Question

to determine whether the method of using herion has changed in Years you use data collected in1980 and again in 2015 as part of an annual survey of arrested drug users. in 1980 you find that 73% of heroin users injected the drug, 12% smoked it, and 15% snorted it. the data from 2015 which were collected from a sample of 500 heorin users, are below. assume you want to test whether the null hypothesis that there has been no change in the distribution of responses in 1980 to 2015 at alpha=0.5

2015 data: injection 330

smoke:80

snort:90

total:500

Answer 1

Solution

The solution is based on Chi-square Test for Goodness of Fit.

Let Oi be the 2015 dat and Ei be the 1980 data

Claim: There has been no change in the distribution of responses in 1980 to 2015

Hypotheses:

Null: H₀: 2015 data are in accordance with the 1980 percentages

Vs

Alternative H_A: H0 is false

Test Statistic:

χ2 = ∑[i = 1,k]{(Oi - Ei)2/Ei} = 13.0229

Under H0, Ei follow the percentages of 1980

With the above terminology, χ2cal =    [Details of calculations follow at the end]

Distribution, Significance Level, α, Critical Value, p-value

Under H0, χ2 ~ χ²_{k – s}, Chi-square distribution with degrees of freedom = k – s,where k = number of classes

and s = number of parameters estimated.

Here, k = 3 and s = 0 since the 1980 percentages are given and hence there is no need for estimation.

p-value = P(χ2k – s > χ2 cal)

Given significance level = α , critical value = χ2crit = upper α% of χ²_{k - s, α} = 7.8147

p-value = 0.0046

Critical value and p-value are obtained using Excel Function: Statistical CHIINV and CHIDIST

Decision

Since, χ²_cal > χ²_crit, or equivalently, since p-value < α, H0 is rejected.

Conclusion

There is not sufficient statistical evidence to conclude that the claim is valid and hence

we conclude that here has been change in the method of using heroin from 1980 to 2015. ANSWER

Calculation Details

Heroin usage method			Injection		Smoke		Snort		Total
Oi - 2015 data			330		80		90		500
pi = 1980 % in decimal form			0.73		0.12		0.15		1
Ei = 500 x pi			365		60		75		500
χ2			3.3561644		6.6666667		3		13.0229
CHECK: ΣOi = ΣEi.		Done		K =		3		S =		0		α =	0.05
χ2cal =	13.0229	DF =		3		χ2crit =		7.8147		p-value =		0.0046

DONE