STANDARD DEVIATION CALCULATIONS

Part A

(a)

  Mean Standard Deviation
1. Distribution Cost 71.26208333 12.92979437
2. Sales 456.5 81.53206891
3.Number of Orders 4393 737.0807873

 

The above values have been calculated as: mean=

And the Standard deviation is calculated as S.D. =  where  is the sample mean and x is sample value while N is the sum of the total terms

4. The Skewness table for Cost:

Cost

Mean

71.26208

Standard Error

2.639283

Median

70.805

Mode

#N/A

Standard Deviation

12.92979

Sample Variance

167.1796

Kurtosis

-0.82859

Skewness

0.28367

Range

41.98

Minimum

52.46

Maximum

94.44

Sum

1710.29

Count

24

 

The Skewness table for Sales:

Sales

Mean

456.5

Standard Error

16.64266

Median

457.5

Mode

#N/A

Standard Deviation

81.53207

Sample Variance

6647.478

Kurtosis

-0.37714

Skewness

0.074334

Range

322

Minimum

301

Maximum

623

Sum

10956

Count

24

 

The Skewness table for Numbers of order received:

Number of orders Received

Mean

4393

Standard Error

150.456

Median

4282.5

Mode

#N/A

Standard Deviation

737.0808

Sample Variance

543288.1

Kurtosis

-0.45946

Skewness

0.052952

Range

2814

Minimum

2921

Maximum

5735

Sum

105432

Count

24

 

 

As can be seen in the tables the Skewness among the cost is the highest.

(b) The Sample proportion of the all the orders which exceed the 4000 is 0.5 and the Standard deviation for the               distribution of the sample proportion is 556.612 as can be found out from the above formula.

 

(c)

As the number of observations in the sample is less than 30, we use t-test to get the population mean as follows:

 

Here,  is the population mean,  is the sample mean,  is the t coefficient in the confidence interval with the sample size N=24 and standard Deviation as S.

(i) From above, =71.26  4.3828

Thus, the population mean interval for cost in millions of dollars is 71.26  4.3828

(ii)From above,=456.542.5078

Thus, the population mean interval for Sales in millions of dollars is 456.542.5078.

(iii) From above,=228.2847

Thus, the population mean interval for Number of orders is 228.2847.Please not the sample size is reduced to 12.

(d) The Sample size should be greater than 30 in order, that point estimate is calculated.

(e) Given population mean=65 (In thousand dollars)

Sample mean of the cost=76.21 (In thousand dollars)

Standard deviation of the cost=12.92

Hence, the Z= =-0.8882, the corresponding Z score is 0.316

Thus, the claim is correct for significance level 0.05.

(f) Z== 0.97

Which is equivalent to the probability of 0.3389, hence the claim is correct.

(g) The t == -2.56

Whereas, the t-value should be lower than 2.50.

Hence, the claim is incorrect.

(h)Given population mean=65 (In thousand dollars)

Sample mean of the cost=68.308 (In thousand dollars)

Standard deviation of the cost=10.78

Hence, the Z= =-0.306, the corresponding Z score is 0.1406

Thus, the claim is correct for significance level 0.05.

The change in the values is due to the change in the mean and the standard deviation because of the reduced sample size.

Part B

a) For A(c)

(i)

t-Test 1-sample
Test Mean

71.26208

Confidence Level

0.95

N

24

Average

71.26208

Test Stdev p 1-sample Stdev
Stdev

12.92979

12.92979

0.922

SE Mean

2.639283

T

0.000

TINV

1.713872

p – One sided

0.5

Accept Null Hypothesis  because p > 0.05 (Means are the same)
p – two sided

1

Accept Null Hypothesis  because p > 0.05 (Means are the same)

 

(ii)

t-Test 1-sample
Test Mean

456.5

Confidence Level

0.99

N

24

Average

456.5

Test Stdev p 1-sample Stdev
Stdev

81.53207

81.53207

0.922

SE Mean

16.64266

T

0.000

TINV

2.499867

p – One sided

0.5

Accept Null Hypothesis  because p > 0.01 (Means are the same)
p – two sided

1

Accept Null Hypothesis  because p > 0.01 (Means are the same)

 

iii)

t-Test 1-sample
Test Mean

4576.417

Confidence Level

0.9

N

12

Average

4576.417

Test Stdev p 1-sample Stdev
Stdev

524.1024

524.1024

0.887

SE Mean

151.2953

T

0.000

TINV

1.36343

p – One sided

0.5

Accept Null Hypothesis  because p > 0.1 (Means are the same)
p – two sided

1

Accept Null Hypothesis  because p > 0.1 (Means are the same)

