# Continuous Random Variables

Site: | Saylor Academy |

Course: | MA121: Introduction to Statistics |

Book: | Continuous Random Variables |

Printed by: | Guest user |

Date: | Friday, December 1, 2023, 5:19 PM |

## Description

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 -score transformations. Finally, it defines tail probabilities and illustrates how to find them.

## Continuous Random Variables

### Learning Objectives

- To learn the concept of the probability distribution of a continuous random variable, and how it is used to compute probabilities.
- To learn basic facts about the family of normally distributed random variables.

### The Probability Distribution of a Continuous Random Variable

For a discrete random variable the probability that assumes one of its possible values on a single trial of the experiment makes good sense. This is not the case for a continuous random variable. For example, suppose denotes the length of time a commuter just arriving at a bus stop has to wait for the next bus. If buses run every minutes without fail, then the set of possible values of is the interval denoted , the set of all decimal numbers between and . But although the number is a possible value of , there is little or no meaning to the concept of the probability that the commuter will wait precisely minutes for the next bus. If anything the probability should be zero, since if we could meaningfully measure the waiting time to the nearest millionth of a minute it is practically inconceivable that we would ever get exactly minutes. More meaningful questions are those of the form: What is the probability that the commuter's waiting time is less than minutes, or is between and minutes? In other words, with continuous random variables one is concerned not with the event that the variable assumes a single particular value, but with the event that the random variable assumes a value in a particular interval.

### Definition

*The probability distribution of a continuous random variable is an assignment of probabilities to intervals of decimal numbers using a function , called density function, in the following way: the probability that assumes a value in the interval is equal to the area of the region that is bounded above by the graph of the equation , bounded below by the -axis, and bounded on the left and right by the vertical lines through and , as illustrated in Figure 5.1 "Probability Given as Area of a Region under a Curve".*

**Figure 5.1** Probability Given as Area of a Region under a Curve

This definition can be
understood as a natural outgrowth of the discussion in Section 2.1.3
"Relative Frequency Histograms" in Chapter 2 "Descriptive Statistics".
There we saw that if we have in view a population (or a very large
sample) and make measurements with greater and greater precision, then
as the bars in the relative frequency histogram become exceedingly fine
their vertical sides merge and disappear, and what is left is just the
curve formed by their tops, as shown in Figure 2.5 "Sample Size and
Relative Frequency Histograms" in Chapter 2 "Descriptive Statistics".
Moreover the total area under the curve is , and the proportion of
the population with measurements between two numbers and is
the area under the curve and between and , as shown in Figure
2.6 "A Very Fine Relative Frequency Histogram" in Chapter 2
"Descriptive Statistics". If we think of as a measurement to
infinite precision arising from the selection of any one member of the
population at random, then is simply the
proportion of the population with measurements between and ,
the curve in the relative frequency histogram is the density function
for , and we arrive at the definition just above.

Every density function must satisfy the following two conditions:

- For all numbers , so that the graph of never drops below the -axis.
- The area of the region under the graph of and above the -axis is .

Because the area of a line
segment is , the definition of the probability distribution of a
continuous random variable implies that for any particular decimal
number, say , the probability that assumes the exact value
is . This property implies that whether or not the endpoints
of an interval are included makes no difference concerning the
probability of the interval.

### Example 1

A random variable has the uniform distribution on the interval : the density function is if is between and and for all other values of , as shown in Figure 5.2 "Uniform Distribution on".

**Figure 5.2** Uniform Distribution on

- Find , the probability that assumes value greater than.
- Find , the probability that assumes value less than or equal to .
- Find , the probability that assumes value between and .

Solution:

- is the area of the rectangle of height and base length , hence is . See Figure 5.3 "Probabilities from the Uniform Distribution on "(a).
- is the area of the rectangle of height and base length , hence is . See Figure 5.3 "Probabilities from the Uniform Distribution on "(b).
- is the area of the rectangle of height and length , hence is . See Figure 5.3 "Probabilities from the Uniform Distribution on "(c).

**Figure 5.3** Probabilities from the Uniform Distribution on

### Example 2

A man arrives at a bus stop at a random time (that is, with no regard for the scheduled service) to catch the next bus. Buses run every 30 minutes without fail, hence the next bus will come any time during the next 30 minutes with evenly distributed probability (a uniform distribution). Find the probability that a bus will come within the next 10 minutes.

