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Chapter 21

Chapter 21. STA 200 Summer I 2011. Previously. A parameter is a number that describes the population. In practice, we won’t know its value. Additionally, the parameter is constant (its value does not change ).

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Chapter 21

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  1. Chapter 21 STA 200 Summer I 2011

  2. Previously • A parameter is a number that describes the population. In practice, we won’t know its value. Additionally, the parameter is constant (its value does not change). • A statistic is a number that describes a sample. A statistic can vary from sample to sample. We will know the value of a statistic. • Sampling variability is the idea that the value of a statistic will vary from one sample to another.

  3. Sampling Distribution of • The sampling distribution of is the distribution of values taken by in all possible samples of the same size. • If the sample size is large enough: • the sampling distribution of is approximately normal • the mean is p • the standard deviation is

  4. 68-95-99.7 Rule • Since the sampling distribution of is approximately normal, the 68-95-99.7 Rule applies. • If we sample repeatedly, about 95% of the values of will be within two standard deviations of p, by the 68-95-99.7 Rule.

  5. 68-95-99.7 Rule • Symbolically, p will be captured by the interval from to 95% of the time. • We can be a little more accurate: for 95% confidence, it’s really 1.96 standard deviations not 2. • Thus, the formula becomes .

  6. One Final Adjustment • In practice, we won’t know the value of p, so we’ll have to guess at it. If we have to guess a specific value, the best guess is . • The formula for a 95% confidence interval is:

  7. Example • In a random sample of 687 college students, 419 indicated that they favored having the option of receiving pass-fail grades for elective courses. • Construct a 95% confidence interval for p, the proportion of all college students who favor the option of pass-fail grades for elective courses.

  8. Other Confidence Levels • The confidence level doesn’t have to be 95%; you can have any confidence level between 0% and 100%. • The other commonly used confidence levels are 90% and 99%.

  9. Generalized Confidence Interval Formula • The generalized formula for confidence intervals is: • z* is called the critical value, which is dependent on the confidence level. • Common Critical Values:

  10. More Examples • Construct a 99% confidence interval for the pass-fail example. • Note: A higher confidence level requires a larger margin of error (and, therefore, a wider interval). A lower confidence level requires a smaller margin of error (and, therefore, a narrower interval).

  11. More Examples (cont.) • An airline wants to determine the proportion of passengers that bring only carry-on luggage. In a random sample of 300 passengers, 31% have only carry-on luggage. • Find a 90% confidence interval for the proportion of all passenger who have only carry-on luggage. • Note: A larger sample size results in a smaller margin of error (and vice versa).

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