Question

An organization believes that the thought of adults ages 55 and over on which industry has the most trustworthy advertising is uniformly distributed. To test this claim, you randomly select 800 adults age 55 and over and ask each which industry has the most untrustworthy advertising. The results are shown in the table. At a = 0.05, can you reject the claim that the distribution is uniform ( ie, that each frequency should be the same - in this case, 160, since there are 800 total adults sampled?) Explain.

Auto Companies: 128

Fast Food Companies 192

Finanical Services Companies 112

Pharmaceutical Companies 152

Soft Drink Companies 216

Answer 1

Solution

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

Goodness of Fit – Uniform distribution

Let Oi and Ei be respectively the observed and expected frequencies of the ith class, i = 1 to k, k being the number of classes given.

Claim: The distribution is uniform.

Hypotheses:

Null: H₀: Observed frequencies are in accordance with the Uniform Distribution Vs

Alternative H_A: H₀ is false

Test Statistic:

χ2 = ∑[i = 1,k]{(Oi - Ei)²/Ei},

Under H₀, Ei follow Uniform distribution and hence Ei’s are the same for all i. So, let Ei = E for all i.

So, χ2 = (1/E)∑[i = 1,k]Oi² - N, where N = total frequency.

Details of calculations

i	1	2	3	4	5	Total
Oi	128	192	112	152	216	800
Oi2	16384	36864	12544	23104	46656	135552
N	800
k	5
E	160
χ2	47.2
s	1
k - s	4
α	0.05
χ2crit	9.487729037
p-value	1.38543E-09
Decision	Reject Ho

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.

p-value = P(χ²_{k – s} > χ²_cal)

Given significance level = α , critical value = χ²_crit = upper α% of χ²_{k - s,}

Critical value and p-value obtained using Excel Function: Statistical CHIINV and CHIDIST are as shown in the above table.

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

the thought of adults ages 55 and over on which industry has the most trustworthy advertising depends on the industry. Answer

DONE