the number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation 2.4. (these values are typical of much of the united states.) the national lightning detection network (nldn) uses automatic sensors to watch for lightning in a random sample of 10 one-square-kilometer plots of land. (a) what are the mean and standard deviation of x, the sample mean number of strikes per square kilometer? (b) explain why you cannot safely calculate the probability that x 5 based on a sample of size 10. (c) suppose the nldn takes a random sample of n 50 square kilometers instead. explain how the central limit theorem allows us to find the probability that the mean number of lightning strikes per square kilometer is less than 5. then calculate this probability. show your work.

Probability is a branch of mathematics that deals with numerical representations of the likelihood of an event occurring or of a proposition being true. The probability of an event is a number between 0 and 1, with 0 approximately denoting impossibility and 1 denoting certainty.

Here,

mu =6

α=2.4

n=50

Σxi=250

Formula used:

z=[tex]\frac{x-u}{\alpha }[/tex]

Step 2: Calculation

[tex]x^{'}[/tex] = Σ[tex]\frac{xi}{n}[/tex] = 250/50 = 5

z = [tex]\frac{x^{'} - u_{x} }{\alpha x^{'} }[/tex] = [tex]\frac{x^{'} - u_{x} }{ \frac{\alpha }{\sqrt{n} } }[/tex] = [tex]\frac{5- 6}{ \frac{2.4}{\sqrt{50} } }[/tex] ≈ -2.95

P([tex]X^{'}[/tex]<5)=P(Z<-2.95)

=0.0016

or 0.16%

Therefore, the probability probability that the mean number of lightning strikes per square kilometer is less than 5 is P([tex]X^{'}[/tex]<5)=0.0016

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