Question

The average number of earthquakes that occur in Los Angeles over one month is 36. Assume the standard deviation is 3.6. if a random sample of 35 months is selected, find the probability that the mean of the sample is between 34 and 37.5.

Answer 1

Answer: 0.9927

Step-by-step explanation:

We assume that  the number of earthquakes that occur in Los Angeles follows normal distribution.

As per given , we have

$\mu=36\\\\ \sigma= 3.6$

Sample size : n= 35

Let $\overline{x}$ be the sample mean.

Now ,

$P(34<\overline{x}<37.5)=P(\dfrac{34-36}{\dfrac{3.6}{\sqrt{35}}}<\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}<\dfrac{37.5-36}{\dfrac{3.6}{\sqrt{35}}})\\\\=P(-3.29<z<2.47)\ \ [\because \ z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}]\\\\ P(z<2.47)-P(z<-3.29)\\\\= P(z<2.47)-(1-P(z<3.29))\\\\=0.9932-(1-0.9995)=0.9927$

Hence, the probability that the mean of the sample is between 34 and 37.5. is 0.9927 .