Question

In: Statistics and Probability

Pls attempt both parts for UPVOTE a) For the following dataset:- A B C 82.95406 48.596...

Pls attempt both parts for UPVOTE

a) For the following dataset:-

A	B	C
82.95406	48.596	62.83925
80.2694	94.88806	69.11351
51.32409	87.00438	5.26083
84.40903	73.14477	67.37821
7.744191	70.83899	30.42249
70.09185	96.19882	35.38787
27.85478	70.86354	10.82541
36.31444	54.53047	94.30487
78.58975	88.44509	91.97403
78.83427	97.59331	67.44993
40.58147	62.05577	67.98824
5.522503	0.005762	28.78233
51.0516	75.53139	82.53751
22.99913	6.099075	16.05481
37.90452	78.80319	33.0078
90.42208	68.23812	86.88297
59.52895	23.34578	8.346984
96.59504	52.17967	75.20052
98.23697	87.31435	97.50355
56.66422	25.66281	27.79151
16.59429	84.47958	61.71686
53.90397	10.89486	93.26763
55.11838	13.11304	75.92159
71.32999	70.36975	10.86584
40.88035	84.11119	97.83293
88.07786	10.15206	76.98687
86.25806	68.54747	98.22674
14.63472	37.58765	68.50834
48.94452	77.09557	45.1666
83.50869	20.72787	33.30376
59.14445	55.82262	96.20811
1.253421	18.14296	71.29829
32.03952	22.48347	1.707322
82.10399	54.66754	71.42761
1.551587	88.15809	13.04672
55.40726	71.10242	10.2861
66.0299	17.13271	90.60817
70.02227	49.47755	9.984934
11.2358	99.71097	2.637771
54.2171	64.7902	28.8158

Please examine Does 68.26% of the data fall within one standard deviation of the mean value? Does 95% of the data fall within 2 standard deviations? Does 99.7% of the data fall within 3 standard deviations? Does the statistical analysis behave as expected?

Please also create a Histogram for data. Use 10 “bins” in your histogram.

b)

For the following dataset:-

A	B
65.77503	64.79644
87.57873	81.42366
69.16423	47.8631
78.7769	74.97734
39.29159	36.33522
83.14534	67.22618
49.35916	36.51458
45.42246	61.71659
83.51742	86.33629
88.21379	81.29251
51.31862	56.87516
2.764132	11.43687
63.2915	69.70683
14.5491	15.05101
58.35385	49.90517
79.3301	81.84772
41.43736	30.40724
74.38735	74.65841
92.77566	94.35162
41.16351	36.70618
50.53694	54.26358
32.39941	52.68882
34.11571	48.051
70.84987	50.85519
62.49577	74.27482
49.11496	58.4056
77.40276	84.34409
26.11119	40.24357
63.02004	57.0689
52.11828	45.84677
57.48353	70.39173
9.698192	30.23156
27.26149	18.74344
68.38577	69.39971
44.85484	34.25214
63.25484	45.59859
41.5813	57.92359
59.74991	43.16158
55.47338	37.86151
59.50365	49.27436

Determine the mean and standard deviations for these new sets and create histograms for them as well. Examine the standard deviation ranges and comment on whether these new sets provide a “Normal” distribution.

Expert Solution

R code:

#Reading the data
x=read.csv(file.choose(),header = FALSE)
v=x$V1
s=sd(x$V1)
m=mean(x$V1)

#Percentage of values that lie within one sd
length(which(v<m+s & v>m-s))/length(v)*100

#Percentage of values that lie within two sd
s1=2*sd(x$V1)
length(which(v<m+s1 & v>m-s1))/length(v)

#Percentage of values that lie within 3 sd
s2=3*sd(x$V1)
length(which(v<m+s2 & v>m-s2))/length(v)

#Check whether data is normal
qqnorm(v)

From the above code, we can see that 58.33333% values lie within one sd from the mean

But , 100% values lie within two sd from the mean.

Also 100% values lie within three sd from the mean.

The reason of this is there is very high standard deviation of the data.

The concept that 68.26,95,99.7 % values lie within one,two,three sd from the mean is only valid if the data is normal.

So, we checked the qqplot of the data i.e. quantile quantile plot which showed the data does not follow normal distribution

Histogram R code:

min(v)
max(v)
hist(v,breaks=c(0,10,20,30,40,50,60,70,80,90,100),main="Histogram of data",xlab="Bins")

Output:

From the histogram also it is evident that the data do not follow normal distribution

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