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Statistical Sampling. Part III – Confidence Intervals. This video is designed to accompany pages 41-76 in Making Sense of Uncertainty Activities for Teaching Statistical Reasoning Van- Griner Publishing Company. Error Due to Sampling. Americans and Their Guns

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## Statistical Sampling

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**Statistical Sampling**Part III – Confidence Intervals**This video is designed to accompany**pages 41-76 in Making Sense of Uncertainty Activities for Teaching Statistical Reasoning Van-Griner Publishing Company**Error Due to Sampling**Americans and Their Guns Title: Poll: Majority of Americans Back Stricter Gun Laws Authors: Sarah Dutton, Jennifer De Pinto, Anthony Salvanto, Fred Backus, Leigh Ann Caldwell Source:CBS News January 17, 2013 As the president outlined sweeping new proposals aimed to reduce gun violence, a new CBS News/New York Times poll found that Americans back the central components of the president's proposals, including background checks, a national gun sale database, limits on high capacity magazines and a ban on semi-automatic weapons. Asked if they generally back stricter gun laws, more than half of respondents - 54 percent - support stricter gun laws …. That is a jump from April - before the Newtown and Aurora shootings - when only 39 percent backed stricter gun laws but about the same as ten years ago. … This poll was conducted by telephone from January 11-15, 2013 among 1,110 adults nationwide. Phone numbers were dialed from samples of both standard land-line and cell phones. The error due to sampling for results based on the entire sample could be plus or minus three percentage points.**Margin of Error**• The 3% is called the “margin of error” for the survey. • You may have heard about this in elementary school. • You may never have heard about it before now. • Our goal is to make sure you understand what it really is and isn’t.**Fundamental Problem**Another, equally well-chosen SRS of n = 1,110 adult Americans, asked the same question, will almost surely yield a different. And it can be quite different from the 0.54seen here.**Sampling Variability**The variability seen in a statistic from sample to sample is called “sampling variability.” Question: Which statistic is correct? Answer: They are all correct! Better How can you estimate the parameter in Question: the face of this variability?**Mathematics to the Rescue**With SRS-type samples, sampling variability is understandable, predictable. With convenience samples it is not. If sample is not a probabilistic sample, then it will be very difficult to do the formal inference with integrity.**Predictable? How?**If you were to do the sampling over and over and plot the different statistics you get …. Then that plot – called a sampling distribution – would exhibit predictable characteristics. In particular, it would be bell-shaped and peak above the parameter from the population.**Predictable? How?**Suppose you take a SRS of size 80 and ask “Are you in favor of same-sex marriages?” Record the proportion who say “Yes.” Put those 80 back in the population and take another sample of size 80. Do this 25 times. A histogram of these 25 sample proportions would have to be bell-shaped and peak above the parameter in the population, which appears to be about 2/3 in this case.**Enormously Useful**Can make quantitative statements about how far is likely to be from p, even if you don’t know p. From the graphic, one can see that 95% of ’s based on samples of size n, will be within of p.**95% Confidence Interval**If you are estimating a proportion p … From a simple random sample of size n … And want 95% “confidence” in your estimate … Then your margin of error is … And a 95% confidence interval for p is**Example Revisited**Americans and Their Guns Title: Poll: Majority of Americans Back Stricter Gun Laws Authors: Sarah Dutton, Jennifer De Pinto, Anthony Salvanto, Fred Backus, Leigh Ann Caldwell Source:CBS News January 17, 2013 As the president outlined sweeping new proposals aimed to reduce gun violence, a new CBS News/New York Times poll found that Americans back the central components of the president's proposals, including background checks, a national gun sale database, limits on high capacity magazines and a ban on semi-automatic weapons. Asked if they generally back stricter gun laws, more than half of respondents - 54 percent - support stricter gun laws …. That is a jump from April - before the Newtown and Aurora shootings - when only 39 percent backed stricter gun laws but about the same as ten years ago. … This poll was conducted by telephone from January 11-15, 2013 among 1,110 adults nationwide. Phone numbers were dialed from samples of both standard land-line and cell phones. The error due to sampling for results based on the entire sample could be plus or minus three percentage points. A 95% confidence interval for p is =0.54 +/- 0.03**No Probabilistic Sample?**Numbers can still be produced from this type of formula since plugging in is easy! But if the samples were not SRS-like, then those numbers are meaningless.**A Technical Interpretation**This kind of plot describes where is likely to fall, relative to p, for any SRS of size n. Since 95% of ’s based on samples of size n, will be within of p, it follows that for 95% of all the ’s observed, the range will be wide enough to cover p.**Parameter**p = 0.79 0.60 0.70 0.80 0.90 Illustration of Interpretation 95% confidence” means to 95% of a long list of these intervals would contain the parameter!**One-Sentence Reflection**Simple formulas are available for the margin of error and associated confidence intervals, provided the data were collected in a simple random sample, or similarly statistically correct fashion.

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