Question

1. a population of test scores is normally distributed with $\mu = 75$ and $\sigma = 8$. find the probability that a randomly selected score is less than 70.
2. heights of adult females are normally distributed with $\mu = 65$ inches and $\sigma = 2.5$ inches. find the probability that a randomly selected woman is taller than 68.2 inches.
3. iq scores follow a normal distribution with $\mu = 100$ and $\sigma = 14$. find the probability that a persons iq is between 88 and 112.
4. the lifetime of a certain brand of batteries is normally distributed with $\mu = 900$ hours and $\sigma = 120$ hours. find the probability that a battery lasts less than 780 hours.
5. weights of adult dogs are normally distributed with $\mu = 50$ lbs and $\sigma = 9$ lbs. find the probability that a dog weighs more than 62 lbs.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
For problem 1: \(x = 70\), \(\mu=75\), \(\sigma = 8\). Then \(z=\frac{70 - 75}{8}=\frac{-5}{8}=-0.625\).
For problem 2: \(x = 68.2\), \(\mu = 65\), \(\sigma=2.5\). Then \(z=\frac{68.2-65}{2.5}=\frac{3.2}{2.5}=1.28\). The probability that \(X>68.2\) is \(P(X > 68.2)=1 - P(X\leq68.2)\).
For problem 3: For \(x_1 = 88\), \(z_1=\frac{88 - 100}{14}=\frac{-12}{14}\approx - 0.86\); for \(x_2 = 112\), \(z_2=\frac{112-100}{14}=\frac{12}{14}\approx0.86\). Then \(P(88<X<112)=P(X < 112)-P(X < 88)\).
For problem 4: \(x = 780\), \(\mu=900\), \(\sigma = 120\). Then \(z=\frac{780 - 900}{120}=\frac{-120}{120}=-1\).
For problem 5: \(x = 62\), \(\mu=50\), \(\sigma = 9\). Then \(z=\frac{62 - 50}{9}=\frac{12}{9}\approx1.33\). The probability that \(X>62\) is \(P(X > 62)=1 - P(X\leq62)\).

Step2: Use the standard normal distribution table

For problem 1: Using the standard - normal table, \(P(X < 70)=P(Z<-0.625)\approx0.266\).
For problem 2: \(P(X\leq68.2)=P(Z\leq1.28) = 0.8997\), so \(P(X > 68.2)=1 - 0.8997=0.1003\).
For problem 3: \(P(X < 112)=P(Z < 0.86)=0.8051\), \(P(X < 88)=P(Z<-0.86)=0.1949\). Then \(P(88 < X < 112)=0.8051-0.1949 = 0.6102\).
For problem 4: \(P(X < 780)=P(Z<-1)=0.1587\).
For problem 5: \(P(X\leq62)=P(Z\leq1.33)=0.9082\), so \(P(X > 62)=1 - 0.9082=0.0918\).

Answer:

1. The probability that a randomly - selected score is less than \(70\) is approximately \(0.266\).
2. The probability that a randomly - selected woman is taller than \(68.2\) inches is approximately \(0.1003\).
3. The probability that a person's IQ is between \(88\) and \(112\) is approximately \(0.6102\).
4. The probability that a battery lasts less than \(780\) hours is approximately \(0.1587\).
5. The probability that a dog weighs more than \(62\) lbs is approximately \(0.0918\).