In [5]:

import numpy as np import matplotlib.pyplot as plt import scipy.stats as stats

In [6]:

#Onesided normal percentile point function #alpha = 0.05 ==> 1-alpha/2 = 0.975 print stats.norm.ppf(0.95) print stats.t.ppf(0.95,1000)

1.64485362695
1.64637881729

In [3]:

#<your calculations here> 1.0 / (1.0/4.0 + 1.0/7.0 + 1.0/6.0)

1.7872340425531914

Let M denotes the size of Linux distribution to be downloaded. The individual speeds of seeders are given by M/4, M/7, and M/6 and these values are constant. Let h denotes total hours requires to download the distribution if all the seeders are used together. The portion of the distribution will be downloaded from each seeder are h*M/4, h*M/7, and h*M/6, respectively. The sum of these values is M. Then we have h*M/4 + h*M/7 + h*M/6 = M => h = 1/(1/4 + 1/7 + 1/6) = 1.787 hours Answer: ~ 1.79 hours

In [4]:

#<your calculations here> # the ammount is downloaded in the first hour with speed of 290Kbps S1 = 290*3600*1000 # the ammount is downloaded in the first hour with speed of 253Kbps S2 = 253*3600*1000 # the ammount is downloaded in the first hour with speed of 269Kbps S3 = 269*3600*1000 # the ammount is downloaded in the first hour with speed of 1008Kbps S4 = 1008*3600*1000 #Total ammount in bit S = S1 + S2 + S3 + S4 #Total ammount in Mbytes SM = S/(8.0*pow(2,20)) print SM

781.059265137

The size of the distribution in Mbytes is Answer: ~781.06 MB

In [8]:

dataset = list() with open("labdata.dat") as datafile: for line in datafile: dataset.append(int(line)) #print dataset print np.mean(dataset) onewayRTT = np.mean(dataset)/2.0 print onewayRTT

31.527
15.7635

In [13]:

data_col1 = list() data_col2 = list() with open("labdata2.dat") as datafile: for line in datafile: data_col1.append(int(line.split()[0])) data_col2.append(int(line.split()[1])) print data_col1 print data_col2

[428, 423, 426, 404, 399, 403, 440, 439, 443, 354, 353, 357, 324, 328, 325, 328, 325, 328, 325, 325, 324, 326, 329, 325, 329, 325, 329, 337, 342, 337, 351, 349, 349]
[541, 525, 525, 440, 440, 441, 444, 445, 440, 451, 451, 455, 404, 399, 399, 436, 437, 437, 446, 445, 448, 378, 377, 376, 390, 390, 388, 405, 404, 407, 371, 372, 373]

In [16]:

print np.mean(dataset) onewayRTT = np.mean(dataset)/2.0 print onewayRTT

31.527
15.7635

The approximate time which is required for a message to go from A to B is calculated as a half of average RTT between A and B Answer: 15.7635 ms

In [6]:

RTT = np.mean(dataset) print RTT

31.527

The typical RTT is calculated by averaging of all 1000 RTT samples Answer: 31.527 ms

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stdev = np.std(dataset) print stdev

19.354463852

How much the RTT varies on average is represented by the standard deviation variable. It is given below Answer: ~ 19.35 ms

In [48]:

plt.hist(dataset) plt.show()

Answer: By ploting histogram of RTT samples from dataset, we get the shape of the distribution of RTT samples in the above figure in which x-basis represents RTT value in milisecond and y-basis represents frequency of appearance of each value

In [6]:

#<your calculations here>

Answer: From the above histogram, it looks like the shape of exponential distribution.

