# Confidence interval help.

Could someone please shed some light on this topic. and explain why if the “An increase in sample size increases degrees of freedom, reducing the reliability factor.” Question and answer attached. Thank you 10 Which of the following changes in sample size and degrees of freedom, respectively, is most likely to result in an increase in the width of the confidence interval for the population mean?      Sample size Degrees of freedom A. Increase Increase B. Increase Decrease C. Decrease Increase D. Decrease Decrease Correct answer = D“Sampling and Estimation,” Richard A. Defusco, Dennis W. McLeavey, Jerald E. Pinto, and David E. Runkel2008 Modular Level I, Vol. 1, pp. 440-441Study Session 3-10-jcalculate and interpret a confidence interval for a population mean, given a normal distribution with 1) a known population variance, 2) an unknown population variance, or 3) an unknown variance and a large sample sizeThe standard error becomes smaller as the sample size increases. An increase in sample size increases degrees of freedom, reducing the reliability factor. A reduction in both the standard error and the reliability factor decrease the width of the confidence interval. A decrease in both would increase the width.

Increasing sample size directly increases DoF. DoF is one minus the sample size (n-1) and is a characteristic for the t-distribution (and chi-squared)

Sorry about that: 10 Which of the following changes in sample size and degrees of freedom, respectively, is most likely to result in an increase in the width of the confidence interval for the population mean? Sample size Degrees of freedom A. Increase Increase B. Increase Decrease C. Decrease Increase D. Decrease Decrease Correct answer = D . An increase in sample size increases degrees of freedom, reducing the reliability factor. A reduction in both the standard error and the reliability factor decrease the width of the confidence interval. A decrease in both would increase the width.

I understand that an increase in n, increases df. How does an increase in the df reduce the reliability factor? How does the reduction in the standard error decrease the width?

If you take a glance at a t-distribution table, you’ll notice that the t-values decrease for a specific alpha as degrees of freedom increase. A smaller t-value makes it much easier to commit a type I error and mistakenly reject the null hypothesis when it shouldn’t. Thus, an increase in sample size diminishes the reliability of the test. Hope that helps.

swellsrf Wrote: ------------------------------------------------------- > If you take a glance at a t-distribution table, > you’ll notice that the t-values decrease for a > specific alpha as degrees of freedom increase. A > smaller t-value makes it much easier to commit a > type I error and mistakenly reject the null > hypothesis when it shouldn’t. > > Thus, an increase in sample size diminishes the > reliability of the test. Hope that helps. I understand the logic here but it still seems completely counter-intuitive.

Seems pretty straightforward. If you lower the sample size, what are your chances of correctly estimating the population paramters? Probably a lot lower. In order to compensate for this, you would need an acceptance range that is wider.

Got it, thanks very much for your help.

bpdulog Wrote: ------------------------------------------------------- >"Seems pretty straightforward. If you lower the > sample size, " ==> Qn asks about increasing the sample size what are your chances of correctly > estimating the population paramters? Probably a > lot lower. ===> So, probability of estimating pop params should be HIGHER In order to compensate for this, you > would need an acceptance range that is wider. ==> So would a lower acceptance range (confidence interval) NOT hurt? Like zero alluded to, “reliabilty” on the greater sample size “reduces” is counter-intuitive… Does someone have a better way of explaining this or some other technique to remember this?

First, let’s get the terminology straight sample size - forget about the T-distribution (or whatever). Increasing the sample size directly decreases the std. error. You can see this mathematically by seeing the Sqrt(n) in the denominator or just intuitively, the more data you get the better you should be able to estimate the parameter you are trying to estimate. degrees of freedom - this is related to sample size almost always but is a parameter of the distribution you are using for your C.I. As pointed out above, the tails of a T-distribution get thinner as df increases, so for a fixed alpha as df increase t[alpha/2] gets smaller. That means that for higher df, the C.I. gets narrower based on the t-stat without regard to the standard error it’s multiplied by. The technical reason for this (which you don’t have to know) is that as your sample size gets bigger the error in using the sample standard deviation with an X-bar instead of a mu in it becomes ever less important. Anyway, increase df => decrease C.I. width increase n => decrease C.I. width and vice versa

thanks Joey.

This may not be technically correct (or maybe it is?), but the way I reasoned this question out was to consider that an increase in sample size or degrees of freedom gives you more info and paints a more accurate picture of the population (as Joey said). More info allows you to narrow down where in the CI it will fall.

^ sure that’s a fine way to think about it.