Please solve the following.

Question

Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.)

𝜇 = 8; 𝜎 = 2 P(7 ≤ x ≤ 11) =

Answer 1

Hi Tamera,

For this problem, we use the classic equation in introductory statistics: z= (x-mu) / sigma. Breaking this down:

z=z-score--look up on z-table (You can find this with a quick Internet search, but I cannot link here as Wyzant reviews posts with links)

x=value(s) you were given

mu=mean

sigma=standard deviation

Here, we were given two x-values, 7 and 11. So I will refer to these as x₁ and x₂ and corresponding z-scores as z₁ and z_2. Thus:

z₁₌(x₁-mu)/sigma

x₁=7

mu=8

sigma=2

z₁₌(7-8)/2

z₁= -0.5

Keep that value in mind. Now we need to find z₂:

z₂=(x₂-mu)/sigma

z₂=(11-8)/2

z₂=1.5

Now, we can go to our z-table and get the probability that z is less than both of these values. Mathematically, this means:

P(7<=x<=11)=P( -0.5<=z<=1.5)

From z-table:

P(Z< -0.5) = 0.3085

P(Z< 1.5) = 0.9332

Now, we were asked for the area between, so we must subtract:

P(7<=x<=11)= 0.9332 - 0.3085 =

P = 0.6247

I hope this helps.