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Forecasting – Simple Linear Regression Applications

STATISTICS FOR MGT DECISIONS FINAL EXAMINATION Forecasting – Simple Linear Regression Applications Interpretation and Use of Computer Output (Results) NAME SECTION A – REGRESSION ANALYSIS AND FORECASTING 1) The management of an international hotel chain is in the process of evaluating the possible sites for a new unit on a beach resort.As part of the analysis, the management is interested in evaluating the relationship between the distance of a hotel from the beach and the hotel’s average occupancy rate for the season.A sample of 14 existing hotels in the area is chosen, and each hotel reports its average occupancy rate.

The management records the hotel’s distance (in miles) from the beach.

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The following set of data is obtained: Distance (miles)0. 10. 10. 20. 30. 40. 40. 50. 60. 7 Occupancy (%)929596908996908385 Continue Distance (miles)0. 70. 80. 80. 90. 9 Occupancy (%)8078767275 Use the computer output to respond to the following questions: a) A simple linear regression was ran with the occupancy rate as the dependent (explained) variable and distance from the beach as the independent (explaining) variable Occpnc=b[pic]+b[pic](Distncy) What is the estimated regression equation?

The regression model is: Occpnc = b[pic] + b[pic](Distncy) The estimated regression equation is: OCCUPNC = 99. 61444 – 26. 703 DISTNCY b) Interpret the meaning behind the values you get for both coefficients b[pic] and b[pic]. b[pic]=99. 61444, represent the y-intercept as well as the starting figure for the distance coverage. This is the amount of distance in miles that the hotel is from a beach. b[pic] = 26. 703, represents the percentage of occupancy a hotel has depending on the distance of the hotel from a beach. c) What sort of relationship exists between average hotel occupancy rate and the hotel’s distance from the beach?

Does this relationship make sense to you? Why or why not? Both distance and occupancy have a direct relationship. This is true because closer the hotel is to the beach, the higher the chance that the hotel’s occupancy will be greater. If a person is going to stay at a hotel, chances are they are on vacation. People on vacation love to spend time on a beach for relaxation purposes, so it would only make sense that a hotel that is closer to the beach will have a higher occupancy rate. d) Interpret the R-Square value in your computer output R-Squared = 0. 848195 = 84. 8195 ) Predict the expected occupancy rate for a hotel that is (i) one mile from the beach in that area, (ii) one and half miles from the beach. i. OCCUPNC = 99. 61444 – 26. 703 (1) = 99. 61444 – 26. 703 = 72. 911 ii. OCCUPNC = 99. 61444 – 26. 703 (1. 5) = 99. 61444 – 40. 055 = 59. 559 f) In your mind, what other variables contribute positively or negatively to hotel occupancy besides distance from the beach? Other variables that contribute positively or negatively to hotel occupancy besides distance from the beach include the distance of restaurants, shopping centers, and airport from the hotel.

The closer theses variables are to the hotel the chances the occupancy rate will be higher. In addition, other variables may include what type of amenities that are offered by the hotel, customer service, and rating of the hotel. g) At a level of significance, ? = 0. 01 or 1 percent test the following pair of hypotheses: H[pic]: b[pic]= 0 H[pic]: b[pic]? 0 On the model: Occpnc=b[pic]+b[pic](Distncy) What is your conclusion and why that particular conclusion? COMPUTER OUTPUT – PART 1 INTERNATIONAL HOTEL REGRESSION FUNCTION & ANOVA FOR OCCPNCY OCCPNCY = 99. 61444 – 26. 703 DISTANCE

R-Squared = 0. 848195 Adjusted R-Squared = 0. 835545 Standard error of estimate = 3. 339362 Number of cases used = 14 Analysis of Variance p-value Source SS df MS F Value Sig Prob Regression 747. 68 1 747. 68390 67. 04880 0. 000002 Residual 133. 82 12 11. 15134 Total 881. 50 13 COMPUTER OUTPUT – PART 1 INTERNATIONAL HOTEL REGRESSION COEFFICIENTS FOR OCCPNCY Two-Sided p-value Variable Coefficient Std Error t Value Sig Prob Constant 99. 61444 1. 4107 51. 31933 0. 000000 DISTANCE -26. 70300 3. 26110 -8. 18833 0. 000002 * Standard error of estimate = 3. 339362 Durbin-Watson statistic = 1. 324282 MULTIPLE REGRESSION 2) You want to find out factors that explain an individual’s weekly savings. You are given a set of data below: Sampled WeeklyHouseFoodEntertain/Weekly IndividualIncomeRentExpenseExpenseSavings Case 1\$25085952520 Case 2\$1907590100 Case 3\$4201401204050 Case 4\$340120130040 Case 5\$2801101003015 Case 6\$310801252525 Case 7\$5201501405580 Case 8\$440175155450 Case 9\$36090852095 Case 10\$3851051353530

