Problem 2 In this problem we are going to consider the total annual flow in the

Question

Problem 2 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 as- sumption 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 fol- lows a Normal distribution; i.e. you may use the Student-t distribution to create the confidence interval using the sample standard deviation 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 or s Year Flow [m3] 2003 2.58 x 10 2004 4.47 x 10 2005 5.66 x 10 2006 5.84 x 10 2007 5.77 x 10 2008 4.29 x 10 2009 3.85 x 10 2010 12.67 x 10 2011 12.74 x 10 2012 2.76 x 10 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 o s. f. Given Figure 1, are you confident that you can estimate the mean total annual stream flow to within 10.1 x 109 [ms] with 95 percent confidence using the number of years that you estimated in part (c)?

Explanation / Answer

Part a

The arithmetic mean Xbar and sample standard deviation for the given data is given as below:

Arithmetic mean = Xbar = 6063000000 (or 6.06*10^9 approximately)

Sample standard deviation = S = 3684086047 (or 3.68*10^9 approximately)

Part b

Now, we have to find out the 95% confidence interval for the mean total annual flow.

The confidence interval formula is given as below:

Confidence interval = Xbar -/+ t*S/sqrt(n)

We have

Xbar =6063000000

S = 3684086047

n = 10

df = n – 1 = 10 – 1 = 9

c = 0.95

t = 2.2622

Confidence interval = 6063000000 -/+ 2.2622*3684086047/sqrt(10)

Confidence interval = 6063000000 -/+ 2635436390.5239

Lower limit = 6063000000 - 2635436390.5239 = 3427563609.48

Upper limit =6063000000 + 2635436390.5239 = 8698436390.52

Confidence interval = (3.43*10^9, 8.70*10^9)

Part c

We are given

Margin of error = E = 0.1*10^9

Confidence level = 95%

Z = 1.96

= 3684086047

Sample size formula is given as below:

n = (Z*/E)^2

n = (1.96*3684086047/0.1*10^9)^2

n = 5213.8161

Required sample size =5214

Part d

Yes, the given figure supports the assumption that the total annual flow is well-modeled as statistically independent from year to year because the figure does not observed any trend and points shown distributed randomly.

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