Question

From your analysis of Desertia, you conclude that citizens there average $1,000 per month in household disposable income. The Minister of Development for Mountania says that disposable income is the same in his country. To see if the data supports this, your company has sampled 100 households from Mountania and obtained data on household disposable income. Use the Data provided to:

a. ​Derive a frequency distribution of income. Divide by 8
b.​Create a histogram of income
c.​Obtain and fully interpret/explain the sample mean and standard deviation
d.​Test the hypothesis, at the 5% significance level, that Mountania also has a mean of $1,000

Disposable Income
602
822.65
842.79
840.95
820.94
958.15
910.1
957.48
995.66
867.17
802.62
1008.77
745.97
905.88
940.84
754.77
960.76
897.26
835.38
925.79
948.25
872.22
719.48
883.08
1028.29
738.2
996.32
919.36
819.94
1004.23
861.31
1067.62
847.87
1041.8
791.61
906.73
1033.35
935.21
983.94
1004.91
993.6
970.68
794.23
948.29
949.45
804.84
936.08
986.73
784.51
922.03
1014.22
963.19
829.64
806.42
860.37
891.54
731.69
888.42
868.95
1025.6
976.48
878.54
871.55
772.61
953.84
830.44
859.37
781.73
880.33
915.49
939.17
867.59
804.17
924.46
847.51
882.95
818.04
807.13
892.76
824.78
977.04
1079.09
895.89
872.66
953.17
994.84
842.77
1033.59
766.47
976.48
846.98
942.39
840.77
856.07
1041.24
782.38
884.3
970.45
878.92 961.69

Answer 1

Part a

A required frequency distribution with eight classes is given as below:

Class	Frequency
602 to 661.64	1
661.64 to721.28	1
721.28 to780.92	6
780.92 to 840.56	18
840.56 to 900.2	28
900.20 to959.84	20
959.84 to 1019.48	18
1019.48 to 1079.12	8
Total	100

*Range for given data is given as 477.09, so divide this value by 8, we get 59.64 approximately. So, class width is 59.64.

*Range = Maximum –minimum = 1079.09 – 602 = 477.09

Part b

A required histogram for the income is given as below:

From above histogram, it is observed that the data for income follows an approximately normal distribution.

Part c

For the study variable disposable income, the sample mean is given as 894.9626 with the standard deviation of 88.59088802. The average disposable income of citizens in Mountania is given as $895 with the standard deviation of $89 approximately.

Part d

Here, we have to use one sample t test for the population mean.

Null hypothesis: H₀: The Mountania citizens have mean disposable income equal to $1,000.

Alternative hypothesis: H_a: The Mountania citizens have mean disposable income different than $1,000.

We are given

Level of significance = 5% or α = 0.05

Test statistic formula is given as below:

t = (Xbar - µ) / [S/sqrt(n)]

From given sample, we have

Xbar = 894.9626

S = 88.59088802

n = 100

df = n – 1 = 100 – 1 = 99

t = (894.9626 – 1000)/[ 88.59088802/sqrt(100)]

t = (894.9626 – 1000)/ 8.8591

t = -11.8565

Critical value = ±1.9842

P-value = 0.0000

P-value < α = 0.05

So, we reject the null hypothesis that The Mountania citizens have mean disposable income equal to $1,000.

There is insufficient evidence that Mountania citizens have mean disposable income of $1,000.