5.2 The uniform distribution  (Page 3/16)

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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

what is confidence interval estimate and its formula in getting it
discuss the roles of vital and health statistic in the planning of health service of the community
given that the probability of
BITRUS
can man city win Liverpool ?
There are two coins on a table. When both are flipped, one coin land on heads eith probability 0.5 while the other lands on head with probability 0.6. A coin is randomly selected from the table and flipped. (a) what is probability it lands on heads? (b) given that it lands on tail, what is the Condi
0.5*0.5+0.5*0.6
Ravasz
It should be a Machine learning terms。
Mok
it is a term used in linear regression
Saurav
what are the differences between standard deviation and variancs?
Enhance
what is statistics
statistics is the collection and interpretation of data
Enhance
the science of summarization and description of numerical facts
Enhance
Is the estimation of probability
Zaini
mr. zaini..can u tell me more clearly how to calculated pair t test
Haai
do you have MG Akarwal Statistics' book Zaini?
Enhance
Haai how r u?
Enhance
maybe .... mathematics is the science of simplification and statistics is the interpretation of such values and its implications.
Miguel
can we discuss about pair test
Haai
what is outlier?
outlier is an observation point that is distant from other observations.
Gidigah
what is its effect on mode?
Usama
Outlier  have little effect on the mode of a given set of data.
Gidigah
How can you identify a possible outlier(s) in a data set.
Daniel
The best visualisation method to identify the outlier is box and wisker method or boxplot diagram. The points which are located outside the max edge of wisker(both side) are considered as outlier.
Akash
@Daniel Adunkwah - Usually you can identify an outlier visually. They lie outside the observed pattern of the other data points, thus they're called outliers.
Ron
what is completeness?
I am new to this. I am trying to learn.
Dom
I am also new Dom, welcome!
Nthabi
thanks
Dom
please my friend i want same general points about statistics. say same thing
alex
outliers do not have effect on mode
Meselu
also new
yousaf
I don't get the example
ways of collecting data at least 10 and explain
Example of discrete variable
Gbenga
I am new here, can I get someone to guide up?
alayo
dies outcome is 1, 2, 3, 4, 5, 6 nothing come outside of it. it is an example of discrete variable
jainesh
continue variable is any value value between 0 to 1 it could be 4digit values eg 0.1, 0.21, 0.13, 0.623, 0.32
jainesh
hi
Kachalla
what's up here ... am new here
Kachalla
sorry question a bit unclear...do you mean how do you analyze quantitative data? If yes, it depends on the specific question(s) you set in the beginning as well as on the data you collected. So the method of data analysis will be dependent on the data collecter and questions asked.
Bheka
how to solve for degree of freedom
saliou
Quantitative data is the data in numeric form. For eg: Income of persons asked is 10,000. This data is quantitative data on the other hand data collected for either make or female is qualitative data.
Rohan
*male
Rohan
Degree of freedom is the unconditionality. For example if you have total number of observations n, and you have to calculate variance, obviously you will need mean for that. Here mean is a condition, without which you cannot calculate variance. Therefore degree of freedom for variance will be n-1.
Rohan
data that is best presented in categories like haircolor, food taste (good, bad, fair, terrible) constitutes qualitative data
Bheka
vegetation types (grasslands, forests etc) qualitative data
Bheka
I don't understand how you solved it can you teach me
solve what?
Ambo
mean
Vanarith
What is the end points of a confidence interval called?
lower and upper endpoints
Bheka
Class members write down the average time (in hours, to the nearest half-hour) they sleep per night.
how we make a classes of this(170.3,173.9,171.3,182.3,177.3,178.3,174.175.3)
Sarbaz
6.5
phoenix
11
Shakir
7.5
Ron
why is always lower class bundry used
Caleb
Assume you are in a class where quizzes are 20% of your grade, homework is 20%, exam _1 is 15%,exam _2 is 15%, and the final exam is 20%.Suppose you are in the fifth week and you just found out that you scored a 58/63 on the fist exam. You also know that you received 6/9,8/10,9/9 on the first
quizzes as well as a 9/11,10/10,and 4.5/7 on the first three homework assignment. what is your current grade in the course?
Diamatu
Abdul