Review of Concept---Confidence Interval (CI).

In the science of statistical inference, we use sample,(x1,x2,...,xn),taking from a population to make inference about that population. For instance,

we use the distribution of sample mean (average),

X=1/n n xi

i=1

to make an inference about the population mean, . AlthoughXis an estimate of , it is only an estimate with uncertainty. A confidence interval about the mean, ,allows us to express the accuracy ofanestimate about mean.

Let do a useful mental exercise that would enhance your insight into the problem. Imagine, if we repeatedly take samples, (x1,x2,...,xn)

,m timesfrom a population, the sample averagescalculated from each of the m samples, would provide a useful information about the distribution of . Intuitively, the mean of these averages, X1, X2, ..., Xm would be a good estimate of the population mean,and a confidence interval can be readily derived from them if m is large enough (Indeed, this is a more advancedstatistical method that does not require an assumption of being normally distributed). You may find it easier to understand if youpretend thatthe population is a urncontaining one million cards on each of them the annual income of a household in 2016 is printed. You do not know the distribution of their incomes but you are allowed to draw 30 cards (n=30)randomly for 25 times (m=25). In each draw, you got an idea about the population characteristics: mean, median, variation, etc. The 25 draws would let you know more about how these population characteristics distribute/vary.

Now, ifall we have is a single sample (m=1), we would have torely onthe assumption that the sample is from a normal population, and then use this informationto construct a confidence interval about the mean by a formula.

Discussion Topic:

The state highway administration wants to know the adequacy of the current speed limit of 65 mph in a section of a state highway. The adequacy is defined as "no more than 5% of vehicles at any time exceeding 80 mph". As a preliminary effort, a small traffic speed data (unit in mph) was collected by a field engineer:

{67.06,82.78,77.14,66.62,75.44,69.63,70.79,88.59, 66.82, 68.95,90.16,76.55, 73.38, 50.27, 61.66,73.46,88.27,69.55,74.54,64.85,72.88, 72.64, 68.57,57.11, 83.07,59.39,75.03, 74.14,59.67,75.63}

With the statistical knowledge that you have learned so far, answer the following questions.

a. Assuming the data are normally distributed, calculate a 90% and 95%confidence intervals (CI)for the population mean.

b. On the basis of the CI obtained in (a), what recommendation can you make?

c. What caveatsyou would like to make? Feel free to express your views about the limitation of the data, and/or thestatistical tools you possess. (Limited to under 100 words).