Question

Using the below question and values how do you calculate the standard deviation without using a calculator (by hand) of (s1^2/n1+s2^2/n2) with the standard deviation answer being 1.4938 and 1.1361
/sqrt(1.4938^2/10 + 1.1361^2/10)
Choose two competing store and compare the prices of 10 items at both stores. Preform a two-sample hypothesis test to try an prove if one store is more expensive than the other based on your sample. Store A item #1-$5.49, #2 $3.77, #3 $0.87, #4 $3.29, #5 $4.79, #6 $1.99, #7 $1.33, #8 $2.19, #9 $2.79, #10 $3.99 Store B items #1 $4.59, #2 3.99, #3 $1.79,  #4 $2.99, #5 $4.59, # $1.79, #7 $1.99, #8 $2.19, #9 $2.79, #10 $3.99

Hypothesis:
H0 ; mu1= mu2
HA: mu 1 > mu2

test statistics:

t = (x1-x2)/sqrt(s1^2/n1+s2^2/n2)
= (3.05 - 3.07)/sqrt(1.4938^2/10 + 1.1361^2/10)
= -0.0337

p value = .4872

FAil to reject the H0

Answer 1

std dev of sample 1

X	(X - X̄)²
5.49	5.954
3.77	0.518
0.87	4.752
3.29	0.058
4.79	3.028
1.99	1.124
1.33	2.958
2.19	0.7
2.79	0.068
3.99	0.884

mean =    ΣX/n =    30.500   /   10.000   =   3.050
                      
sample variance =    Σ(X - X̄)²/(n-1)=   20.0824   /   9.0000   =   2.2314
                      
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   2.2314   =       1.4938

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std dev of sample 2

X	(X - X̄)²
4.59	2.310
3.99	0.846
1.79	1.638
2.99	0.006
4.59	2.310
1.79	1.638
1.99	1.166
2.19	0.8
2.79	0.078
3.99	0.846

mean =    ΣX/n =    30.700   /   10.000   =   3.070
                      
sample variance =    Σ(X - X̄)²/(n-1)=   11.6160   /   9.0000   =   1.2907
                      
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   1.2907   =       1.1361
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H0 ; mu1= mu2
HA: mu 1 > mu2  

std error , SE =    √(s1²/n1+s2²/n2) = √(1.4938^2/10 + 1.1361^2/10) = 0.5935

t = (x1-x2)√(s1^2/n1+s2^2/n2) = (x1-x2)√std error
= (3.05 - 3.07)/√0.5935
= -0.0337

DF = min(n1-1 , n2-1 )=   9
p-value=0.5131

p-value >α, so

FAil to reject the H0