# 5.2 The uniform distribution  (Page 3/16)

 Page 3 / 16

## Try it

Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. Let X = the time, in minutes, it takes a student to finish a quiz. Then X ~ U (6, 15).

Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Then find the probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes.

P ( x >8) = 0.7778

P ( x >8 | x>7) = 0.875

Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs to fix a furnace is uniformly distributed between 1.5 and four hours. Let x = the time needed to fix a furnace. Then x ~ U (1.5, 4).

1. Find the probability that a randomly selected furnace repair requires more than two hours.
2. Find the probability that a randomly selected furnace repair requires less than three hours.
3. Find the 30 th percentile of furnace repair times.
4. The longest 25% of furnace repair times take at least how long? (In other words: find the minimum time for the longest 25% of repair times.) What percentile does this represent?
5. Find the mean and standard deviation

e. $\mu =\frac{a+b}{2}$ and $\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}$
$((\mu , \frac{1.5+4}{2}), 2.75)$ hours and $\sigma =\sqrt{\frac{{\left(4–1.5\right)}^{2}}{12}}=0.7217$ hours

## Try it

The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Let X = the time needed to change the oil on a car.

1. Write the random variable X in words. X = __________________.
2. Write the distribution.
3. Graph the distribution.
4. Find P ( x >19).
5. Find the 50 th percentile.
1. Let X = the time needed to change the oil in a car.
2. X ~ U (11, 21).
3. P ( x >19) = 0.2
4. the 50 th percentile is 16 minutes.

## Chapter review

If X has a uniform distribution where a < x < b or a x b , then X takes on values between a and b (may include a and b ). All values x are equally likely. We write X U ( a , b ). The mean of X is $\mu =\frac{a+b}{2}$ . The standard deviation of X is $\sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}$ . The probability density function of X is $f\left(x\right)=\frac{1}{b-a}$ for a x b . The cumulative distribution function of X is P ( X x ) = $\frac{x-a}{b-a}$ . X is continuous.

The probability P ( c < X < d ) may be found by computing the area under f ( x ), between c and d . Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height.

## Formula review

X = a real number between a and b (in some instances, X can take on the values a and b ). a = smallest X ; b = largest X

X ~ U (a, b)

The mean is $\mu =\frac{a+b}{2}$

The standard deviation is

Probability density function: $f\left(x\right)=\frac{1}{b-a}$ for $a\le X\le b$

Area to the Left of x : P ( X < x ) = ( x a ) $\left(\frac{1}{b-a}\right)$

Area to the Right of x : P ( X > x ) = ( b x ) $\left(\frac{1}{b-a}\right)$

Area Between c and d : P ( c < x < d ) = (base)(height) = ( d c ) $\left(\frac{1}{b-a}\right)$

Uniform: X ~ U ( a , b ) where a < x < b

• pdf: $f\left(x\right)=\frac{1}{b-a}$ for a ≤ x ≤ b
• cdf: P ( X x ) = $\frac{x-a}{b-a}$
• mean µ = $\frac{a+b}{2}$
• standard deviation σ $=\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}$
• P ( c < X < d ) = ( d c ) $\left(\frac{1}{b–a}\right)$

## References

McDougall, John A. The McDougall Program for Maximum Weight Loss. Plume, 1995.

Use the following information to answer the next ten questions. The data that follow are the square footage (in 1,000 feet squared) of 28 homes.

 1.5 2.4 3.6 2.6 1.6 2.4 2 3.5 2.5 1.8 2.4 2.5 3.5 4 2.6 1.6 2.2 1.8 3.8 2.5 1.5 2.8 1.8 4.5 1.9 1.9 3.1 1.6

The sample mean = 2.50 and the sample standard deviation = 0.8302.

The distribution can be written as X ~ U (1.5, 4.5).

What type of distribution is this?

In this distribution, outcomes are equally likely. What does this mean?

It means that the value of x is just as likely to be any number between 1.5 and 4.5.

What is the height of f ( x ) for the continuous probability distribution?

What are the constraints for the values of x ?

1.5 ≤ x ≤ 4.5

Graph P (2< x <3).

What is P (2< x <3)?

0.3333

What is P (x<3.5| x <4)?

What is P ( x = 1.5)?

zero

What is the 90 th percentile of square footage for homes?

Find the probability that a randomly selected home has more than 3,000 square feet given that you already know the house has more than 2,000 square feet.

0.6

Use the following information to answer the next eight exercises. A distribution is given as X ~ U (0, 12).

What is a ? What does it represent?

What is b ? What does it represent?

b is 12, and it represents the highest value of x .

What is the probability density function?

What is the theoretical mean?

six

What is the theoretical standard deviation?

Draw the graph of the distribution for P ( x >9).

Find P ( x >9).

Find the 40 th percentile.

4.8

Use the following information to answer the next eleven exercises. The age of cars in the staff parking lot of a suburban college is uniformly distributed from six months (0.5 years) to 9.5 years.

What is being measured here?

In words, define the random variable X .

X = The age (in years) of cars in the staff parking lot

Are the data discrete or continuous?

The interval of values for x is ______.

0.5 to 9.5

The distribution for X is ______.

Write the probability density function.

f ( x ) = $\frac{1}{9}$ where x is between 0.5 and 9.5, inclusive.

Graph the probability distribution.

1. Sketch the graph of the probability distribution.
2. Identify the following values:
1. Lowest value for $\overline{x}$ : _______
2. Highest value for $\overline{x}$ : _______
3. Height of the rectangle: _______
4. Label for x -axis (words): _______
5. Label for y -axis (words): _______

Find the average age of the cars in the lot.

