Question

1. a population of exam scores is normally distributed with μ = 80 and σ = 6. find the probability that a randomly selected score is less than 77.
2. heights of adult males are normally distributed with μ = 70 inches and σ = 3 inches. find the probability that a randomly selected man is taller than 74.5 inches.
3. iq scores follow a normal distribution with μ = 100 and σ = 15. find the probability that a persons iq is between 92 and 118.
4. a brand of light bulb lasts an average of μ = 1200 hours with σ = 100 hours. find the probability that a bulb lasts less than 1075 hours.
5. weights of adult cats are normally distributed with μ = 10 lbs and σ = 2 lbs. find the probability that a cat weighs more than 11.8 lbs.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x - \mu}{\sigma}\)
For the first problem: \(\mu = 80\), \(\sigma=6\), \(x = 77\)
\(z=\frac{77 - 80}{6}=\frac{-3}{6}=- 0.5\)
Using the standard normal distribution table, \(P(Z\lt - 0.5)=0.3085\)

Step2: Calculate the z - score for the second problem

\(\mu = 70\), \(\sigma = 3\), \(x = 74.5\)
\(z=\frac{74.5-70}{3}=\frac{4.5}{3}=1.5\)
\(P(Z\gt1.5)=1 - P(Z\leqslant1.5)\)
From the standard normal table, \(P(Z\leqslant1.5)=0.9332\)
\(P(Z\gt1.5)=1 - 0.9332 = 0.0668\)

Step3: Calculate the z - scores for the third problem

For \(x = 92\): \(z_1=\frac{92 - 100}{15}=\frac{-8}{15}\approx - 0.53\)
For \(x = 118\): \(z_2=\frac{118 - 100}{15}=\frac{18}{15}=1.2\)
\(P(-0.53\lt Z\lt1.2)=P(Z\lt1.2)-P(Z\lt - 0.53)\)
From the standard normal table, \(P(Z\lt1.2)=0.8849\), \(P(Z\lt - 0.53)=0.2981\)
\(P(-0.53\lt Z\lt1.2)=0.8849 - 0.2981=0.5868\)

Step4: Calculate the z - score for the fourth problem

\(\mu = 1200\), \(\sigma = 100\), \(x = 1075\)
\(z=\frac{1075 - 1200}{100}=\frac{-125}{100}=-1.25\)
From the standard normal table, \(P(Z\lt - 1.25)=0.1056\)

Step5: Calculate the z - score for the fifth problem

\(\mu = 10\), \(\sigma = 2\), \(x = 11.8\)
\(z=\frac{11.8 - 10}{2}=\frac{1.8}{2}=0.9\)
\(P(Z\gt0.9)=1 - P(Z\leqslant0.9)\)
From the standard normal table, \(P(Z\leqslant0.9)=0.8159\)
\(P(Z\gt0.9)=1 - 0.8159 = 0.1841\)

Answer:

1. \(0.3085\)
2. \(0.0668\)
3. \(0.5868\)
4. \(0.1056\)
5. \(0.1841\)