large as sample for size to not be a factor on variability. • Large random samples almost always give an estimate that is close to the truth.

  14. Margin of error • “If we took many samples using the same method we used to get this one sample, 95% of the samples would give us a result within plus or minus x percentage points of the truth about the population.” • “We are 95% confident that …” • Quick equation: (n is the size of the sample)

  15. Margin of error • In many surveys (i.e., Gallup), the margin of error is rounded to the nearest whole number. • To cut the margin of error in half, we must use a sample size 4 times as large.

  16. Try these • Margin of error for sample sizes… • 1000 • 150 • 520 Answers: Sample size 1000 = margin of error 0.03=3% Sample size 150 = margin of error 0.08=8% Sample size 520 = margin of error 0.04=4%

  17. Confidence Statements • Contains 2 parts: • Margin of error – how close the sample statistic lies to the population parameter. • Level of confidence – what percent of all possible samples satisfy the margin of error.

  18. Confidence Statements • The conclusion applies to the population, not to the sample. • The conclusion is never completely certain. • You can use other confidence levels (not just 95%). • If not given, the confidence level in most studies is 95%.

  19. Example… • According to Gallup, “57% of Americans have bought a lottery ticket in the last 12 months”. This claim was made about the population of 200 million adults. 1523 people were chosen at random to be surveyed. • 1/√n = 1/√1523 = 0.026 ≈ 0.03

  20. Statements… • We are 95% confident that 57% of US adults bought a lottery ticket in the past 12 months, plus or minus 3%. • We are 95% confident that 54% - 60% of US adults bought a lottery ticket in the past 12 months.

  21. Homework • Page 86, #2.25-2.29 • Page 94-95, #2.35-2.39

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