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Large Sample Confidence Interval for p Lecture 15

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Large Sample Confidence Interval for p Lecture 15

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    1. 1 Large Sample Confidence Interval for p Lecture 15 William F. Hunt, Jr. Statistics 361 Section 7.3

    2. 2 Sampling dist. for proportions Variables which are Yes/No summarized with a proportion which sample you select will indicate which proportion you get. Need to know the distribution of the sample proportion.

    3. 3 Notation

    4. 4 The Basic Paradigm.

    5. 5 Sampling Different samples => Different statistics Each sample could have a different proportion Depends on which particular individuals we have chosen. How varied?

    6. 6 St. Dev of sample proportion

    7. 7 St. Dev of sample proportion

    8. 8 St. Dev of sample proportion

    9. General Properties of the Sampling Distribution of p: (page 304 Text) 1. p = p 2. sp = 3. If both np > 10 and n(1-p) > 10, the sampling distribution is approximately normal.

    10. 10 Variability Variability depends on the population proportion. Sample that is 80/20 is more predictable than one which is 50/50. Depends on sample size Larger sample=> less variability

    11. 11 Proportion in Population

    12. 12 Proportion in Population

    13. Proportions For a large sample the sample proportion will have a normal distribution centered around the population proportion p with standard deviation Can use normal to model proportions if we have a large sample.

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    15. Estimation Margin of Error (MOE) numeric indication of distance a statistic may be from true parameter Use the sampling distribution to determine the MOE This is where the Margin of error comes in. You might be familiar with the margin of error in polls you may have seen in the news. The margin of error gives us an idea of how far from the parameter the statistic might be. Since we know how the statistic acts we can predict how far it might be from the true population parameter using the sampling distribution we discussed in the previous section. Before we examine the method to find the MOE we should review a few things about the normal distribution.This is where the Margin of error comes in. You might be familiar with the margin of error in polls you may have seen in the news. The margin of error gives us an idea of how far from the parameter the statistic might be. Since we know how the statistic acts we can predict how far it might be from the true population parameter using the sampling distribution we discussed in the previous section. Before we examine the method to find the MOE we should review a few things about the normal distribution.

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    17. How much should the margin of error be? Should depend on the standard error Too small may miss the value, too big may not be useful. how often will it contain the true parameter What percentage of samples will work? How large should the margin of error be? We could make the margin of error very large. That would not be very helpful So if a statistician told you that the proportion of people who support raising taxes is between 0 and 100%. That would not really be helpful. We instead need to choose a margin of error that will contain the true parameter. How large should the margin of error be? We could make the margin of error very large. That would not be very helpful So if a statistician told you that the proportion of people who support raising taxes is between 0 and 100%. That would not really be helpful. We instead need to choose a margin of error that will contain the true parameter.

    18. How often do we capture the parameter Confidence level-Percentage of possible samples for which margin of error works 90% confidence: We capture the true parameter in 90% of the possible samples. Based on sampling distribution of statistic (Z scores) The proportion of times that we do capture the parameter is called the confidence level. For instance if we might have 90% confidence or 80% confidence. 90% confidence means that in 90% of the possible samples we would capture the true population parameter. We will be able to determine the proportion of times that we will capture the parameter by using the normal distribution and the Z-scores. The proportion of times that we do capture the parameter is called the confidence level. For instance if we might have 90% confidence or 80% confidence. 90% confidence means that in 90% of the possible samples we would capture the true population parameter. We will be able to determine the proportion of times that we will capture the parameter by using the normal distribution and the Z-scores.

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    20. Margin of error We can make our formula more specific by filling in the formula for the standard error of the statistic. This will give us a margin of error that we can use to determine how far from the true parameter the statistic might be. However, there is a small problem. We can make our formula more specific by filling in the formula for the standard error of the statistic. This will give us a margin of error that we can use to determine how far from the true parameter the statistic might be. However, there is a small problem.

    21. Margin of error This formula needs the true population proportion. That true proportion is unknown.This formula needs the true population proportion. That true proportion is unknown.

    22. Margin of error Put in sample proportion since we dont know true value of p Since we dont know the true population proportion. We will instead use our best guess at this proportion the sample proportion. We can use this formula as the margin of error for a population proportion.Since we dont know the true population proportion. We will instead use our best guess at this proportion the sample proportion. We can use this formula as the margin of error for a population proportion.

