## Continuous Random Variables

First, this section talks about how to describe continuous distributions and compute related probabilities, including some basic facts about the normal distribution. Then, it covers how to compute probabilities related to any normal random variable and gives examples of using $z$-score transformations. Finally, it defines tail probabilities and illustrates how to find them.

### Basic

1. Find the value of $\mathrm{z}^{*}$ that yields the probability shown.

a. $P\left(Z < \mathrm{z}^{*}\right)=0.0075$
b. $P\left(Z < \mathrm{z}^{*}\right)=0.9850$
c. $P\left(Z > \mathrm{z}^{*}\right)=0.8997$
d. $P\left(Z > \mathrm{z}^{*}\right)=0.0110$

3. Find the value of $\mathrm{z}^{*}$ that yields the probability shown.

a. $P\left(Z < \mathrm{z}^{*}\right)=0.1500$
b. $P\left(Z < \mathrm{z}^{*}\right)=0.7500$
c. $P\left(Z > \mathrm{z}^{*}\right)=0.3333$
d. $P\left(Z > \mathrm{z}^{*}\right)=0.8000$

5. Find the indicated value of $Z$. (It is easier to find $-z_{c}$ and negate it.)

a. $z 0.025$
b. $z 0.20$

7. Find the value of $x^{*}$ that yields the probability shown, where $X$ is a normally distributed random variable $X$ with mean $83$ and standard deviation $4$.

a. $P\left(X < \mathrm{x}^{*}\right)=0.8700$
b. $P\left(X > \mathrm{x}^{*}\right)=0.0500$

9. $X$ is a normally distributed random variable $X$ with mean $15$ and standard deviation $0.25$. Find the values $x_{L}$ and $x_{R}$ of $X$ that are symmetrically located with respect to the mean of $X$ and satisfy $P\left(x_{L} < X < x_{R}\right) = 0.80$. (Hint. First solve the corresponding problem for $Z$.)

### Applications

11. Scores on a national exam are normally distributed with mean $382$ and standard deviation $26$.

a. Find the score that is the $50$th percentile.
b. Find the score that is the $90$th percentile.

13. The monthly amount of water used per household in a small community is normally distributed with mean $7,069$ gallons and standard deviation $58$ gallons. Find the three quartiles for the amount of water used.

15. Scores on the common final exam given in a large enrollment multiple section course were normally distributed with mean $69.35$ and standard deviation $12.93$. The department has the rule that in order to receive an A in the course his score must be in the top $10 \%$ of all exam scores. Find the minimum exam score that meets this requirement.

17. Tests of a new tire developed by a tire manufacturer led to an estimated mean tread life of $67,350$ miles and standard deviation of $1,120$ miles. The manufacturer will advertise the lifetime of the tire (for example, a " $50,000$ mile tire") using the largest value for which it is expected that $98 \%$ of the tires will last at least that long. Assuming tire life is normally distributed, find that advertised value.

19. The weights $X$ of eggs produced at a particular farm are normally distributed with mean $1.72$ ounces and standard deviation $0.12$ ounce. Eggs whose weights lie in the middle $75 \%$ of the distribution of weights of all eggs are classified as "medium". Find the maximum and minimum weights of such eggs. (These weights are endpoints of an interval that is symmetric about the mean and in which the weights of $75 \%$ of the eggs produced at this farm lie.)

21. All students in a large enrollment multiple section course take common in-class exams and a common final, and submit common homework assignments. Course grades are assigned based on students' final overall scores, which are approximately normally distributed. The department assigns a $C$ to students whose scores constitute the middle $2 / 3$ of all scores. If scores this semester had mean $72.5$ and standard deviation $6.14$, find the interval of scores that will be assigned a $C$.

23. A machine for filling $2$-liter bottles of soft drink delivers an amount to each bottle that varies from bottle to bottle according to a normal distribution with standard deviation $0.002$ liter and mean whatever amount the machine is set to deliver.
a. If the machine is set to deliver $2$ liters (so the mean amount delivered is $2$ liters) what proportion of the bottles will contain at least $2$ liters of soft drink?
b. Find the minimum setting of the mean amount delivered by the machine so that at least $99 \%$ of all bottles will contain at least $2$ liters.