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I'm having some trouble with this problem. The way I'd go about solving it is:

$56.2 +- 2.576 * \frac{53.6}{\sqrt5}$

However this does not give me the correct answer which is:

$56.2 +- 15.07$.

Can some one help me out?

EDIT:

I worked the problem out with the guidance from below:

56 +- 4.604 * (7.3212/sqrt(5))

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  • $\begingroup$ You have to use square root of 53.6 $\endgroup$ – user83346 Oct 19 '17 at 4:23
  • $\begingroup$ 99% CI's are in many cases meaningless, you could as well use 100% interval: $(-\infty, \infty)$, in both cases it would be very wide and would not provide you much information. $\endgroup$ – Tim Oct 19 '17 at 7:54
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The key to this problem is that they say you are working with $s^2$, which is an estimate of the population variance. You have applied a z-distribution to the problem, which would be correct if you had the actual population variance ($\sigma^2$) . However, when you are working with an estimate of the population variance you must apply a student's t-distribution to the problem.

The form of the t-distribution equation is similar to the z-distribution equation except that you replace the "z" with a "t", and the population standard deviation with the sample standard deviation (s):

$t = \frac{xbar - \mu}{\frac{s}{\sqrt{n}}}$

where $s$ is the square root of the sample variance, $n$ is the sample size, and $t$ is a value to be calculated from a t-table. You will need two things to find the t-value to plug into your equation:

  1. Confidence interval - in this case you are asked for 99% confidence, meaning that your total acceptable Type I error rate ($\alpha$) is 1-.99 = .01. That error rate must be split over both ends of the t-distribution, thus the heading of the t - table column you seek should read .005.
  2. Degrees of freedom ($n$). In this case, your Degrees of Freedom will be 5-1 = 4

Go to a t-table with a t($\alpha$/2=.005, DOF=4), and you will have enough information to find your answer.

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  • $\begingroup$ So just to clarify: If you are given an estimate you ALWAYS have to use the t-statistic? Also, when do you seek 0.01/2 or 0.005 vs 0.01? Because some questions I simply use the value of 2.576. Is that only without degrees of freedom? $\endgroup$ – Koalafications Oct 19 '17 at 2:38
  • $\begingroup$ Yes, if the sample is 'small'. The t-distribution converges on the normal distribution as the number of samples grows. By the time you have 50 samples from an infinite population, the difference is usually irrelevant. $\endgroup$ – eSurfsnake Oct 19 '17 at 5:28

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