Question

The theater of the city of Mayaguez has a popular concert in schedule and has decided to sells tickets by phone. For that, it needs to decide how many operators to hire for the sale. On one hand, it doesn't want the customers to wait too long, but on the other hand the operators are expensive. The data the manager has collected from previous concerts is as follows.

Operators Wait Time

4 385

5    335

6 383

7 344

8 288

Specifically, we want to find whether there is a significant linear correlation between the variables.

In this case, if management wants a lower waiting time, it would to need to have:

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	30	1735	10	6374.0	-185.00
mean	6.00	347.00	SSxx	SSyy	SSxy

sample size ,   n =   5          
here, x̅ = Σx / n=   6.00   ,     ȳ = Σy/n =   347.00  
                  
SSxx =    Σ(x-x̅)² =    10.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   -185.0          
                  
estimated slope , ß1 = SSxy/SSxx =   -185.0   /   10.000   =   -18.5000
                  
intercept,   ß0 = y̅-ß1* x̄ =   458.0000          
                  
so, regression line is   Ŷ =   458.0000   +   -18.5000   *x
                  
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    2951.500          
                  
std error ,Se =    √(SSE/(n-2)) =    31.366          
                  
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   -0.7328          
                  
R² =    (Sxy)²/(Sx.Sy) =    0.5369

correlation hypothesis test                  
Ho:   ρ = 0       tail=   2  
Ha:   ρ ╪ 0              
n=   5              
alpha,α =    0.01              
correlation , r=   -0.7328              
t-test statistic = r*√(n-2)/√(1-r²) =        -1.865          
DF=n-2 =   3              
p-value =    0.1590              
Decison:   P value > α, So, Do not reject Ho  

Hence no significant correlation

Thanks in advance!

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