Case 11\$2058010505 Case 12\$26565951515 Case 13\$19550801020 Case 14\$25090100250 Case 15\$4801401604545 A multiple regression was ran with WEEKLY SAVINGS as the DEPENDENT VARIABLE and the rest as the INDEPENDENT VARIABLES. SAVINGS = b[pic][pic]+ b[pic]INCOME + b[pic]RENT + b[pic]FOOD + b[pic]ENTERT a) What is the estimated multiple regression equation? SAVINGS = 23. 14156 + 0. 591446 INCOME – 0. 341793 RENT – 1. 119734 FOOD – 0. 907868 ENTERT b) What relationship exists between (i) SAVINGS and INCOME? , SAVINGS and RENT? , SAVINGS and FOOD expense, SAVINGS and ENTERTAINMENT expense?

There are no direct relationship between saving and income, savings and rent, savings and food expense, and savings and entertainment expense. c) Which of the independent (explaining) variables are (is) significant in the multiple regression and which ones are (is) not significant (use ? = 0. 05 level of significance). Are the results in line with Maslow hierarchy of needs? Explain. COMPUTER OUTPUT PART I WEEKLY SAVINGS REGRESSION FUNCTION & ANOVA FOR SAVINGS SAVINGS = 23. 14156 + 0. 591446 INCOME – 0. 341793 RENT – 1. 119734 FOOD – 0. 907868 ENTERT R-Squared = 0. 917562 Adjusted R-Squared = 0. 70454 Standard error of estimate = 10. 9635 Number of cases used = 12 Analysis of Variance p-value Source SS df MS F Value Sig Prob Regression 9364. 86 4 2341. 21 19. 47795 0. 000677 Residual 841. 39 7 120. 198 Total 10206. 250 11 COMPUTER OUTPUT PART II WEEKLY SAVINGS REGRESSION COEFFICIENTS FOR SAVINGS Two-Sidedp-value Variable Coefficient Std Error t Value Sig Prob Constant 23. 14156 18. 34071 1. 26176 0. 247451 INCOME 0. 59145 0. 07388 8. 00526 0. 000091 RENT -0. 4179 0. 19849 -1. 72199 0. 128743 * FOOD -1. 11973 0. 24633 -4. 54565 0. 002650 ENTERT -0. 90787 0. 32460 -2. 79689 0. 026643 * indicates that the variable is marked for leaving Standard error of estimate = 10. 9635 Durbin-Watson statistic = 1. 683103 3) REGRESSION ANALYSIS A business person is trying to estimate the relationship between the price of good X and the sales of good Y of certain groups of staples. Tests in similar cities throughout the country have yielded the data below: PRICE (X)SALES (Y) \$2010,300 \$259,100 \$308,200 \$356,500 \$405,100 \$502,300

A simple linear regression of a model SALES(Y) = b[pic] + b[pic]PRICE(X) Was run and the computer output is shown below: PRICE OF X / SALES OF Y REGRESSION FUNCTION & ANOVA FOR SALES(Y) SALES(Y) = 15907. 14 – 269. 7143 PRICE(X) R-Squared = 0. 994999 Adjusted R-Squared = 0. 993749 Standard error of estimate = 230. 9143 Number of cases used = 6 Analysis of Variance p-value Source SS df MS F Value Sig Prob Regression 4. 24350E+07 1 4. 24350E+07 795. 83480 0. 000009 Residual 213285. 70000 4 53321. 43000 Total 4. 26483E+07 5

PRICE OF X / SALES OF Y REGRESSION COEFFICIENTS FOR SALES(Y) Two-Sidedp-value Variable Coefficient Std Error t Value Sig Prob Constant 15907. 14000 332. 34250 47. 86370 0. 000001 PRICE(X) -269. 71430 9. 56076 -28. 21054 0. 000009 * Standard error of estimate = 230. 9143 Durbin-Watson statistic = 1. 687953 QUESTIONS a) What is the estimated equation of the model: SALES(Y) = b[pic] + b[pic]PRICE(X)? SALES(Y) = 15907. 14 – 269. 7143 PRICE(X) b) What sort of relationship exists between SALES OF Y and the PRICE OF X? Does this relationship make sense? Why or why not?