In [24]:

def conf_func(X, clevel): avg = np.mean(X) stdev = np.std(X) alpha = 1 - clevel n = len(X) temp = stats.norm.ppf(1 - alpha/2)*stdev/(pow(n,1/2.0)) #number of measurement = 1000 >> 30 c1, c2 = avg - temp, avg + temp return c1, c2 print conf_func(dataset, 0.90)

(30.520280682916461, 32.533719317083538)

Calculate [c1, c2] for a confidence level of 90% Answer: [c1, c2] = [30.520280682916461, 32.533719317083538]

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print conf_func(dataset, 0.95)

(30.327419945158617, 32.726580054841385)

Calculate [c1, c2] for a confidence level of 95% Answer: [c1, c2] = [30.327419945158617, 32.726580054841385]

In [25]:

print "Confidence Intervals for confidence level of 90%: [c1,c2] = ",conf_func(dataset, 0.90) print "Confidence Intervals for confidence level of 95%: [c3,c4] = ",conf_func(dataset, 0.95)

Confidence Intervals for confidence level of 90%: [c1,c2] = (30.520280682916461, 32.533719317083538)
Confidence Intervals for confidence level of 95%: [c3,c4] = (30.327419945158617, 32.726580054841385)

Answer 01: Confidence Interval is the range of values so defined that is a specified probability that the value of a parameter lies within it or probability of including the actual value, the mean value in this case. The size of this range affects the precison of measurement. 90% confidence means that there is a 90% chance that the actual mean is within that interval. Increasing the confidence level to 95% means that we are increasing the probability that the actual mean is in the resulting interval. That's the reason why the interval is larger for 95% than for a 90% interval. Answer 02: Increasing the confidence level means we are inscreasing size of precision domain. The wider interval implies that we know less about the actual mean. The higher our confidence about the mean, the less precise is the information we seem to have.

In [14]:

print conf_func(data_col1, 0.95)

(351.85107458580217, 365.05801632328871)

The range of values that we can expect the real transmission time to be is represented by confidence intervals Answer: (351.85107458580217, 365.05801632328871)

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def num_rep(X, err_rang, clevel): avg = np.mean(X) alpha = 1.0 - clevel err = err_rang/2.0 stdev = np.std(X) temp = (stats.norm.ppf(1 - alpha/2)*stdev)/(avg*err) n = temp**2 return n print num_rep(data_col1, 0.03, 0.95)

The number of repetitions of the experiment are required to be sure that the average transmission time is within 3% of the average is given as below Answer: 228

In [20]:

# Values for original TCP stack print "Original TCP Stack" print "Mean = ", np.mean(data_col1) print "Standard Deviation = ",np.std(data_col1) # Values for modified TCP stack print "Modified TCP Stack" print "Mean = ", np.mean(data_col2) print "Standard Deviation = ",np.std(data_col2)

Original TCP Stack
Mean = 358.454545455
Standard Deviation = 41.3990857088
Modified TCP Stack
Mean = 426.666666667
Standard Deviation = 43.0670137898

To compare two versions of TCP stack before and after modifying, we compute mean and standard deviatoin Answer: After modifying, new TCP stack increases the average of transmission time compared to the original one, from 358.45 to 426.67 (time units)

In [19]:

# Means of Difference Approach d = list() d = list(np.array(data_col2) - np.array(data_col1)) print "List d =",d print "Average of d=",np.mean(d) print "Standard Deviation of d=", np.std(d) print "Number of degrees of freedom =",len(d) print "Confidence Intervals [c1,c2] = ",conf_func(d,0.95) # Difference of Mean Approach avg_dif = np.mean(data_col2) - np.mean(data_col1) print "Difference of Average=", avg_dif n1 = len(data_col1) n2 = len(data_col2) s1 = np.std(data_col1) s2 = np.std(data_col2) temp1 = s1**2/n1 + s2**2/n2 std_comb = pow(temp1,1/2.0) print "Combined standard deviation = ",std_comb temp2 = pow(s1**2/n1,2)/(n1-1) + pow(s2**2/n2,2)/(n2-1) n_dof = temp1**2/temp2 print "Number of degrees of freedom", n_dof temp3 = stats.norm.ppf(0.975)*std_comb/pow(n_dof,1/2.0) c1, c2 = avg_dif - temp3,avg_dif + temp3 print c1, c2,

