Public Policy and Political Science A building inspector enforces building, elec

Question

Public Policy and Political Science A building inspector enforces building, electrical, mechanical, plumbing, and energy code requirements for the safety and health of people a certain city, county, or state. In Santa Cruz County, the probability that a building inspector will find at least one code violation at a commercial building is 0.25. Suppose 30 commercial buildings are selected at random. a. Find the mean, variance, and standard deviation of the number of commercial buildings with at least one violation. b. Find the probability that the number of commercial buildings with at least one violation will be within one standard deviation of the mean. c. Find the probability that the number of commercial buildings with at least one violation will be more than the two standard deviations from the mean. d. Suppose the actual number of commercial buildings with has at least one violation is 10. Is there any evidence to suggest that code violations are found in more than 25% of commercial buildings? Justify your answer.

Explanation / Answer

a. Here Probability that a building inspector will find at least one code violation at a commercial building p = 0.25

Sample size n = 30

Mean = n * p = 30 * 0.25 = 7.50

Variance 2= n * p * (1-p) = 30 * 0.25 * 0.75 = 5.625

Std. Deviation = sqrt (np(1-p) ) = sqrt (5.625) = 2.372

b. Here we have to calculate probability of values 1 standard deviation away from mean

Pr ( - <= X <= + ) = Pr ( 7.50 - 2.372 <= X <= 7.50 + 2.372) = Pr ( 5.128 <= X <= 9.872)

as the function is discrete so the given probability

Pr ( - <= X <= + ) = Pr(X =6) + Pr( X = 7) + Pr( X =8) + Pr( X=9)

= 0.1455 + 0.1662+ 0.1593 + 0.1298 = 0.601

so 60% probability that values will be in between 1 standard deviation around mean.

(c) Here we have to calculate probability of values 2 standard deviation away from mean

Pr ( - 2 <= X <= + 2 ) = Pr ( 7.50 - 2* 2.372 <= X <= 7.50 + 2 * 2.372) = Pr ( 2.756 <= X <= 12.244)

as the function is discrete so the given probability

Pr ( - 2 <= X <= +2 ) = Pr(X =3,4,...11,12) = Pr( X <= 12) - Pr( X < 3) =

= 0.9784 - 0.0106 = 0.9678

so 96.78% probability that values will be in between 2 standard deviation around mean.

(iv) Here

Null Hypothesis : H0: p = 0.25

Alternative Hypothesis : Ha : p > 0.25

so here sample p = 10/30 = 0.3333

so standard error of sampling se = sqrt [ 0.25 * 0.75/ 30] = 0.079

so Test Statisitc

Z = (p - 0.25)/ se = (0.08333)/ 0.079 = 1.0548

so p - value = Pr( p> 0.3333 ; 0.25 ; 0.079) = 1 - (1.0548) = 0.1458

so it is not significant. so we cannot reject the null and claim that there is no statistical significant evidence that code violations are found in more than 25% of commercial buildings.

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