There is a direct relationship between Sales of Y and the Price of X. The lower the price the higher are the sales. This makes sense because if the price is lower, a person will purchase more items. c) What can you say about GOOD Y and GOOD X (a good can be an item, a commodity, etc. ). Name a pair of good X and good y that can display this kind of relationship. Suppose the price of candy is \$0. 50/lb, the sales of the candy versus the same type of candy that is \$0. 80/lb would yield more sales because of the price. The price of the candy directly affects sales in this instance because a person would buy more candy at \$0. 0/lb versus \$0. 80/lb. 4) REGRESSION ANALYSIS A business person is trying to estimate the relationship between the price of good X and the sales of good Z of certain groups of staples. Tests in similar cities throughout the country have yielded the data below: PRICE (X)SALES (Z) \$153300 \$203900 \$254750 \$305500 \$406550 \$507250 A simple linear regression of a model SALES (Z) = b[pic] + b[pic]PRICE(X) Was run and the computer output is shown below: PRICE OF X / SALES OF Z REGRESSION FUNCTION & ANOVA FOR SALES(Y) SALES(Z) = 1740. 686 + 115. 5882 PRICE(X) R-Squared = 0. 977573 Adjusted R-Squared = 0. 71966 Standard error of estimate = 255. 2152 Number of cases used = 6 Analysis of Variance p-value Source SS df MS F Value Sig Prob Regression 1. 13565E+07 1 1. 13565E+07 174. 35450 0. 000190 Residual 260539. 20000 4 65134. 80000 Total 1. 16171E+07 5 PRICE OF X / SALES OF Z REGRESSION COEFFICIENTS FOR SALES(Z) p-value Variable Coefficient Std Error t Value Sig Prob Constant 1740. 68600 282. 52800 6. 16111 0. 003522 PRICE(X) 115. 58820 8. 75381 13. 20434 0. 000190 *

Standard error of estimate = 255. 2152 Durbin-Watson statistic = 1. 240299 QUESTIONS a) What is the estimated equation of the model: SALES(Z) = b[pic] + b[pic]PRICE(X)? SALES (Z) = 1740. 686 + 115. 5882 PRICE(X) b) What sort of relationship exists between SALES OF Z and the PRICE OF X? Does this relationship make sense? Why or why not? There is a direct relationship between Sales of Y and the Price of X. The higher the price the higher are the sales. This makes sense as it relates to supply and demand. The higher the demand and for the product and unavailability of the product, the price will go up even though sales may he same due to the price increase the sales amount will be higher. c) What can you say about GOOD Z and GOOD X (a good can be an item, a commodity, etc. ). Give an example of good X and good Z that can display this kind of relationship A prime example that displays this kind of relationship is gas. The price of gas has been going up for sometime now. The demand for it is high, but the supply of is low. Therefore, even though the amount of sales may stay constant, the dollar amount will be higher because the price is higher. Chi-Squared Test #1

M&M, makers of Chocolate Candies, conducted a national poll in which more than ten million people indicated their preference for a new color. The tally of this poll resulted in the replacement of tan-colored M&Ms with a new blue color. In the brochure “Colors,” made available by M&MS Consumer Affairs, the distribution of colors for the plain candies is as follows: BROWNYELLOWREDORANGEGREENBLUE 30%20%20%10%10%10% In a follow-up study two years later, samples of 1-pound bags were used to determine whether the reported percentages were still valid. The following results were obtained (observed) for one sample of 506 plain candies

BROWNYELLOWREDORANGEGREENBLUE 17713579413638 Use a level of significance ( = 0. 05 to determine whether these data support the percentages reported by the company Hint: To obtain the Expected Number of multiply the sample value (506) by each color’s probability, i. e. , E = BROWNYELLOWREDORANGEGREENBLUE 30% (506)20%(506)20%(506)10%(506)10%(506)10%(506) Then compute the Chi-Squared. H[pic]: f[pic], f[pic], f[pic], f[pic], f[pic], f[pic] hold previous year’s patterns or percentages H[pic]: At least one frequency differs from the previous year’s pattern or percentages E = 506/6 = 84. 33 [pic]=(177 –84. 33)[pic]/84. 33+(135 – 84. 33)[pic]/84. 33 + (79 – 84. 33)[pic]/84. 33+(41 – 84. 33)[pic]/84. 33)+(36 – 84. 33)[pic]/84. 33+(38 – 84. 33)[pic]/84. 33) ([pic]=101. 937 + 30. 49217 + 0. 333215 + 22. 24069 + 27. 67367 + 25. 4293 ([pic]=208. 106. This is the computed ([pic]-value. ( = 0. 05 d. f. = 6 – 1 = 5. Go to ([pic]-tables at ( = 0. 05, and d. f. = 5, you will get CRITICAL ([pic]-value = 11. 070. Since Computed ([pic]-value is greater than Critical ([pic]-value REJECT NULL H[pic]:P[pic] = P[pic] = P[pic] = P[pic] = P[pic] ALTERNATIVE H[pic]: At least one P is different is correct

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