Solution:

The graph of the density function is a horizontal line above the interval from to and is the -axis everywhere else. Since the total area under the curve must be , the height of the horizontal line is . See Figure 5.4 "Probability of Waiting At Most Minutes for a Bus". The probability sought is . By definition, this probability is the area of the rectangular region bounded above by the horizontal line , bounded below by the -axis, bounded on the left by the vertical line at (the -axis), and bounded on the right by the vertical line at . This is the shaded region in Figure 5.4 "Probability of Waiting At Most Minutes for a Bus". Its area is the base of the rectangle times its height, . Thus .

**Figure 5.4** Probability of Waiting At Most Minutes for a Bus

### Normal Distributions

Most people have heard of the "bell curve". It is the graph of a specific density function that describes the behavior of continuous random variables as different as the heights of human beings, the amount of a product in a container that was filled by a high-speed packing machine, or the velocities of molecules in a gas. The formula for contains two parameters and that can be assigned any specific numerical values, so long as is positive. We will not need to know the formula for , but for those who are interested it is

where and is the base of the natural logarithms.

Each different choice of specific numerical values for the pair and gives a different bell curve. The value of determines the location of the curve, as shown in Figure 5:5 "Bell Curves with ". In each case the curve is symmetric about .

**Figure 5.5** Bell Curves with and Different Values of

The value of determines whether the bell curve is tall and thin or short and squat, subject always to the condition that the total area under the curve be equal to . This is shown in Figure 5.6 "Bell Curves with ", where we have arbitrarily chosen to center the curves at .

**Figure 5.6** Bell Curves with and Different Values of

### Definition

*The probability distribution corresponding to the density function for the bell curve with parameters and is called the normal distribution with mean and standard deviation .*

### Definition

*A continuous random variable whose probabilities are described by the normal distribution with mean and standard deviation is called a normally distributed random variable, or a normal random variable for short, with mean and standard deviation .*

Figure 5.7 "Density Function for a Normally Distributed Random Variable with Mean" shows the density function that determines the normal distribution with mean and standard deviation . We repeat an important fact about this curve:

The density curve for the normal distribution is symmetric about the mean.

**Figure 5.7** Density Function for a Normally Distributed Random Variable with Mean and Standard Deviation

### Example 3

Heights of 25 -year-old men in a certain region have mean inches and standard deviation inches. These heights are approximately normally distributed. Thus the height of a randomly selected 25 -year-old man is a normal random variable with mean and standard deviation . Sketch a qualitatively accurate graph of the density function for . Find the probability that a randomly selected 25-year-old man is more than inches tall.

Solution:

The distribution of heights looks like the bell curve in Figure 5.8 "Density Function for Heights of 25 Year-Old Men". The important point is that it is centered at its mean, , and is symmetric about the mean.

**Figure 5.8** Density Function for Heights of 25-Year-Old Men

Since the total area under the curve is , by symmetry the area to the right of is half the total, or . But this area is precisely the probability , the probability that a randomly selected 25 year-old man is more than inches tall.

We will learn how to compute other probabilities in the next two sections.

This text was adapted by Saylor Academy under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License without attribution as requested by the work's original creator or licensor.

### Key Takeaways

- For a continuous random variable the only probabilities that are computed are those of taking a value in a specified interval.
- The probability that take a value in a particular interval is the same whether or not the endpoints of the interval are included.
- The probability , that take a value in the interval from to , is the area of the region between the vertical lines through and , above the -axis, and below the graph of a function called the density function.
- A normally distributed random variable is one whose density function is a bell curve.
- Every bell curve is symmetric about its mean and lies everywhere above the -axis, which it approaches asymptotically (arbitrarily closely without touching).

### Exercises

### Basic

1. A continuous random variable has a uniform distribution on the interval . Sketch the graph of its density function.

3. A continuous random variable has a normal distribution with mean and standard deviation . Sketch a qualitatively accurate graph of its density function.

5. A continuous random variable has a normal distribution with mean . The probability that takes a value greater than is . Use this information and the symmetry of the density function to find the probability that takes a value less than . Sketch the density curve with relevant regions shaded to illustrate the computation.

