Confidence Interval Alpha Error
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19 March, 2015 What do significance levels and P values mean in hypothesis tests? What is statistical significance anyway? In this post, I’ll continue to focus on concepts and graphs to help you gain a more intuitive understanding of how hypothesis tests work in confidence interval alpha beta statistics. To bring it to life, I’ll add the significance level and P value to
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the graph in my previous post in order to perform a graphical version of the 1 sample t-test. It’s easier to understand confidence interval and alpha level when you can see what statistical significance truly means! Here’s where we left off in my last post. We want to determine whether our sample mean (330.6) indicates that this year's average energy cost is significantly different from
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last year’s average energy cost of $260. The probability distribution plot above shows the distribution of sample means we’d obtain under the assumption that the null hypothesis is true (population mean = 260) and we repeatedly drew a large number of random samples. I left you with a question: where do we draw the line for statistical significance on the graph? Now we'll add in the significance level and the P value, which are the 95 confidence interval alpha decision-making tools we'll need. We'll use these tools to test the following hypotheses: Null hypothesis: The population mean equals the hypothesized mean (260). Alternative hypothesis: The population mean differs from the hypothesized mean (260). What Is the Significance Level (Alpha)? The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference. These types of definitions can be hard to understand because of their technical nature. A picture makes the concepts much easier to comprehend! The significance level determines how far out from the null hypothesis value we'll draw that line on the graph. To graph a significance level of 0.05, we need to shade the 5% of the distribution that is furthest away from the null hypothesis. In the graph above, the two shaded areas are equidistant from the null hypothesis value and each area has a probability of 0.025, for a total of 0.05. In statistics, we call these shaded areas the critical region for a two-tailed test. If the population mean is 260, we’d expect to obtain a sample mean that falls in the critical region 5% of the time. The critical region defines how far away
the sample data and the population parameter is inferred (or estimated) from this sample statistic. Let me say that again: Statistics are calculated,
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parameters are estimated. We talked about problems of obtaining the value of
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the parameter earlier in the course when we talked about sampling techniques. Another area of inferential statistics formula 95 confidence interval is sample size determination. That is, how large of a sample should be taken to make an accurate estimation. In these cases, the statistics can't be used since the http://blog.minitab.com/blog/adventures-in-statistics/understanding-hypothesis-tests:-significance-levels-alpha-and-p-values-in-statistics sample hasn't been taken yet. Point Estimates There are two types of estimates we will find: Point Estimates and Interval Estimates. The point estimate is the single best value. A good estimator must satisfy three conditions: Unbiased: The expected value of the estimator must be equal to the mean of the parameter Consistent: The value of the estimator approaches https://people.richland.edu/james/lecture/m170/ch08-int.html the value of the parameter as the sample size increases Relatively Efficient: The estimator has the smallest variance of all estimators which could be used Confidence Intervals The point estimate is going to be different from the population parameter because due to the sampling error, and there is no way to know who close it is to the actual parameter. For this reason, statisticians like to give an interval estimate which is a range of values used to estimate the parameter. A confidence interval is an interval estimate with a specific level of confidence. A level of confidence is the probability that the interval estimate will contain the parameter. The level of confidence is 1 - alpha. 1-alpha area lies within the confidence interval. Maximum Error of the Estimate The maximum error of the estimate is denoted by E and is one-half the width of the confidence interval. The basic confidence interval for a symmetric distribution is set up to be the point estimate minus the maximum error of the estimate is less
test AP formulas FAQ AP study guides AP calculators Binomial Chi-square f Dist Hypergeometric Multinomial Negative binomial Normal Poisson t Dist Random numbers Probability Bayes rule Combinations/permutations Factorial Event http://stattrek.com/estimation/margin-of-error.aspx counter Wizard Graphing Scientific Financial Calculator books AP calculator review Statistics AP study guides Probability Survey sampling Excel Graphing calculators Book reviews Glossary AP practice exam Problems and solutions Formulas Notation Share with Friends Margin of Error In a confidence interval, the range of values above and below the sample statistic is called the margin of error. For example, suppose confidence interval we wanted to know the percentage of adults that exercise daily. We could devise a sample design to ensure that our sample estimate will not differ from the true population value by more than, say, 5 percent (the margin of error) 90 percent of the time (the confidence level). How to Compute the Margin of Error The margin of error can be confidence interval alpha defined by either of the following equations. Margin of error = Critical value x Standard deviation of the statistic Margin of error = Critical value x Standard error of the statistic If you know the standard deviation of the statistic, use the first equation to compute the margin of error. Otherwise, use the second equation. Previously, we described how to compute the standard deviation and standard error. How to Find the Critical Value The critical value is a factor used to compute the margin of error. This section describes how to find the critical value, when the sampling distribution of the statistic is normal or nearly normal. The central limit theorem states that the sampling distribution of a statistic will be nearly normal, if the sample size is large enough. As a rough guide, many statisticians say that a sample size of 30 is large enough when the population distribution is bell-shaped. But if the original population is badly skewed, has multiple peaks, and/or has outliers, researchers like the sample size to be even larger. When the sampling distribution is nearly normal, the