East-west beer price comparison

	A	B	C	D	E	F	G	H	I	J	K	L	M
1	East-west beer price comparison using NHST
2
3	Assumptions:
4	1	Beer prices in east-end and west-end normally distributed with uknown means mu_E and mu_W
5	2	Variance = sigma^2 =	5	(assumed to be known)
6
7	Step 1
8	H_0	same mean			H_A	means are different
9	Δ =	0			Δ =	-2.62		An assmption about the effect size we're interested in detecting
10
11
12	Step 2											-1.64	0.05050258347
13	alpha =	0.05										-3.3375	0.0004226786094
14	z_\alpha	-1.644853625
15					beta =	0.2
16					statistical power =	0.8
17					z_\beta	0.8416212327
18
19
20	Step 3										-1.644853625	0.8416212327
21	Need to choose sample size n and cutoff c required to guarantee the chosen error rates \alpha and \beta
22		c = -1.644853625*SE(n)			c = Δ + 0.8416212327*SE(n)				A. Table solution method
23	solving these two equations simultanously for n and c we find								n	SE(n)	-1.644853625*SE(n)	Δ + 0.8416212327*SE(n)
24	n =	9							8	1.118033989	-1.839002259	-1.679038856
25									9	1.054092553	-1.733827958	-1.732853326
26									10	1	-1.644853625	-1.778378767
27			SE(8) =	1.054092553					11	0.9534625892	-1.568306396	-1.81754564
28									12	0.9128709292	-1.501539057	-1.851708443
29									13	0.8770580193	-1.442632062	-1.881849349
30	c =	-1.733827958	(via alpha req)	c=	-1.732853326	(via beta req)			14	0.8451542547	-1.39015504	-1.908700234
31									15	0.8164965809	-1.343017361	-1.932819141
32	Data samples collected from the two populations:								16	0.790569415	-1.300370968	-1.954639994
33		x_E	x_W						17	0.7669649888	-1.261545142	-1.974505981
34		7.7	11.8						18	0.7453559925	-1.226001506	-1.992692571
35		5.9	10						19	0.7254762501	-1.19330224	-2.009423784
36		7	11						20	0.7071067812	-1.163087152	-2.024883919
37		4.8	8.6
38		6.3	8.3
39		6.3	9.4						B. Formula solution method
40		5.5	8						n =	9.00669719
41		5.4	6.8
42		6.5	8.5						C. Solution using SymPy method
43									https://live.sympy.org/?evaluate=from%20sympy%20import%20%0Afrom%20sympy.stats%20import%20P%2C%20E%2C%20variance%2C%20Die%2C%20Normal%2C%20cdf%2C%20density%2C%20std%0A%23--%0An%2C%20c%20%3D%20symbols(%27n%20c%27)%0Asigmasq%20%3D%205%20%20%23%20%0ASE%20%3D%20sqrt(sigmasq%2Fn%20%2B%20sigmasq%2Fn)%0ASE%0A%23--%0A%23%20We%20solve%20for%20c%20and%20n%20simultaneously%20for%20fixed%20significance%20and%20power%20level%0A%23%20solve%20%20eqn1%20%20%20%20and%20%20%20%20%20%20%20eqn2%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20for%20unknowns%20n%20and%20c%20%0Asolve(%5B%20%20c%2B1.644853625SE%2C%20c%2B2.62%20-%200.8416212327SE%5D%2C%20%20%20%20%20%20%20%20%20%20%20%20%20n%2C%20%20%20%20c%20)%0A%23--%0A%23%20So%20we%20need%20sample%20size%20n%3D9%2C%20and%20cutofff%0Ac%20%3D%20-1.644853625SE.subs(n%2C9).n()%0Ac%0A%23--%0A
44	sum x	55.4	82.4
45	\bar{x}	6.155555556	9.155555556		d =	-3
46					z =	-2.846049894
47
48	Step 4
49					Decision =	reject H_0	because d < c	(equivalently because z < z_\alpha)
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51
52
53	Step 5 -- computing the p-value
54					p-value =	0.002213262929	statistically significant since p-value < alpha-level chosen
55
56
57	Step 5 -- Confidence interval of the effect size
58		\gamma	0.1	= 90% confidence interval
59		1-\gamma/2	0.95
60		z_{1-gamma/2)	1.644853625
61		variance of CI	1.111111111
62		sqrt of variance of CI	1.054092553
63		CI centre =	-3
64		CI lower =	-4.733827958
65		CI upper =	-1.266172042
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