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In: Statistics and Probability

Question 1 Find the correlation coefficients between salest and all other independent variables and test the...

Question 1

  1. Find the correlation coefficients between salest and all other independent variables and test the significance of population correlation coefficient using the value of r calculated for sales and price alpha value of 0.04.
  2. Estimate the regressions for sales and advertising expenses, sales and number of staff and report and interpret the results.
  3. Check the significance or insignificance of all independent variables using alpha 0.05 and state whether the regression model is overall fit or not?  
  4. Find the predicted value of sales for price = 121, advertising = 330 and staff = 104. Write down first five estimated values
  5. For the given data set below, test the hypothesis that the means of sales, advertising expenses and staff are same by using 0.03 level of significance. Report the Excel output as well.
Sales Advertising Price
157.7301 114.9536 187
121.1506 141.0268 186.3123
106.4625 153.3305 186.2964
127.5707 110.1133 136.1812
174.9113 159.3815 159.1798
100.1009 143.0154 182.8766
130.0771 153.8483 158.7549
158.8319 123.3517 177.6487
166.3804 151.2224 126.3865
149.9134 104.4393 157.523
141.8844 120.7072 126.7821
152.5924 110.5488 106.5236
102.5648 120.516 130.9825
137.5087 138.0106 153.2031
131.2427 122.6614 165.1192
129.1159 104.0119 112.1737
166.0645 181.9792 146.8946
100.4567 122.4835 167.7717
114.7491 122.964 127.3822
122.9083 126.5962 138.4009
183.8643 182.8872 168
101.4258 128.4283 127.2866
118.4291 152.4145 160.04
152.4756 184.5918 111.4037
150.6595 166.8079 111.1329
116.7936 132.5331 166.7229
107.7582 116.9184 137.8592
119.5469 124.1774 142.0437
133.1358 171.3747 150.8799
108.9371 137.8592 127.8202
178.1877 128.5637 138.8044
157.0583 142.5853 156.0096
150.625 183.9174 145.7502
127.2361 120.9435 185.4388
144.2846 117.768 117.7202
161.6968 178.6099 101.8665
149.0532 165.2546 165.0051
152.3349 140.1559 120.0726
145.2139 137.5114 113.7561
161.2481 104.6571 103.5632
141.587 182.3801 166.6539
137.6707 113.2118 104.9544
186.6548 117.1865 107.9175
165.5812 162.9686 141.3799
110.8753 125.3696 139.901
136.8317 120.2691 177.3965
149.8364 129.1319 169.6674
107.4874 175.1715 106.89
181.2995 180.2666 154.6262
100.7248 131.4843 131.7259
101.848 186.2991 182.935
175.2219 186.8301 163.9616

Solutions

Expert Solution

(i)

Sales Advertising Price
Sales 1.000
Advertising .249 1.000
Price -.130 .151 1.000

r = -0.130

The hypothesis being tested is:

H0: = 0

H1: ≠ 0

Source SS   df   MS F p-value
Regression 540.5628 1   540.5628 0.86 .3586
Residual 31,481.2638 50   629.6253
Total 32,021.8266 51  

The p-value is 0.3586.

Since the p-value (0.3586) is greater than the significance level (0.04), we fail to reject the null hypothesis.

Therefore, we cannot conclude that the relationship is significant.

(ii) The regression equation is:

y = 156.8587 - 0.1260*x

variables coefficients std. error    t (df=50) p-value
Intercept 156.8587
Price -0.1260 0.1359 -0.927 .3586

The regression equation is:

y = 104.5359 + 0.2410*x

variables coefficients std. error    t (df=50) p-value
Intercept 104.5359
Advertising 0.2410 0.1327 1.816 .0754

(iii)

variables coefficients std. error    t (df=49) p-value
Intercept 125.0632
Advertising 0.2661 0.1335 1.993 .0519
Price -0.1662 0.1336 -1.244 .2194

The hypothesis being tested is:

H0: β1 = β2 = 0

H1: At least one βi ≠ 0

The p-value is 0.0519.

Since the p-value (0.0519) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, we cannot conclude that the slope is significant.

The hypothesis being tested is:

H0: β2 = 0

H1: β2 ≠ 0

The p-value is 0.2194.

Since the p-value (0.2194) is greater than the significance level (0.05), we fail to reject the null hypothesis.

Therefore, we cannot conclude that the slope is significant.

(iv)

Predicted values for: Sales
Advertising Price Predicted
114.9536 187.0000 124.5652928
141.0268 186.3123 131.6169137
153.3305 186.2964 134.8932083
110.1133 136.1812 131.7246678
159.3815 159.1798 141.0105944

(v) The hypothesis being tested is:

H0: µ1 = µ2 = µ3

Ha: Not all means are equal

Mean n Std. Dev
138.61153 52 25.057513 Sales
141.41712 52 25.864454 Advertising
144.85727 52 25.845311 Price
141.62864 156 25.555002 Total
ANOVA table
Source SS    df MS F    p-value
Treatment 1,017.730953 2 508.8654763 0.78 .4616
Error 1,00,206.281846 153 654.9430186
Total 1,01,224.012798 155

The p-value is 0.4616.

Since the p-value (0.4616) is greater than the significance level (0.03), we fail to reject the null hypothesis.

Therefore, we cannot conclude that not all means are equal.

Please give me a thumbs-up if this helps you out. Thank you!

orchestra answered 1 year ago
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