In this problem we are going to consider the total annual flow in the Minnesota

Question

In this problem we are going to consider the total annual flow in the Minnesota River measured at Jordan, MN. Table 1 gives ten years of data. Answer the following three questions based on the data in Table 1 alone, and the assumption that the total annual flow is well-modelled as statistically independent from year to year. a. Compute and report the arithmetic mean, x, and sample standard deviation, s, for these ten observed values. b. Establish the two-sided 95 percent confidence interval for the mean total annual flow. Although this cannot be demonstrated using only 10 observed values, for the purposes of this problem you may assume that the underlying population approximately follows a Normal distribution; i.e. you may use the Student-t distribution to create the confidence interval using the sample standard deviation. c. It is desired to estimate the mean total annual stream flow to within plusminus 0.1 times 10^9 [m^3] with 95 percent confidence. Approximately how many total years of observation will be required to achieve this goal? Assume that the sample standard deviation will be approximately unchanged; i.e. sigma almostequalto s. Table 1: Total annual flow in the Minnesota River at Jordan, MN. These annual data were computed using the daily data from USGS site #05330000. Consider the complete available data set for the total annual flow of the Minnesota River at Jordan, MN, as shown in Figure 1. Answer the following three questions based on the data in this figure. Be brief in your responses but, use complete sentences. d. Does Figure 1 support or contradict the assumption that the total annual flow is well-modelled as statistically independent from year to year? e. Does Figure 1 support or contradict the assumption that the sample standard deviation will be approximately unchanged; i.e. sigma almostequalto s. f. Given Figure 1, are you confident that you can estimate the mean total annual stream flow to within plusminus 0.1 times 10^9 [m^3] with 95 percent confidence using the number of years that you estimated in part (c)?

Explanation / Answer

a. arithmatic mean of the sample xbar= 6.063 x 109

standard deviation s = 3.684 x 109

b. t - value for 95% confidence interval for dF = 10-1 = 9

tcritical = +- 2.262

so 95 % confidence interval

xbar - 2.262 * (s/n) <= xbar <= xbar + 2.262 * (s/n)

6.063 x 109 - 2.262 * (3.684 x 109 /10) <=  xbar <= 6.063 x 109 + 2.262 * (3.684 x 109 /10)

3.4278 x 109 <=  xbar <=  8.698 x 109

c. How many observations required to have confidence interval less than +- 0.1 x 109

so for that 0.1 x 109 >= tcritical * (s/n)

0.1 x 109 >= 2.262 * (3.684 x 109 /n)

n >= 2.262 * 3.684 /0.1

n >= 83.33

n > 6944 or n = 6945 years

for d. , e. and f. please provide figure 1, i will do it then. Thank you.

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