7. A continuous random variable has a normal distribution with mean . The probability that takes a value less than is . Use this information and the symmetry of the density function to find the probability that takes a value greater than . Sketch the density curve with relevant regions shaded to illustrate the computation.

9. The figure provided shows the density curves of three normally distributed random variables , , and . Their standard deviations (in no particular order) are , , and . Use the figure to identify the values of the means , , and and standard deviations , , and of the three random variables.

### Applications

11. Dogberry's alarm clock is battery operated. The battery could fail with equal probability at any time of the day or night. Every day Dogberry sets his alarm for 6:30 a.m. and goes to bed at 10:00 p.m. Find the probability that when the clock battery finally dies, it will do so at the most inconvenient time, between 10:00 p.m. and 6:30 a.m.

13. The amount of orange juice in a randomly selected half-gallon container varies according to a normal distribution with mean ounces and standard deviation ounce.

a. Sketch the graph of the density function for .b. What proportion of all containers contain less than a half gallon ( ounces)? Explain.

c. What is the median amount of orange juice in such containers? Explain.

## Probability Computations for General Normal Random Variables

### Learning Objective

- To learn how to compute probabilities related to any normal random variable.

If is any normally distributed normal random variable then Figure 12.2 "Cumulative Normal Probability" can also be used to compute a probability of the form by means of the following equality.

If is a normally distributed random variable with mean and standard deviation , then

where denotes a standard normal random variable. can be any decimal number or can be any decimal number or .

The new endpoints and are the -scores of and as defined in Section 2.4.2 in Chapter 2 "Descriptive Statistics".

Figure 5.14 "Probability for an Interval of Finite Length" illustrates the meaning of the equality geometrically: the two shaded regions, one under the density curve for and the other under the density curve for , have the same area. Instead of drawing both bell curves, though, we will always draw a single generic bell-shaped curve with both an -axis and a -axis below it.

**Figure 5.14** Probability for an Interval of Finite Length

### Example 9

Let be a normal random variable with mean and standard deviation . Compute the following probabilities.

a. .Solution:

a. See Figure 5.15 "Probability Computation for a General Normal Random Variable".

**Figure 5.15**Probability Computation for a General Normal Random Variablea. See Figure 5.16 "Probability Computation for a General Normal Random Variable".

**Figure 5.16** Probability Computation for a General Normal Random Variable

### Example 10

The lifetimes of the tread of a certain automobile tire are normally distributed with mean 37,500 miles and standard deviation 4,500 miles. Find the probability that the tread life of a randomly selected tire will be between 30,000 and 40,000 miles.

Let denote the tread life of a randomly selected tire. To make the numbers easier to work with we will choose thousands of miles as the units. Thus , and the problem is to compute . Figure 5.17 "Probability Computation for Tire Tread Wear" illustrates the following computation:

**Figure 5.17** Probability Computation for Tire Tread Wear

Note that the two -scores were rounded to two decimal places in order to use Figure 12.2 "Cumulative Normal Probability".

### Example 11

Scores on a standardized college entrance examination (*CEE*)
are normally distributed with mean 510 and standard deviation 60. A
selective university considers for admission only applicants with *CEE* scores over 650. Find percentage of all individuals who took the *CEE* who meet the university's *CEE* requirement for consideration for admission.

Solution:

Let denote the score made on the *CEE* by a randomly selected individual. Then is normally distributed with mean 510 and standard deviation 60. The probability that lie in a particular interval is the same as the proportion of all exam scores that lie in that interval. Thus the solution to the problem is , expressed as a percentage. Figure 5.18 "Probability Computation for Exam Scores" illustrates the following computation:

**Figure 5.18** Probability Computation for Exam Scores

The proportion of all *CEE* scores that exceed 650 is , hence or about do.

### Key Takeaway

### Exercises

### Basic

1. is a normally distributed random variable with mean and standard deviation . Find the probability indicated.

3. is a normally distributed random variable with mean and standard deviation . Find the probability indicated.

5. is a normally distributed random variable with mean and standard deviation . Find the probability indicated.

7. is a normally distributed random variable with mean and standard deviation . Use Figure 12.2 "Cumulative Normal Probability" to find the first probability listed. Find the second probability using the symmetry of the density curve. Sketch the density curve with relevant regions shaded to illustrate the computation.

