<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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. μ = a + b 2 and σ = ( b a ) 2 12
μ 1.5 + 4 2 2.75 hours and σ = ( 4 1.5 ) 2 12 = 0.7217 hours

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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. This graph shows a uniform distribution. The horizontal axis ranges from 405 to 525. The distribution is modeled by a rectangle extending from x = 447 to x = 521.
  4. P ( x >19) = 0.2
  5. the 50 th percentile is 16 minutes.
Got questions? Get instant answers now!

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 μ = a + b 2 . The standard deviation of X is σ = ( b a ) 2 12 . The probability density function of X is f ( x ) = 1 b a for a x b . The cumulative distribution function of X is P ( X x ) = x a b a . X is continuous.

The graph shows a rectangle with total area equal to 1. The rectangle extends from x = a to x = b on the x-axis and has a height of 1/(b-a).

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 μ = a + b 2

The standard deviation is σ = ( b  –  a ) 2 12

Probability density function: f ( x ) = 1 b a for a X b

Area to the Left of x : P ( X < x ) = ( x a ) ( 1 b a )

Area to the Right of x : P ( X > x ) = ( b x ) ( 1 b a )

Area Between c and d : P ( c < x < d ) = (base)(height) = ( d c ) ( 1 b a )

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

  • pdf: f ( x ) = 1 b a for a ≤ x ≤ b
  • cdf: P ( X x ) = x a b a
  • mean µ = a + b 2
  • standard deviation σ = ( b a ) 2 12
  • P ( c < X < d ) = ( d c ) ( 1 b a )

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 2.0
3.5 2.5 1.8 2.4 2.5 3.5 4.0
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?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

What are the constraints for the values of x ?

1.5 ≤ x ≤ 4.5

Got questions? Get instant answers now!

What is P (2< x <3)?

0.3333

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

What is P ( x = 1.5)?

zero

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!


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?

Got questions? Get instant answers now!

What is b ? What does it represent?

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

Got questions? Get instant answers now!

What is the probability density function?

Got questions? Get instant answers now!

What is the theoretical mean?

six

Got questions? Get instant answers now!

What is the theoretical standard deviation?

Got questions? Get instant answers now!

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

This graph shows a uniform distribution. The horizontal axis ranges from 0 to 12. The distribution is modeled by a rectangle extending from x = 0 to x = 12. A region from x = 9 to x = 12 is shaded inside the rectangle.
Got questions? Get instant answers now!

Find the 40 th percentile.

4.8

Got questions? Get instant answers now!


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?

Got questions? Get instant answers now!

In words, define the random variable X .

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

Got questions? Get instant answers now!

Are the data discrete or continuous?

Got questions? Get instant answers now!

The interval of values for x is ______.

0.5 to 9.5

Got questions? Get instant answers now!

The distribution for X is ______.

Got questions? Get instant answers now!

Write the probability density function.

f ( x ) = 1 9 where x is between 0.5 and 9.5, inclusive.

Got questions? Get instant answers now!

Graph the probability distribution.

  1. Sketch the graph of the probability distribution.
    This is a blank graph template. The vertical and horizontal axes are unlabeled.
  2. Identify the following values:
    1. Lowest value for x ¯ : _______
    2. Highest value for x ¯ : _______
    3. Height of the rectangle: _______
    4. Label for x -axis (words): _______
    5. Label for y -axis (words): _______
Got questions? Get instant answers now!

Find the average age of the cars in the lot.

μ = 5

Got questions? Get instant answers now!

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.
    Blank graph with vertical and horizontal axes.
  2. Find the probability. P ( x <4) = _______
Got questions? Get instant answers now!

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.
    This is a blank graph template. The vertical and horizontal axes are unlabeled.
  2. Find the probability. P ( x <4| x <7.5) = _______
  1. Check student’s solution.
  2. 3.5 7
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Find the third quartile of ages of cars in the lot. This means you will have to find the value such that 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.
    Blank graph with vertical and horizontal axes.
  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
Got questions? Get instant answers now!

