# How many individuals must be measured to determine the true mean?

I saw this was asked earlier, but I also have the same question. This was the question: "Anatomy: The human height is the distance from head to toe. When populations share a genetic background and other environmental factors such as diet, the average height is frequently characteristic within a group. In a large group of Caucasians, how many individuals must be measured to determine the true mean height of a male within 0.5 inches at the 10% level of significance, given that the standard deviation is 1.5 inches."

What I'm confused about is the 10% level of significance...this is not the confidence level is it? I'm also going to assume the 0.5 is the interval of confidence as in plus/minus 0.5? Should I be using this equation: $n > (\frac{\sigma \frac{z_\alpha}{2})}{E})^2$? Any help/pointers would be appreciated!

EDIT: would the E value be the 0.5? and the 10% is alpha, making $\frac{\alpha}{2}= 0.05$ giving it a Z score of 1.65? $$n > \frac{(1.65)(1.5)}{0.5}$$ $$n > 4.95$$ ?

• Just one (if that individual happens to be at the true mean! ;) – Alexis Apr 21 '15 at 15:50

I am not quite sure about the meaning of the question.

First of, the question asks for the mean height of "a" male in a large group of Caucasians. I guess this means that the mean height of all male individuals in the specified large group is required.

Strictly speaking, and only for one exact point in time (considering that height varies as a function of age/ time of day/ etc.), one would need a perfect instrument for measuring the exact size of the individuals in question, and importantly, one would have to measure all individuals belonging to the population (i.e. the large group of Caucasian males) in order to determine the truly true mean height. This is because all measuring is only approximation and any measured values in terms of the classic test theory is an addition of a true value plus measurement error.

So, in order not to painstakingly measure every individual and considering the lack of a perfect instrument which does not give a measurement error, one would need to take samples in order to estimate the true value of the mean height of the population of Caucasian males.

Of course, the larger the sample, the better. If you have a large sample, its mean height will be a much better estimate of the population's mean height than if you draw a small sample, in which the sampling error will be larger. The size of the sample and the reliability of your instrument will influence the standard error of the parameter of interest (i.e. the mean of the height). The larger the sample and the more reliable your instrument, the smaller the standard error becomes. You can estimate the standard error (which is a characteristic of the sampling distribution, see this useful linkg ) from the standard deviation of the measurements in your sample.

So. Now turning to your question concerning the 10%. Together with the abovementioned standard error, and a chosen alpha (which in this case appears to be 10%), you can construct confidence intervals. That is, the 10 per cent level of significance is not the confidence interval but rather a characteristic of the confidence interval. You can construct any confidence interval you like between 0 and 100 percent. The higher the value, the larger the confidence interval.

Commonly, 95 per cent confidence intervals are used in sciences such as psychology. Let us assume you calculate a 90 per cent confidence interval (because of the alpha of 10 %) for the mean height you measured in your sample, which ranges from 170 to 190 centimeters. Sometimes it is taught in statistics courses, that this means that with a probability of 90 % the population's mean height lies between 170 and 190 centimeters. This is not correct. Rather it means that if one took an infinite amount of samples from the population in question, 90% of the sample means would lie between 170 and 190 centimeters.

So, in conclusion:

1. You can only calculate the true mean height of the large Caucasian male population,

• if you have a perfect measuring instrument that measures without error,
• if you measure all individuals of the population
• and if you measure all individuals at the same time (because height varies, see above; this point makes the definition of true mean height quite questionable).

, everything else will give you only an approximation of the true mean height of the population.

2. The 10% level of significance is a characteristic of the confidence intervals that you chose. It is not the confidence interval itself.

• Thank you for taking the time to reply, it most certainly helped me understand the concept! @bunsenbaer – Brooke Dec 7 '14 at 0:29
• I am happy to hear that, although I am sorry not to have given you a solution to the conrete question you have asked ;) – bunsenbaer Dec 8 '14 at 11:21
• There indeed is much useful information in this answer. Unfortunately, some key elements are just plain wrong. The worst is the interpretation of a confidence interval of [170,190] as "90% of the sample means would lie between 170 and 190 centimeters." For correct interpretations please search our site. – whuber Feb 13 '15 at 14:36
• Thank you @whuber for pointing out the worst one of my mistakes (apparently another interpretation of CIs being tought in statistics courses that is incorrect). I will be very glad to know which other aspects are incorrect and will check out the link you supplied. – bunsenbaer Feb 14 '15 at 15:29
• As I said, much of this answer is useful. However, upon rereading the entire answer and then reviewing the question, my opinion remains that this answer does not address the question. I think it would be more helpful to respond to the issues related to finding an appropriate sample size needed to estimate the mean to a given precision, and specifically to help interpret the symbols in the formula given in the question. Enough time has elapsed that you should not be concerned about disclosing the full answer to a homework question. – whuber Feb 14 '15 at 15:35