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Confidence Intervals

Confidence Intervals. Ap Statistics Newell. Statistical Inference. Methods for drawing conclusions about a population based on sample data. From a sample we can get: Mean Proportion Standard Deviation s. Quantitative Data. Categorical Data (usually).

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Confidence Intervals

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  1. Confidence Intervals Ap Statistics Newell

  2. Statistical Inference Methods for drawing conclusions about a population based on sample data. From a sample we can get: Mean Proportion Standard Deviation s Quantitative Data Categorical Data (usually)

  3. Amount of “wiggle room” you’re willing to give Based on confidence level Sample mean or Sample proportion

  4. Interpretation… … of the confidence level, C “We expect that, in repeated samples using the same method, the confidence intervals created would capture the true population parameter C% of the time.” … of the confidence interval “We can be C% confident that the true population parameter is between the end points of the interval.”

  5. To calculate confidence intervals we’ll need… • Sampling distribution based on sample mean or proportion and • A sample statistic (mean or proportion) • Standard deviation: Means: Proportions: Standard Error

  6. We’ll also need… Critical Values That is, the critical value is the z-score that cuts off the appropriate tails in the normal distribution.

  7. The critical value for a 90% confidence interval will cut off 10% of the tails. That’s 5% in each tail. So we need to find the z-score that cuts off the bottom 95% of data. Use invNorm(.95)… z* = 1.645

  8. Find the critical values for… • 92% confidence interval. z* = 1.751 • 94% confidence interval. z* = 1.881 • 98% confidence interval. z* = 2.326 • 95% confidence interval. z* = 1.96

  9. Confidence interval Estimate ± z*(standard error)

  10. To find a confidence interval… • Ask whether you have a sample mean and standard deviation or a sample proportion. • Find the standard deviation (or standard error) using the appropriate formula. • Find the value of z*. • Create the margin of error. • Put it all together.

  11. Example As part of a study to evaluate appropriate vitamin C levels in corn soy blend (CSB), measurements were taken on samples of CSB produced in a factory. The following data are the amounts of vitamin C measured in milligrams per 100 grams of blend for a random sample of size 8 from a production run. Suppose the population standard deviation is known to be 7.1913 grams. 26 31 23 22 11 22 14 31

  12. Construct a 95% confidence interval for m. 1. Are you working with mean or proportion? What is the sample statistic? sample mean = 22.5 • Find the standard error using the appropriate formula. • Find the value of z*. invNorm(.975) = 1.96

  13. Construct a 95% confidence interval for m. 4. Create the margin of error margin of error = z*(standard error) = 1.96(2.5425) = 4.9833 5. Put it all together. 22.5 ± 4.9833 (17.5167, 27.4833) Interpretation: We can be 95% confident that the true mean amount of vitamin C in corn soy blend is between 17.5167 and 27.4833 milligrams per 100 grams of blend.

  14. Example Have efforts to promote equality for women gone far enough in the United States? A poll on this issue by the cable network MSNBC contacted 1019 adults. Overall, 54% of the sample answered “Yes.” Find a 90% confidence interval for the proportion in the adult population who would say “Yes” if asked.

  15. Assumptions and Conditions What must be true in order to make sure we can run this? • Random: The sample should be an SRS of the population (or it should be reasonable to assume this) • Normal: The sampling distribution must be approximately normal Means: Sample size must be large enough for CLT to be in effect OR the original population must be approx. normal Proportions: n ≥10 and n(1-)≥10 • Independent: 10% rule must be in effect

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