The reason for this is that a t distribution has greater variability in its tails than a standard normal distribution. Confidence intervals provide us with a way to estimate a population parameter. As we saw in the first two problems, here we also have different levels of confidence. This range of values is typically an estimate, along with a margin of error that we add and subtract from the estimate. Mathematically, the formula for the confidence interval is represented as, Thus we will again use a table of t-scores. Courtney K. Taylor, Ph.D., is a professor of mathematics at Anderson University and the author of "An Introduction to Abstract Algebra. The level of confidence gives a measurement of how often, in the long run, the method used to obtain our confidence interval captures the true population parameter. We use the following formula to calculate a confidence interval for a difference in population means: Confidence interval = (x 1 – x 2) +/- t*√((s p 2 /n 1) + (s p 2 /n 2)) where: In the second two problems the population standard deviation is unknown. Attached to every interval is a level of confidence. The 95% confidence interval for the true population mean weight of turtles is [292.36, 307.64]. We will see that the method we use to construct a confidence interval about a mean depends on further information about our population. If we know that 0.2 cm is the standard deviation of the tail lengths of all newts in the population, then what is a 90% confidence interval for the mean tail length of all newts in the population? The value of, Here we do not know the population standard deviation, only the sample standard deviation. There are 24 degrees of freedom, which is one less than sample size of 25. The difference between these two problems is that the level of confidence is greater in #2 than what it is for #1. Parametric and Nonparametric Methods in Statistics, know the value of the population standard deviation, the population standard deviation is unknown, B.A., Mathematics, Physics, and Chemistry, Anderson University. The reason for this is that in order to be more confident that we did indeed capture the population mean in our confidence interval, we need a wider interval. How Large of a Sample Size Do Is Needed for a Certain Margin of Error? Example: Reporting a confidence interval “We found that both the US and Great Britain averaged 35 hours of television watched per week, although there was more variation in the estimate for Great Britain (95% CI = 34.02, 35.98) than for the US (95% CI = 33.04, 36.96).” If you are asked to report the confidence interval, you should include the upper and lower bounds of the confidence interval. Below we will look at several examples of confidence intervals about a population mean. If we find that that 0.2 cm is the standard deviation of the tail lengths of the newts in our sample the population, then what is a 95% confidence interval for the mean tail length of all newts in the population? The formula for confidence interval can be calculated by subtracting and adding the margin of error from and to sample mean. Specifically, the approach that we take depends on whether or not we know the population standard deviation or not. If we know that 0.2 cm is the standard deviation of the tail lengths of all newts in the population, then what is a 95% confidence interval for the mean tail length of all newts in the population? When we use a table of, Here we do not know the population standard deviation, only the sample standard deviation. Thus we will use a table of t-scores. Rather than say that the parameter is equal to an exact value, we say that the parameter falls within a range of values. A Confidence Interval is a range of values we are fairly sure our true value lies in. It is helpful when learning about statistics to see some examples worked out. We start with a simple random sample of 25 a particular species of newts and measure their tails. By working through countless examples of how to create confidence intervals for the difference of population means, we will learn to recognize when to use a z-test or t-test and when to pool or not based on the sample data provided. We begin by analyzing each of these problems.

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