9. is a normally distributed random variable with mean and standard deviation . The probability that takes a value in the union of intervals will be denoted or . Use Figure 12.2 "Cumulative Normal Probability" to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the density curve you need to use Figure 12.2 "Cumulative Normal Probability" only one time for each part.

### Applications

11. The amount of beverage in a can labeled ounces is normally distributed with mean ounces and standard deviation ounce. A can is selected at random.

b. Find the probability that the can contains between and ounces.

13. The
systolic blood pressure of adults in a region is normally
distributed with mean and standard
deviation . A person is considered
"prehypertensive" if his systolic blood pressure is between and
. Find the probability that the blood
pressure of a randomly selected person is prehypertensive.

15.
Heights of adult men are normally distributed with mean
inches and standard deviation inches. Juliet, who is
inches tall, wishes to date only men who are taller than she but within
inches of her height. Find the probability that the next man she
meets will have such a height.

17. A
regulation golf ball may not weigh more than ounces. The
weights of golf balls made by a particular process are normally
distributed with mean ounces and standard deviation
ounce. Find the probability that a golf ball made by this process will
meet the weight standard.

19. The
amount of non-mortgage debt per household for households in a
particular income bracket in one part of the country is normally
distributed with mean and standard deviation .
Find the probability that a randomly selected such household has
between and in non-mortgage debt.

21. The
distance from the seat back to the front of the knees of seated adult
males is normally distributed with mean inches and standard
deviation inches. The distance from the seat back to the back
of the next seat forward in all seats on aircraft flown by a budget
airline is inches. Find the proportion of adult men flying with
this airline whose knees will touch the back of the seat in front of
them.

23. The useful life of a particular make and type of automotive tire is normally distributed with mean , miles and standard deviation miles.

a. Find the probability that such a tire will have a useful life of between and miles.

b.
Hamlet buys four such tires. Assuming that their lifetimes are
independent, find the probability that all four will last between
and miles. (If so, the best tire will have no more
than miles left on it when the first tire fails.) Hint: There
is a binomial random variable here, whose value of comes from
part (a).

25. The lengths of time taken by students on an algebra proficiency exam (if not forced to stop before completing it) are normally distributed with mean minutes and standard deviation minutes.

a. Find the proportion of students who will finish the exam if a -minute time limit is set.

b.
Six students are taking the exam today. Find the probability that all
six will finish the exam within the -minute limit, assuming that
times taken by students are independent. Hint: There is a binomial
random variable here, whose value of comes from part (a).

27. A regulation hockey puck must weigh between and ounces. In an alternative manufacturing process the mean weight of pucks produced is ounce. The weights of pucks have a normal distribution whose standard deviation can be decreased by increasingly stringent (and expensive) controls on the manufacturing process. Find the maximum allowable standard deviation so that at most of all pucks will fail to meet the weight standard. (Hint: The distribution is symmetric and is centered at the middle of the interval of acceptable weights).

## Areas of Tails of Distributions

### Learning Objective

- To learn how to find, for a normal random variable and an area , the value of so that or that , whichever is required.

### Definition

*The left tail of a density curve of a continuous random variable cut off by a value of is the region under the curve that is to the left of , as shown by the shading in Figure 5.19 "Right and Left Tails of a Distribution" (a). The right tail cut off by is defined similarly, as indicated by the shading in Figure 5.19 "Right and Left Tails of a Distribution"(b).*

**Figure 5.19** Right and Left Tails of a Distribution

The probabilities tabulated in Figure 12.2 "Cumulative Normal Probability" are areas of *left* tails in the standard normal distribution.

### Tails of the Standard Normal Distribution

At times it is important to be able to solve the kind of problem illustrated by Figure 5.20. We have a certain specific area in mind, in this case the area of the shaded region in the figure, and we want to find the value of that produces it. This is exactly the reverse of the kind of problems encountered so far. Instead of knowing a value of and finding a corresponding area, we know the area and want to find . In the case at hand, in the terminology of the definition just above, we wish to find the value that cuts off a left tail of area in the standard normal distribution.

The idea for solving such a problem is fairly simple, although sometimes its implementation can be a bit complicated. In a nutshell, one reads the cumulative probability table for in reverse, looking up the relevant area in the interior of the table and reading off the value of from the margins.

