Park Acreage A state executive
Park Acreage A state executive claims that the average number ofacres in western Pennsylvania state parks is less than 2000 acres.A random sample of 12 parks is selected, and the number of acres isshown. 2075 2065 2174 1934 2265 1936 2277 1968 2183 2029 19762151
a. At = 0.01, is there enough evidence to support the claim? Usethe P-value method
b. Find the 99% confidence interval of the true mean.
Answer:
Mean X̅ = Σ Xi / nX̅ = 25033 / 12 = 2086.0833
Sample Standard deviation SX = √ ( (Xi – X̅ )2 / n – 1 )SX = √ ( 165338.9162 / 12 -1 ) = 122.6002
To Test :-
H0 :- µ >= 2000H1 :- µ < 2000
Test Statistic :-t = ( X̅ – µ ) / (S / √(n) )t = ( 2086.0833 – 2000 ) / ( 122.6002 / √(12) )t = 2.4323
Decision based on P valueP – value = P ( t > 2.4323 ) = 0.9834Reject null hypothesis if P value < α = 0.01 level ofsignificanceP – value = 0.9834 > 0.01 ,hence we fail to reject nullhypothesisConclusion :- Fail to reject nullhypothesis
There is insufficient evidence to support the claim that theaverage number of acres in western Pennsylvania state parks is lessthan 2000 acres.
Part b)
Confidence IntervalX̅ ± t(α/2, n-1) S/√(n)t(α/2, n-1) = t(0.01 /2, 12- 1 ) = 3.1062086.0833 ± t(0.01/2, 12 -1) * 122.6002/√(12)Lower Limit = 2086.0833 – t(0.01/2, 12 -1) 122.6002/√(12)Lower Limit = 1976.1569Upper Limit = 2086.0833 + t(0.01/2, 12 -1) 122.6002/√(12)Upper Limit = 2196.009799% Confidence interval is ( 1976.1569 , 2196.0097)
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