Question

1. Results of the Youth Risk Behavior Surveillance Survey indicate that 15% of young adults are heavy drinkers, 65% are moderate drinkers, and 20% are non-drinkers. The University of South Dakota conducts its own survey, and finds the following:      

	Heavy Drinkers	Moderate Drinkers	Non-Drinkers
Number of Students	83	91	46

Is the distribution of alcohol consumers on the University of South Dakota campus approximately normal? Use α = .05 as your level of significance.

Answer 1

Solution

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

Hypotheses:

Null: H₀: Observed frequencies of are in accordance with the given expected percentages. Vs

Alternative H_A: H₀ is false

Test Statistic:

χ² = ∑[i = 1,k]{(Oi - Ei)²/Ei},

where Oi = given observed frequencies as brought by The University of South Dakota’s own survey,

Ei = Expected frequencies are as per the given results of the Youth Risk Behavior Surveillance Survey of young adults

Under H₀, E_i = Total Oi x proportion as per the given results of the Youth Risk Behavior Surveillance Survey

With the above terminology, χ²_cal = 94.758   [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, all proportions are given and so s = 0 and k = 3.

p-value = P(χ²_{k – s} > χ²_cal)

Given significance level = α = 0.05 , critical value = χ²_crit = upper 5% of χ²₃ = 11.345

p-value = 2.08E-20

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 sufficient statistical evidence to conclude that the distribution of alcohol consumers on the University of South Dakota campus is NOT normal Answer

Details of calculations

Drinker Type	Heavy	Moderate	Non-drinker	Total
Oi	83	91	46	220
Normal Proportion	0.15	0.65	0.2	1
Ei	33	143	44	220
Chi-square	75.757576	18.90909	0.09091	94.75758

DONE