List d = [113, 102, 99, 36, 41, 38, 4, 6, -3, 97, 98, 98, 80, 71, 74, 108, 112, 109, 121, 120, 124, 52, 48, 51, 61, 65, 59, 68, 62, 70, 20, 23, 24]
Average of d= 68.2121212121
Standard Deviation of d= 36.4564975925
Number of degrees of freedom = 33
Confidence Intervals [c1,c2] = (61.608650343377938, 74.815592080864491)
Difference of Average= 68.2121212121
Combined standard deviation = 10.3990850039
Number of degrees of freedom 63.9004102198
65.6624076375 70.7618347867

Answer: The confidence intervals for means of difference approach are [c1,c2] = [61.608650343377938, 74.815592080864491] and the confidence intervals for difference of means approach are [c3,c4] = [65.6624076375, 70.7618347867]. These values are different. Although, two mean values in both approaches are the same but the differences between the standard deviation values and number of degrees of freedom when calculating distribution function (n # n_dof) are reasons why these values are different.

In [134]:

#<your calculations here> #net1, net2, net3 = list(), list(), list() net1 = [129,139,142,138,133] net2 = [144,132,135,145,139] net3 = [142,147,144,149,138] #print net1[:], net2[:], net3[:] matrix = [net1,net2,net3] print matrix # First way print "Pre-built Function:",stats.f_oneway(net1,net2,net3) # Second way mean_list = list() n = 5 k = 3 for j in range(k): mean_list.append(np.mean(matrix[:][j])) SSA = 0 SSE = 0 SST = 0 # Calculate variation due to effects of alternatives for j in range(k): SSA += n*(mean_list[j] - np.mean(mean_list))**2 print "SSA =",SSA # Calculate variation due to errors in measurements for j in range(k): for i in range(n): SSE += (matrix[j][i] - mean_list[j])**2 print "SSE =",SSE # Calculate difference between each measurement and overall mean for j in range(k): for i in range(n): SST += (matrix[j][i] - np.mean(mean_list))**2 #print SSA + SSE, STT = SSA + SSE print "STT =",SST # Degrees of freedom df_SSA = k-1 df_SSE = k*(n-1) df_SST = k*n-1 # Mean square values sa_sq = SSA/df_SSA se_sq = SSE/df_SSE # Comparing variances F_cacluated = sa_sq/se_sq F_calculated2 = (SSA/SSE)*k*(n-1)/(k-1) #print "F_calculated2 =",F_calculated2 print "F_calcuated =",F_cacluated

[[129, 139, 142, 138, 133], [144, 132, 135, 145, 139], [142, 147, 144, 149, 138]]
Pre-built Function: F_onewayResult(statistic=3.0534550195573615, pvalue=0.084726918790441252)
SSA = 156.133333333
SSE = 306.8
STT = 462.933333333
F_calcuated = 3.05345501956

Answer: Comparing the F value calculated from the set of computation above to F value got from the F-distribution table, we see that F_calculated = 3.0535 < F_tabulated[0.95;2;12] = 3.89. Therefore, with 95% confidence that differences among the alternatives are not statistically significant

In [6]:

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Answer: Using F-distribution table, we can get the F value with 90% confidence as F_tabulated_90 = 2.81 which is smaller than F_calculated = 3.0535. Therefore, it can be concluded that with 90% confidence, the differences among the alternatives are statistically significant.

In [6]:

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Answer: In the experimental design, we need to know if our new design can have better performce than other alternatives. In order to do that we need a technique for determining whether any changes we see are due to random fluctuations in the measurements or they are statistically significant. As shown above, ANOVA can help us but we need to carefully choice the confidence level because this value can affect comparision results. In previous computation for example, with confidence level of 95% we cannot see any difference or improvement between three alternatives but confidence level of 90% does. However, ANOVA doesn't tell us where the differences occur. Therefore, we may need to use contrast method to find more detailed comparisions between those alternatives.

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