μ = 5

Find the probability that a randomly chosen car in the lot was less than four years old.

1. Sketch the graph, and shade the area of interest.
2. Find the probability. P ( x <4) = _______

Considering only the cars less than 7.5 years old, find the probability that a randomly chosen car in the lot was less than four years old.

1. Sketch the graph, shade the area of interest.
2. Find the probability. P ( x <4| x <7.5) = _______
1. Check student’s solution.
2. $\frac{3.5}{7}$

What has changed in the previous two problems that made the solutions different?

Find the third quartile of ages of cars in the lot. This means you will have to find the value such that $\frac{3}{4}$ , or 75%, of the cars are at most (less than or equal to) that age.

1. Sketch the graph, and shade the area of interest.
2. Find the value k such that P ( x < k ) = 0.75.
3. The third quartile is _______
1. Check student's solution.
2. k = 7.25
3. 7.25

Calculate theta if one minus theta times hundred percent is equal to ninety five cI
what's poison distribution
what's poissan distribution?
Abdulhakim
It's Poisson. It's a discrete probability distribution that is used to find the probability of an event x happening within a fixed interval of time. E. g. If a shop sells an average of 5 goods on Saturdays, what is the probability that the shop will sell 7 goods on a particular Saturday.
samoyo
which kind of work do statistics do
how so I know the right answers
?
a. l. bowley definition
what are events in statistics
Like a roll of a dice! Or a coin toss. Or a gender reveal party!
what is statistics
can anyone explain it better for me
the science of statistics deal with the collection, analysis, interpretation and presentation of data
saquib
I am also studying statistics
saquib
Correlation regression, explain it to me in short.
guillio
correlation is used to find relationship between two and dependent ), regression used for predicting the future by analyzing past data
Arun
correlation is used to find relationship between two variables
Arun
dependent and independent eg. profit is dependent on sales
Arun
Statistics has been designed as the mathematical science of making decisions and drawing conclusions from data in situations of uncertainty. It includes the designings of experiments, collection, organization, summarization snd interpretation of numerical data.
Aliya
excellent Aliya..... good...Arun....
IRFAN
The degree or strength of relationship(interdependence) between the variables is called "correlation ". Examples: heights and weights of children, ages of husbands and ages of wives at the time of their marriages, marks of students in mathematics and in statistics.
Aliya
The dependence of one variable (dependent variable) one one or more independent variables ( independent variables) is called "regression ".
Aliya
simply regression and multiple regression are the types of regression.
Aliya
IRFAN HAIDER thanks
Aliya
hi
nabil
I need help with a math problem
nabil
shoot
umair
9. The scatterplot below relates wine consumption (in liters of alcohol from wine per person per year) and death rate from heart disease (in deaths per 100,000 people) for 19 developed countries.
nabil
For questions e. and f. use the equation of the Least-Square Regression LSR line is: y = −22.97x+260.56 e. Circle the correct choice and fill in the blank in the following statement: As wine consumption increases by 1 liter of alcohol per person per year, the predicted death
nabil
Rate from heart disease increases/decreases by ______deaths per ________people.
nabil
is a scientific study of collection analysis interpretation and also presenting it by researchers.
Murtala
frequency distribution
noun STATISTICS a mathematical function showing the number of instances in which a variable takes each of its possible values.
Robin
ok
Common language-- taking a bunch of information and seeing if it is related or not to other info
Mandy
Does standard deviation have measuring unit?
Mohamed
yes, the measuring unit of the data you are looking at, for example centimetres for height.
Emma
thanks
Mohamed
is that easy to plot a graph between three axis?
Mohamed
yes we can but we do not have that much effective tools. If the graph is normal or less complicated then it is plotted effectively otherwise it will give you nightmare.
umair
whats the difference between discrete and contineous data
umar
Discrete variables are variables that can assume finite number of values. Continuous variables are variables that can assume infinite number of values
Mike
i will give you an example: {0,4,84} it contains discrete or limited values like it can also contain boolean values{true,false} or {0,1} and continuous are like {1,2,3,4,5......} , {0,0.1,0.2,0.3,0.4...........}
umair
a no. of values which are countable are called discrete variables on the other hand, a no. of values which are not countable are called continuous variables
Aliya
Yup, I would like to support Mr.Umair's argument by saying that it can only apply if we have a 3-D graph,otherwise a plane graph will not apply at all
festus
Aliya and Mike thnks to both of you ❤❤
umar
what's variance
what's case control study?
Shakilla
hi
Noman
?
Sulaiman
what is covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.[1] If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, (i.e., the variables tend to show simila
Robin
Economics department, faculty of social sciences, NOUN. You are required to calculate: the covariance and State whether the covariance is positive or negative. (11½ marks) Observation E D 1 15 17.24 2 16 15.00 3 8 14.91 4 6 4.50 5 15 18.00 6 12 6.29 7 12 19.23 8 18 18.69 9 12 7.21 10 20 4
Florence
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
Robin
what is the purpose of statistics and why it is important that statistics to be a solo and one complete field?
to organize,analyze and interpret information in order to make decision
Berema
what is noun?
so simple. the name of any person,place or thing.
Edu-info
Using the Chi-square test, two coins were flipped a hundred times. What will be the chances of getting a head and getting a tale? Given observed values is 62 heads and 38 tails. Expected value is 50 heads, 50 tails. Is the difference due to chance or a significant error? a. Draw your hypothesis
how can I win
what is difference between the blocking and confounding