    23. Example A random sample of 100 undergraduate students from a large university found that 78 of them had used the university librarys website to find resources for a class. Find the margin of error for the true proportion of all undergraduates who had used the librarys website to find resources for a class. Use 95% confidence level. Lets take an example. Libraries are changing rapidly with new technology. Many libraries are devoting more resources to their websites and helping students find research materials using websites. In this example we see the results of a survey of 100 undergraduates. Of these undergraduates we see that 78% of them have used the library website to find resources for a class. Lets take an example. Libraries are changing rapidly with new technology. Many libraries are devoting more resources to their websites and helping students find research materials using websites. In this example we see the results of a survey of 100 undergraduates. Of these undergraduates we see that 78% of them have used the library website to find resources for a class.

    24. Example A random sample of 100 undergraduate students from a large university found that 78 of them had used the university librarys website to find resources for a class. Find the margin of error for the true proportion of all undergraduates who had used the librarys website to find resources for a class. Use 95% confidence level. In this case pi-hat is given by 78/100 or 0.78. In other words 78% of the sampled students had used the library website. But what percentage of the whole population had used the website? We can find the margin of error using the formula from before. In this case pi-hat is given by 78/100 or 0.78. In other words 78% of the sampled students had used the library website. But what percentage of the whole population had used the website? We can find the margin of error using the formula from before.

    25. Example 95% confidence In this example we want 95% confidence. That means we need to find the point from a normal distribution that has 2.5% above it. Using the table we see that is 1.96.In this example we want 95% confidence. That means we need to find the point from a normal distribution that has 2.5% above it. Using the table we see that is 1.96.

    26. Example 95% confidence =>1.96 In this example we want 95% confidence. That means we need to find the point from a normal distribution that has 2.5% above it. Using the table we see that is 1.96.In this example we want 95% confidence. That means we need to find the point from a normal distribution that has 2.5% above it. Using the table we see that is 1.96.

    27. Example Plugging in the appropriate values the Z score is 1.96 and the sample proportion is 0.78. The sample size is 100. running the numbers through the calculator we find Plugging in the appropriate values the Z score is 1.96 and the sample proportion is 0.78. The sample size is 100. running the numbers through the calculator we find

    28. Example running the numbers through the calculator we find the margin of error would be 1.96 times 0.041 or 0.08. or 8%running the numbers through the calculator we find the margin of error would be 1.96 times 0.041 or 0.08. or 8%

    29. Example The proportion of students who have used the university library to find resources for a class is 78% 8% So we could say that the proportion of students who have used the university library to find resources for a class is 78% 8% So we could say that the proportion of students who have used the university library to find resources for a class is 78% 8%

    30. Confidence Interval Statistic Margin of Error Common way to present estimate. (interval estimation) This interval should contain the true population parameter. We often use the term confidence interval to talk about the statistic plus or minus the margin of error. In many research settings and publications we present the confidence interval. This interval should contain the true population parameter. We often use the term confidence interval to talk about the statistic plus or minus the margin of error. In many research settings and publications we present the confidence interval. This interval should contain the true population parameter.

    31. Confidence Interval

    32. Example A random sample of 100 undergraduate students from a large university found that 78 of them had used the university librarys website to find resources for a class. Find a 95% confidence interval for the true proportion of all undergraduates who had used the librarys website to find resources for a class. So revisiting our example from the previous section we can turn the margin of error we had into a confidence interval.So revisiting our example from the previous section we can turn the margin of error we had into a confidence interval.

    33. Example Recall that our margin of error from before was 8%. We can add and subtract this from the statistic to get an interval. Typically we write the interval as the two endpoints. In this case it would be 0.7 to 0.86 or in other words 70% to 86%.Recall that our margin of error from before was 8%. We can add and subtract this from the statistic to get an interval. Typically we write the interval as the two endpoints. In this case it would be 0.7 to 0.86 or in other words 70% to 86%.

    34. Example We are 95% confident that the true proportion of undergraduates who used the librarys website is between 70% and 86% So in conclusion we say We are 95% confident that the true proportion of undergraduates who used the libraries website is between 70% and 86% So in conclusion we say We are 95% confident that the true proportion of undergraduates who used the libraries website is between 70% and 86%

    35. Class Problem In a study of a particular wafer inspection process, 356 dies were examined and 201 of these passed the probe. Assuming a stable process, calculate a 95% two-sided confidence interval for the proportion of all dies that pass the probe. Section 7.3 page 310 problem 21Section 7.3 page 310 problem 21

    36. Confidence Interval for p (page 305) Show confidence interval for pi from text page 305Show confidence interval for pi from text page 305

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