Question

The manager of an amusement park would like to be able to predict daily attendance in order to develop more accurate plans about how much food to order and how many ride operators to hire. After some consideration, he decided that the following three factors are critical: Yesterday’s attendance Weekday or weekend Predicted weather He then took a random sample of 36 days. For each day, he recorded the attendance, the previous day’s attendance, day of the week, and weather forecast(mostly sunny, rain, cloudy). The first independent variable is interval, but the other two are nominal. a. Create the three indicator variables you need. b. Conduct a regression analysis. c. Is this model valid? Explain. d. Can we conclude that weather is a factor in determining attendance? e. Do these results provide sufficient evidence that weekend attendance is, on average, larger than weekday attendance? f. Do these results provide sufficient evidence that mostly sunny attendance is, on average, larger than cloudy attendance?

Attendance	Yest Att	day of the week	weather forecast
7882	8876	2	1
6115	7203	2	3
5351	4370	2	3
8546	7192	1	1
6055	6835	2	3
7367	5469	2	1
7871	8207	2	1
5377	7026	2	3
5259	5592	2	1
4915	3190	2	3
6538	7012	2	3
6607	5434	2	3
5118	3764	2	3
6077	7575	2	3
4475	6047	2	3
3771	4430	2	3
6106	5697	2	3
7017	3928	1	2
5718	5552	2	3
5966	3142	1	2
8160	8648	1	2
4717	3397	2	3
7783	7655	2	3
5124	5920	2	3
7495	7831	1	2
5848	6355	2	3
5166	3529	2	3
4487	4220	2	3
7320	7526	2	1
6925	4083	1	1
8133	6382	1	1
7929	6459	2	3
7291	3432	1	2
5419	8077	2	3
3634	3353	2	3
6859	3803	1	2
		1   weekend	1   mostly sunny
		2   weekdays	2   rain
			3   cloudy

Answer 1

A.

Since "day of the week" can assume two values (1 and 2), we will need only one variable e.g.

A = 0 ; if weekend

A = 1 ; if weekday

Since "weather forecast" can assume three values (1, 2 and 3), we will need two variable e.g.

_B1 = 1, if sunny; B₁ = 0, otherwise.

_B2 = 1, if rain ; B₂ = 0, otherwise.

if both the variables gets 0 then it is cloudy.

B.

Summary of regression analysis:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.824568883
R Square	0.679913843
Adjusted R Square	0.638612404
Standard Error	790.6786272
Observations	36
ANOVA
	df	SS	MS	F	Significance F
Regression	4	41166949.31	10291737.33	16.4622311	2.47624E-07
Residual	31	19380353.44	625172.6915
Total	35	60547302.75
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	4619.366527	695.8111859	6.638821882	2.02115E-07	3200.250257	6038.482797	3200.250257	6038.482797
Yest Att	0.383998499	0.080450863	4.773081158	4.11184E-05	0.219917882	0.548079116	0.219917882	0.548079116
A	-1207.558126	586.0985572	-2.060332876	0.047844232	-2402.914014	-12.20223744	-2402.914014	-12.20223744
B1	988.54631	410.8190614	2.406281506	0.022266169	150.6753102	1826.41731	150.6753102	1826.41731
B2	541.7985097	685.57383	0.790284702	0.435365659	-856.438535	1940.035555	-856.438535	1940.035555

C.

Since the significance of F-statistics (i.e. the p-value) = 2.476*10^-7 < 0.05 (if $\alpha$ = 0.05)

Thus model is significant.

D.

The p-value for B1 = 0.022 < 0.05

Thus, the weather is a significant factor in determining attendance.

Since for B2, p-value = 0.435 > 0.05 i.e not significant. It means knowing if a day is sunny is more helpful than knowing if it is rainy.

E.

Regression Equation:

Attendance = 4619.366 + 0.384*Yest. Att. -1207.558*A + 988.546*B1 + 541.7985097*B2

For the weekend:

A = 0

Attendance = 4619.366 + 0.384*Yest. Att. + 988.546*B1 + 541.7985097*B2

For the weekday:

day of the week = 1

Attendance = 4619.366 + 0.384*Yest. Att. + 988.546*B1 + 541.7985097*B2 - 1207.558

Thus if all the other factors are constant, attendance on a weekend will be more as compared to a weekday on average.

F.

Attendance = 4619.366 + 0.384*Yest. Att. -1207.558*A + 988.546*B1 + 541.7985097*B2

For mostly sunny:

B1 = 1

B2 = 0

Attendance = 4619.366 + 0.384*Yest. Att. -1207.558*A + 988.546

For cloudy:

B1 and B2  = 0

Attendance = 4619.366 + 0.384*Yest. Att. -1207.558*A

Thus if all the other factors are constant, attendance on a sunny day will be more as compared to a cloudy day on average.

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