Questions & Answers

frequency destribution
the Reply
प्रायिकता सिध्दान्त पर आधारित प्रतिदर्श सिध्दान्त का विकास किसने किया?
Lokesh Reply
7.The following data give thenumber of car thefts that occurred in a city in the past 12 days. 63711438726915 Calculate therange, variance, and standard deviation.
Mitu Reply
express the confidence interval 81.4% ~8.5% in interval form
Xx Reply
a bad contain 3 red and 5 black balls another 4 red and 7 black balls, A ball is drawn from a bag selected at random, Find the probability that A is red?
Shazain Reply
The information is given as, 30% of customers shopping at SHOPNO will switch to DAILY SHOPPING every month on the other hand 40% of customers shopping at DAILY SHOPPING will switch to other every month. What is the probability that customers will switch from A to B for next two months?
sharmin Reply
Calculate correlation coefficient, where SP(xy) = 144; SS(x) = 739; SS(y) = 58. (2 Points)
Ashfat Reply
The information are given from a randomly selected sample of age of COVID-19 patients who have already survived. These information are collected from 200 persons. The summarized information are as, n= 20; ∑x = 490; s^2 = 40. Calculate 95% confident interval of mean age.
Ashfat
The mode of the density of power of signal is 3.5. Find the probability that the density of a random signal will be more than 2.5.
Ashfat
The average time needed to repair a mobile phone set is 2 hours. If a customer is in queue for half an hour, what is the probability that his set will be repaired within 1.6 hours?
Ashfat
A quality control specialist took a random sample of n = 10 pieces of gum and measured their thickness and found the mean 9 and variance 0.04. Do you think that the mean thickness of the spearmint gum it produces is 8.4
nazrul Reply
3. The following are the number of mails received in different days by different organizations: Days (x) : 23, 35, 38, 50, 34, 60, 41, 32, 53, 67. Number of mails (y) : 18, 40, 52, 45, 32, 55, 50, 48, 26, 25. i) Fit a regression line of y on x and test the significance of regression. ii) Estimate y
Atowar Reply
The number of problem creating computers of two laboratories are as follows: Number of computers: 48, 6, 10, 12, 30, 11, 49, 17, 10, 14, 38, 25, 15, 19, 40, 12. Number of computers: 12, 10, 26, 11, 42, 11, 13, 12, 18, 5, 14, 38. Are the two laboratories similar in respect of problem creating compute
Tamim Reply
Is the severity of the drug problem in high school the same for boys and girls? 85 boys and 70 girls were questioned and 34 of the boys and 14 of the girls admitted to having tried some sort of drug. What can be concluded at the 0.05 level?
Ashfat Reply
null rejected
Pratik
a quality control specialist took a random sample of n=10 pieces of gum and measured their thickness and found the mean 7.6 and standered deviation 0.10. Do you think that the mean thickness of the spearmint gum it produces is 7.5?
Shanto Reply
99. A one sample, one-tail t-test is conducted and the test statistic value is calculated to be 2.56. The degrees of freedom for the test are 10. Which of the following conclusions for the test would be correct? a
Niaz Reply
A one sample, one-tail t-test is conducted and the test statistic value is calculated to be 2.56. The degrees of freedom for the test are 10. Which of the following conclusions for the test would be correct?
Niaz
what is null Hypothesis
Niaz
what is null Hypothesis
Niaz
when median is greater than mode?
Hafiza Reply
hello
Amaano
is this app useful
Worthy
little bit 😭
G-
oh
Worthy
when tail is positive
Jungjoon
define hypothesis
Worthy
I'm struggling to type it's on my laptop...statistics
Yoliswa
types of averages .mean median mode quarantiles MCQ question
Rupa Reply

Get the best Introductory statistics course in your pocket!





Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introductory statistics' conversation and receive update notifications?

Ask