Question

The number of cell phones per 100 residents in countries in Europe is given in table #9.3.9 for the year 2010.  The number of cell phones per 100 residents in countries of the Americas is given in table #9.3.10 also for the year 2010 ("Population reference bureau," 2013).  Find the 98% confidence interval for the different in mean number of cell phones per 100 residents in Europe and the Americas.

Table #9.3.9: Number of Cell Phones per 100 Residents in Europe

100	76	100	130	75	84
112	84	138	133	118	134
126	188	129	93	64	128
124	122	109	121	127	152
96	63	99	95	151	147
123	95	67	67	118	125
110	115	140	115	141	77
98	102	102	112	118	118
54	23	121	126	47

Table #9.3.10: Number of Cell Phones per 100 Residents in the Americas

158	117	106	159	53	50
78	66	88	92	42	3
150	72	86	113	50	58
70	109	37	32	85	101
75	69	55	115	95	73
86	157	100	119	81	113
87	105	96

1. ) State the random variable and the parameter in words.

b.) If needed, state the null and alternative hypotheses and the level of significance and don't "bump" them into one line (refer to the textbook)

Ho :

HA :

a =

c.) State and check the assumptions for a hypothesis test

d.) Find the sample statistic, test statistic, and p-value, confidence interval

e.) Conclusion

Since the p-value……..fail to reject or reject   H o or confidence intervals...

f.) Interpretation ( statistical and real world)

Answer 1

Solution

Final answers in the stipulated format are given below. Back-up Theory and Details of calculations follow at the end.

Part (a)

Random variables

X = Number of Cell Phones per 100 Residents in Europe     

Y = Number of Cell Phones per 100 Residents in Americas Answer 1

Parameter in words

λ₁ = average number of Cell Phones per 100 Residents in Europe

λ₂ = average number of Cell Phones per 100 Residents in Americas Answer 2

Part (b)

Hypotheses:

Null H₀ : λ₁ = λ₂   Vs H_A : λ₁ ≠ λ₂ Answer 3

Part (c)

Assumption

Difference between estimates of λ₁ and λ₂ follows N(0, 1). Answer 4

Validity:

Since the sample sizes are large enough, by Central Limit Theorem, the above assumption holds valid. Answer 5

Part (d)

Sample statistic: estimates of λ₁ and λ₂ 108.15 and 87.21 respectively. Answer 6

Test statistic: Z = = (m₁ – m₂)/√[m{(1/n₁) + (1/n₂)}] = 9.9646 Answer 7

where m₁ and m₂ are averages based on n₁ and n₂ observations respectively and

m = (n₁m₁ + n₂m₂)/( n₁ + n₂)

p-value < 0.0001 Answer 8

98% Confidence interval: [16.40, 26.19] Answer 9

Part (e)

Conclusion

Since the p-value is less than the significance level we reject H₀ or confidence interval does not hold zero Answer 10

Part (f)

Interpretation (statistical): λ₁ ≠ λ₂ Answer 11

Interpretation (real world)

The mean number of Cell Phones per 100 Residents in Europe is different from that in Americas. Answer 12

DONE

Back-up Theory and Details of calculations

Test

Let X = Number of Cell Phones per 100 Residents in Europe

      Y = Number of Cell Phones per 100 Residents in Americas

Then, X ~ Poisson (λ₁) and Y ~ Poisson (λ₂)

Hypotheses:

Null H₀ : λ₁ = λ₂   Vs H_A : λ₁ ≠ λ₂

Test Statistic:

Let m₁ = sample estimate of λ₁ and m₂ = sample estimate of λ₂

Z = (m₁ – m₂)/(√m) if m₁ and m₂ are based on single observations and m = (m₁ + m₂)/2

   = (m₁ – m₂)/√[m{(1/n₁) + (1/n₂)}] if m₁ and m₂ are averages based on n₁ and n₂ observations

      respectively and m = (n₁m₁ + n₂m₂)/( n₁ + n₂)

Calculations:

n1	53
n2	39
m1	108.1509
m2	87.2051
Zcal	9.9646
α	0.01
Zcrit	2.5758
p-value	0.00E+00
1 - (α/2)	0.995
1- α	0.99
1-pvalue	1.00E+00
(1/n1)	0.0189
(1/n2)	0.0256
sum	0.0445
sqrt(msum)	2.1020
n1m1+n2m2	9133
n1+n2	92
m	99.2717

Distribution, Critical Value, p-value:

Under H₀ , distribution of Z can be approximated by N(0, 1). Hence, given α as the level of significance,

Critical Value = upper (α/2)% point of N(0, 1) and p-value = P(Z > | Zcal |).

Using Excel Functions: Statistical NORMSINV and NORMSDIST the above are found to be as shown in the above table.

Decision Criterion (Rejection Region):

Reject H₀, if | Zcal | > Zcrit or p-value < α.

Decision:

Since | Zcal | > Zcrit or equivalently p-value < α, H₀ rejected.

Conclusion:

There is enough evidence to suggest that the mean number of Cell Phones per 100 Residents are different between Europe and Americas..

Confidence interval

100(1 - α) % Confidence Interval for (λ₁ - λ₂) is: (Mean X – Mean Y) ± (Z_α/2)√[m{(1/n) + (1/m)}]

where m = (nMeanX + mMeanY)/(n + m), where

Z_α/2 is the upper (α /2)% point of N(0, 1), n and m are the two sample sizes.

Calculations

n	53
m	39
m1	108.5
m2	87.21
m	99.47489
d = m1 - m2	21.29
α	0.02
1 - (α/2)	0.99
Zα/2	2.3263
(1/n)+(1/m)	0.0445
m{(1/n) + (1/m)}	4.4275
sqrt	2.1042
MoE	4.8950
LB	16.3950
UB	26.1850