**Figure 5.20** * Value that Produces a Known Area*

### Example 12

Find the value of as determined by Figure 5.20: the value that cuts off a left tail of area in the standard normal distribution. In symbols, find the number such that .

Solution:

The number that is known, , is the area of a left tail, and as already mentioned the probabilities tabulated in Figure 12.2 "Cumulative Normal Probability" are areas of left tails. Thus to solve this problem we need only search in the interior of Figure 12.2 "Cumulative Normal Probability" for the number . It lies in the row with the heading and in the column with the heading . This means that , hence .

### Example 13

Find the value of as determined by Figure 5.21: the value that cuts off a right tail of area in the standard normal distribution. In symbols, find the number such that .

**Figure 5.21** * Value that Produces a Known Area*

Solution:

The important distinction between this example and the previous one is that here it is the area of a *right* tail that is known. In order to be able to use Figure 12.2 "Cumulative Normal Probability" we must first find that area of the *left* tail cut off by the unknown number . Since the total area under the density curve is , that area is . This is the number we look for in the interior of Figure 12.2 "Cumulative Normal Probability". It lies in the row with the heading and in the column with the heading . Therefore .

### Definition

*The value of the standard normal random variable that cuts off a right tail of area is denoted . By symmetry, value of that cuts off a left tail of area is . See Figure 5.22 "The Numbers".*

The previous two examples were atypical because the areas we were looking for in the interior of Figure 12.2 "Cumulative Normal Probability" were actually there. The following example illustrates the situation that is more common.

### Example 14

Find and , the values of that cut off right and left tails of area in the standard normal distribution.

Solution:

Since cuts off a left tail of area and Figure 12.2 "Cumulative Normal Probability" is a table of left tails, we look for the number in the interior of the table. It is not there, but falls between the two numbers and in the row with heading . The number is closer to than is, so for the hundredths place in we use the heading of the column that contains , namely, , and write .

The answer to the second half of the problem is automatic: since , we conclude immediately that .

We could just as well have solved this problem by looking for first, and it is instructive to rework the problem this way. To begin with, we must first subtract from to find the area of the left tail cut off by the unknown number . See Figure 5.23

"Computation of the Number ". Then we search for the area in Figure 12.2 "Cumulative Normal Probability". It is not there, but falls between the numbers and in the row with heading . Since is closer to than is, we use the column heading above it, , to obtain the approximation . Then finally .

**Figure 5.23** Computation of the Number

### Tails of General Normal Distributions

The problem of finding the value of a general normally distributed random variable that cuts off a tail of a specified area also arises. This problem may be solved in two steps.

Suppose is a normally distributed random variable with mean and standard deviation . To find the value of that cuts off a left or right tail of area in the distribution of :

1. find the value of that cuts off a left or right tail of area in the standard normal distribution;

2. is the -score of ; compute using the destandardization formula

In short, solve the corresponding problem for the standard normal distribution, thereby obtaining the -score of , then destandardize to obtain .

### Example 15

Find such that , where is a normal random variable with mean and standard deviation .

Solution:

All the ideas for the solution are illustrated in Figure 5.24 "Tail of a Normally Distributed Random Variable". Since is the area of a left tail, we can find simply by looking for in the interior of Figure 12.2 "Cumulative Normal Probability". It is in the row and column with headings and , hence . Thus is standard deviations above the mean, so

**Figure 5.24** Tail of a Normally Distributed Random Variable

### Example 16

Find such that , where is a normal random variable with mean and standard deviation .

Solution:

The situation is illustrated in Figure 5.25 "Tail of a Normally Distributed Random Variable". Since is the area of a right tail, we first subtract it from to obtain , the area of the complementary left tail. We find by looking for in the interior of Figure 12.2 "Cumulative Normal Probability". It is not present, but lies between table entries and . The entry with row and column headings and is closer to than the other entry is, so . Thus is standard deviations below the mean, so

**Figure 5.25** Tail of a Normally Distributed Random Variable

### Example 17

Scores on a standardized college entrance examination *(CEE)* are normally distributed with mean and standard deviation . A selective university decides to give serious consideration for admission to applicants whose *CEE* scores are in the top of all *CEE* scores. Find the minimum score that meets this criterion for serious consideration for admission.