 

A(d). The table is similar to A(c)(iii)

A(e).The table is similar to A(c)(i)

A(f)

Test Mean

3577

Confidence Level

0.95

N

7

Average

3577

Test Stdev p 1-sample Stdev
Stdev

399.6523

399.6523

0.846

SE Mean

151.0544

T

0.000

TINV

1.94318

p – One sided

0.5

Accept Null Hypothesis  because p > 0.05 (Means are the same)
p – two sided

1

Accept Null Hypothesis  because p > 0.05 (Means are the same)

 

A(g) The table is similar to A(c)(ii)

A(h).

Test Mean

68.30833

Confidence Level

0.99

N

12

Average

68.30833

Test Stdev p 1-sample Stdev
Stdev

10.78041

10.78041

0.887

SE Mean

3.112035

T

0.000

TINV

2.718079

p – One sided

0.5

Accept Null Hypothesis  because p > 0.01 (Means are the same)
p – two sided

1

Accept Null Hypothesis  because p > 0.01 (Means are the same)

 

 

Part-C

(a)

i)In order to get the model of Cost vs. Sales, we using the least square method, we have the:

Coefficient of Sales=0.13354757

Intercept of Cost= 10.29761784

So, we have: Cost= (0.13354757) sales+10.29761784

ii)In order to get the model of Cost vs. Number of orders Received, we using the least square method, we have the:

Coefficient of Number of orders Received=0.016117564

Intercept of Cost= 0.457625305

So, we have: Cost= (0.016117564) Number of orders Received+0.457625305

(b) The model in part (ii) describes the cost better than (i) because, the coefficient of determination is stronger in the previous one (0.844200772) than the latter case (0.709162344).While the slope of the curves shows the relationship between the two variable the strength of the relationship is shown by the value of Coefficient of determination.

(c)In order to get the model of Cost vs. Sales andNumber of orders Received, we using the least square method, we have the:

Coefficient of Sales=0.047113872

Coefficient of Number of orders Received=0.011946926

Intercept of Cost= -2.728246583

So, we have: Cost= (0.047113872) sales+ (0.011946926) Number of orders Received -2.728246583

(d) For part a (i)

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.842118

R Square

0.709162

Adjusted R Square

0.695942

Standard Error

7.129671

Observations

24

ANOVA

 

df

SS

MS

F

Significance F

Regression

1

2726.822

2726.822

53.64357

2.47E-07

Residual

22

1118.309

50.83221

Total

23

3845.13

     

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

10.29762

8.449998

1.218653

0.235883

-7.22661

27.82184

-7.22661

27.82184

Sales

0.133548

0.018234

7.324177

2.47E-07

0.095733

0.171362

0.095733

0.171362

 

For part a(ii)

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.918804

R Square

0.844201

Adjusted R Square

0.837119

Standard Error

5.218274

Observations

24

ANOVA

 

df

SS

MS

F

Significance F

Regression

1

3246.062

3246.062

119.2074

2.39E-10

Residual

22

599.0683

27.23038

Total

23

3845.13

     

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

0.457625

6.571883

0.069634

0.945114

-13.1716

14.08688

-13.1716

14.08688

Number of orders Received

0.016118

0.001476

10.91821

2.39E-10

0.013056

0.019179

0.013056

0.019179

 

For part c

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.935914

R Square

0.875936

Adjusted R Square

0.86412

Standard Error

4.766166

Observations

24

ANOVA

 

df

SS

MS

F

Significance F

Regression

2

3368.087

1684.044

74.1336

3.04E-10

Residual

21

477.043

22.71633

Total

23

3845.13

     

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

-2.72825

6.15788

-0.44305

0.66226

-15.5343

10.07777

-15.5343

10.07777

Sales

0.047114

0.020328

2.317693

0.030644

0.00484

0.089388

0.00484

0.089388

Number of orders Received

0.011947

0.002249

5.313123

2.87E-05

0.007271

0.016623

0.007271

0.016623

 

References

1. Rumsey D., Statistics for dummies.

2.Rumsey D., Intermediate Statistics for dummies.

3.SharbBoslough and Andrew Watters, Statistics in a nutshell

4. John Buglear, Stats means Business

5.“Statistics in excel” from http://www.ehow.com

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