# Confidence interval for mean of non-normal distribution based on a small sample

The following problem appears in chapter 7 of Probability and Statistics for Engineers and Scientists by Ross:

Suppose that a random sample of nine recently sold houses in a certain city resulted in a sample mean price of \$222,000, with a sample standard deviation of \$22,000. Give a 95 percent upper confidence interval for the mean price of all recently sold houses in this city.

If we were sampling from a normal distribution, then we could construct a confidence interval based on the fact that $$\sqrt{n} (\bar X - \mu)/S$$ has a $$t$$-distribution. (Here $$\bar X$$ is the sample mean, $$S$$ is the sample standard deviation, and $$n$$ is the size of the sample.) Alternatively, if $$n$$ were large, then we could construct a confidence interval based on the fact that $$\sqrt{n} (\bar X - \mu)/S$$ is approximately normally distributed. However, in this problem we are sampling from a non-normal distribution and the sample size is small. So what can be done?

I don't need a solution worked out in full detail -- I'd just like to know what approach the book expects us to take.

I don't have Ross's book at hand. Without the context of what has been covered recently, it is difficult to know what approach the author expects you to take.

Perhaps you are supposed to assume that the nine observations are from a nearly normal population. Then you could use the t-interval you mentioned.

Perhaps use of a nonparametric procedure is intended. Then one possibility would be the one-sided nonparametric confidence interval that often accompanies a Wilcoxon signed-rank test. Here is an example of that using R statistical software. (Other statistical software will do much the same thing, but the syntax of the code will be somewhat different.)

Given the following nine (fictional) housing prices in thousands of dollars,

y
## 410.4 413.3 430.4 460.4 473.9 508.3 533.6 546.7 555.3


we find that the 95% upper bound (for the population median) is $521,000, as shown below: summary(y); sd(y) Min. 1st Qu. Median Mean 3rd Qu. Max. 410.4 430.4 473.9 481.4 533.6 555.3 ## 56.95459 # standard deviation  wilcox.test(y, conf.int=T, alte="less") Wilcoxon signed rank test data: y V = 45, p-value = 1 alternative hypothesis: true location is less than 0 95 percent confidence interval: -Inf 520.95 sample estimates: (pseudo)median 482  Note: (1) Using R software, the corresponding t-interval gives the bound \$516,700. It is not surprising that the t and Wilcoxon bounds are about the same because I generated my fake housing prices from a normal distribution. (2) Another nonparametric procedure uses the confidence interval corresponding to a sign test. From Miniab 17 software the bound for the population median is \$537,200.

• Thanks! At this point in the book, nonparametric procedures have not been introduced. So I bet the intention is that the reader should assume the nine observations are from a nearly normal distribution, as you suggested. Very helpful answer. – eternalGoldenBraid Feb 11 at 8:46
• I agree with your guess. (Remember the nonparametric example for later in the course.) – BruceET Feb 11 at 8:49