Solution:

Let denote the score made on the *CEE* by a randomly selected individual. Then is normally distributed with mean and standard deviation . The probability that lie in a particular interval is the same as the proportion of all exam scores that lie in that interval. Thus the minimum score that is in the top of all *CEE* is the score that cuts off a right tail in the distribution of of area ( expressed as a proportion). See Figure 5.26 "Tail of a Normally Distributed Random Variable".

**Figure 5.26** Tail of a Normally Distributed Random Variable

Since is the area of a right tail, we first subtract it from to obtain , the area of the complementary left tail. We find by looking for in the interior of Figure 12.2 "Cumulative Normal Probability". It is not present, and lies exactly half-way between the two nearest entries that are, and . In the case of a tie like this, we will always average the values of corresponding to the two table entries, obtaining here the value . Using this value, we conclude that is standard deviations above the mean, so

### Example 18

All boys at a military school must run a fixed course as fast as they can as part of a physical examination. Finishing times are normally distributed with mean minutes and standard deviation minutes. The middle of all finishing times are classified as "average". Find the range of times that are average finishing times by this definition.

Solution:

Let denote the finish time of a randomly selected boy. Then is normally distributed with mean and standard deviation . The probability that lie in a particular interval is the same as the proportion of all finish times that lie in that interval. Thus the situation is as shown in Figure 5.27 "Distribution of Times to Run a Course". Because the area in the middle corresponding to "average" times is , the areas of the two tails add up to in all. By the symmetry of the density curve each tail must have half of this total, or area each. Thus the fastest time that is "average" has -score , which by Figure 12.2 "Cumulative Normal Probability" is , and the slowest time that is "average" has -score . The fastest and slowest times that are still considered average are

and

**Figure 5.27** Distribution of Times to Run a Course

A boy has an average finishing time if he runs the course with a time between and minutes, or equivalently between minutes seconds and minutes seconds.

### Key Takeaways

- The problem of finding the number so that the probability is a specified value is solved by looking for the number in the interior of Figure 12.2 "Cumulative Normal Probability" and reading from the margins.
- The problem of finding the number so that the probability is a specified value is solved by looking for the complementary probability in the interior of Figure 12.2 "Cumulative Normal Probability" and reading from the margins.
- For a normal random variable with mean and standard deviation , the problem of finding the number so that is a specified value (or so that is a specified value ) is solved in two steps: (1) solve the corresponding problem for with the same value of , thereby obtaining the -score, , of ; (2) find using .
- The value of that cuts off a right tail of area in the standard normal distribution is denoted .

### Exercises

### Basic

1. Find the value of that yields the probability shown.

a.b.

c.

d.

3. Find the value of that yields the probability shown.

5. Find the indicated value of . (It is easier to find and negate it.)

7. Find the value of that yields the probability shown, where is a normally distributed random variable with mean and standard deviation .

9. is a normally
distributed random variable with mean and standard
deviation . Find the values and of
that are symmetrically located with respect to the mean of and
satisfy . (Hint. First
solve the corresponding problem for .)

### Applications

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

a. Find the score that is the th percentile.b. Find the score that is the th percentile.

13. The monthly amount of water used per household in a small community is normally distributed with mean gallons and standard deviation 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 and standard deviation . The
department has the rule that in order to receive an A in the course his
score must be in the top 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
miles and standard deviation of miles. The
manufacturer will advertise the lifetime of the tire (for example, a "
mile tire") using the largest value for which it is expected
that of the tires will last at least that long. Assuming tire
life is normally distributed, find that advertised value.

19. The weights of eggs
produced at a particular farm are normally distributed with mean
ounces and standard deviation ounce. Eggs whose
weights lie in the middle 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 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 to
students whose scores constitute the middle of all scores. If
scores this semester had mean and standard deviation ,
find the interval of scores that will be assigned a .

### Additional Exercises

23. A machine for filling -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 liter and mean whatever amount the machine is set to deliver.

a. If the machine is set to deliver liters (so the mean amount delivered is liters) what proportion of the bottles will contain at least liters of soft drink?b. Find the minimum setting of the mean amount delivered by the machine so that at least of all bottles will contain at least liters.