I'm trying to understand how uncertain an estimation of a demand distribution of a good is. If I take a random sample of n people from the representative consumer population who give me their personal cutoff/choke prices (assuming they accurately report them), I can construct a demand curve with n points representing an estimation of the proportion of the population who would buy the good at that price.

Given that I used n independent pieces of information to calculate n points, should I be using a t-distribution with 1 degree of freedom to calculate the confidence interval for each point? And, if I want to reduce this interval by increasing the degrees of freedom, I either need to take more samples, or increase the granularity of the demand curve to get more d.f. at each point?

  • $\begingroup$ Could you clarify how you would construct a demand curve? Do your n points constitute the "curve", or would you having calculated the n points use regression to fit a smooth curve to them? $\endgroup$ Commented Jun 18, 2014 at 21:01
  • $\begingroup$ n "choke price" points can be aggregated into a bar chart (the "demand curve"), where each bar represents the proportion of people in the sample who would buy the good at that price. It is a cumulative distribution. You can, of course, smooth this discrete bar chart into a smooth continuous curve, but as it doesn't add new information (and the application I'm using this for doesn't require it) I don't bother. $\endgroup$
    – Escher
    Commented Jun 19, 2014 at 13:04
  • $\begingroup$ A regression doesn't add new information, but it could perhaps make more effective use of the information you have. However, I have answered your question on the basis that each estimated point simply reflects the sample proportion of people who would buy the good at that price. $\endgroup$ Commented Jun 20, 2014 at 10:03

1 Answer 1


You should use a $t$-distribution with $n-1$ degrees of freedom (or if $n$ is large the normal distribution which provides a good approximation to the $t$-distribution). A larger sample, increasing $n$, would narrow the confidence interval both because it increases the degrees of freedom (although once $n$ increases beyond 30 this effect is quite small) and because it reduces the standard error.

For a given price the sample contains, say, $m$ people who would buy the good at that price. Thus the proportion of the population who would buy the good at that price can be estimated as $m/n$. This applies to any price, not just to the $n$ cutoff prices given by the people in the sample. So you can calculate any number of points, taking a point to be identified by a price and its associated estimated proportion.

The number of points that are or could be calculated has no relevance to the degrees of freedom used in calculating a confidence interval for the proportion at any one price (the situation is quite different from a regression in which $n$ observations are used to jointly estimate $k$ parameters, implying $n-k$ degrees of freedom). The degrees of freedom here depend only on the size $n$ of the sample used to estimate that proportion and its standard error, from which we deduct 1 because use of the same sample to estimate both the proportion and its standard error implies a loss of 1 degree of freedom.

The value of $t_{n-1}$ at the desired confidence level times the estimated standard error gives the half-width of the confidence interval.


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