Question

7.40 the time required to cook a pizza at a neighborhood pizza joint is normally distributed with a mean of 12 minutes and a standard deviation of 2 minutes. find the time for each event. show your work. a. highest 5 percent b. lowest 50 percent c. middle 95 percent d. lowest 80 percent

Explanation:

Step1: Recall the z - score formula

The formula for a z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation. We will find the z - scores corresponding to the given percentiles from the standard normal distribution table and then solve for $x$.

Step2: a. For the highest 5 percent

The area to the left of the z - score is $1 - 0.05=0.95$. Looking up in the standard - normal table, the z - score $z\approx1.645$. Using the z - score formula $z=\frac{x-\mu}{\sigma}$, and substituting $\mu = 12$, $\sigma = 2$, and $z = 1.645$, we get $1.645=\frac{x - 12}{2}$. Solving for $x$: $x=12+1.645\times2=12 + 3.29 = 15.29$ minutes.

Step3: b. For the lowest 50 percent

The area to the left of the z - score is 0.5. The z - score corresponding to an area of 0.5 in the standard - normal table is $z = 0$. Using the z - score formula $z=\frac{x-\mu}{\sigma}$, and substituting $\mu = 12$, $\sigma = 2$, and $z = 0$, we get $0=\frac{x - 12}{2}$, so $x = 12$ minutes.

Step4: c. For the middle 95 percent

The area in the two tails is $1-0.95 = 0.05$, so the area in each tail is $\frac{0.05}{2}=0.025$. The z - score corresponding to an area of $0.025$ to the left is $z=-1.96$, and the z - score corresponding to an area of $0.975$ (since $0.025$ is in the left - tail and $0.95$ is in the middle, so $0.025+0.95 = 0.975$) to the left is $z = 1.96$.
For $z=-1.96$, $-1.96=\frac{x_1 - 12}{2}$, so $x_1=12-1.96\times2=12 - 3.92 = 8.08$ minutes.
For $z = 1.96$, $1.96=\frac{x_2 - 12}{2}$, so $x_2=12 + 1.96\times2=12+3.92 = 15.92$ minutes.

Step5: d. For the lowest 80 percent

The area to the left of the z - score is 0.8. Looking up in the standard - normal table, the z - score $z\approx0.84$. Using the z - score formula $z=\frac{x-\mu}{\sigma}$, and substituting $\mu = 12$, $\sigma = 2$, and $z = 0.84$, we get $0.84=\frac{x - 12}{2}$. Solving for $x$: $x=12+0.84\times2=12 + 1.68 = 13.68$ minutes.

Answer:

a. 15.29 minutes
b. 12 minutes
c. 8.08 minutes and 15.92 minutes
d. 13.68 minutes