Q:

Assume the random variable X is normally distributed with mean muequals50 and standard deviation sigmaequals7. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded. ​P(55less than or equalsXless than or equals70​)

Accepted Solution

A:
Answer: P [ 55 ≤ X ≤ 70 ] = 23.55 %Step-by-step explanation:We know:μ₀ = mean of population              σ  = standard deviation   We begin for calculating the probability of X ≤ 55 soStep 1:μ₀  = 50              σ = 7×₁    =   ( μ - μ₀ ) ÷ σ   ⇒  ×₁  = (55-50) ÷ 7     ⇒  ×₁  = 5 ÷ 7    ×₁ = 0.7143We look z table and we have to interpolatevalue (closest smaller Than    ×₁   0.71   and closest bigger than ×₁   0.72The associated area for these point are:   0.7611 and 0.7642Taking diferences and by rule of three 0.01             ⇒  0.00310.0042        ⇒      Δ          Δ = 0.0013     and  Area for ×₁  = 0.7624That is the probability of  X ≤ 55 Now we have to find the probability of X ≤ 70We procede as in step 1 μ₀  = 50             σ = 7×₂    =   ( μ - μ₀ ) ÷ σ   ⇒  ×₂  = (70-50) ÷ 7     ⇒  ×₂  = 20 ÷ 7    ×₂ = 2.8571We look z table and we have to interpolate for 2.85    ⇒ 0.9978 (area)              2.8571 - 2.85  = 0.007 for 2.86    ⇒ 0.9979 (area) Therefore by rule of three we have0.01      ⇒    0.00010.007   ⇒        Δ                       Δ = 0.00007And area for      ×₂ = 2.8571      is  equal to 0.99787   and the probability of X ≤ 0.99787   or    99.79Now if we look the annex drawing (the area we are looking for is enclose for the values x₁   and ×₂  (pink area) and is the diference between these two areasSo   P [ 55 ≤ X ≤ 70 ]  is 99.79 % - 76.24 %  = 23.55