MATH SOLVE

5 months ago

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