Quantitative Survey Methods Ch 6

Assume that the cost of a wedding is normally distributed with a mean of $29,858 and a standard deviation of $5,600. For a wedding to be among the 5% most expensive, how much would it have to cost (to the nearest whole number)?

$39,069

The mean cost of domestic airfares in the United States rose to an all-time high of 385 per ticket. Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of 110. What is the cost for the 3% highest domestic airfares? (rounded to nearest dollar)

$592 or more

The time needed to complete a final examination in a particular college course is normally distributed with a mean of 80 minutes and a standard deviation of 10 minutes. Answer the following questions. a.What is the probability of completing the exam in one hour or less (to 4 decimals)?

.0228

Assume that the cost of a wedding is normally distributed with a mean of $29,858 and a standard deviation of $5,600. What is the probability that a wedding costs less than $20,000 (to 4 decimals)?

.0392

The mean cost of domestic airfares in the United States rose to an all-time high of 385 per ticket (Bureau of Transportation Statistics website, November , ). Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of 110. What is the probability that a domestic airfare is $550 or more (to 4 decimals)?

.0668

The mean cost of domestic airfares in the United States rose to an all-time high of 385 per ticket. Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of 110. What is the probability that a domestic airfare is $250 or less (to 4 decimals)?

.1099

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night. Assume that room rates are normally distributed with a standard deviation of $55. What is the probability that a hotel room costs less than $140 per night (to 4 decimals)?

.1223

The port of South Louisiana, located along 54 miles of the Mississippi River between New Orleans and Baton Rouge, is the largest bulk cargo port in the world. The U.S. Army Corps of Engineers reports that the port handles a mean of 4.5 million tons of cargo per week. Assume that the number of tons of cargo handled per week is normally distributed with a standard deviation of .82 million tons. What is the probability that the port handles between 3 million and 4 million tons of cargo per week (to 4 decimals)?

.2373

Delta Airlines quotes a flight time of hours, minutes for its flights from Cincinnati to Tampa. Suppose we believe that actual flight times are uniformly distributed between hours and hours, minutes. What is the probability that the flight will be more than 10 minutes late (to 2 decimals)?

.25

The time needed to complete a final examination in a particular college course is normally distributed with a mean of 80 minutes and a standard deviation of 10 minutes. What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes (to 4 decimals)?

.2858

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night. Assume that room rates are normally distributed with a standard deviation of $55. a. What is the probability that a hotel room costs $225 or more per night (to 4 decimals)?

.3513

Assume that the cost of a wedding is normally distributed with a mean of $29,858 and a standard deviation of $5,600. What is the probability that a wedding costs between $20,000 and $30,000 (to 4 decimals)?

.4709

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night. Assume that room rates are normally distributed with a standard deviation of $55. What is the probability that a hotel room costs between $200 and $300 per night (to 4 decimals)?

.4885

Delta Airlines quotes a flight time of hours, minutes for its flights from Cincinnati to Tampa. Suppose we believe that actual flight times are uniformly distributed between hours and hours, minutes. What is the probability that the flight will be no more than 5 minutes late (to 2 decimals)?

.50

The mean cost of domestic airfares in the United States rose to an all-time high of 385 per ticket. Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of 110. What if the probability that a domestic airfare is between $300 and $500 (to 4 decimals)?

.6323

A random variable is normally distributed with a mean of 50 and a standard deviation of 5. What is the probability that the random variable will assume a value between and (to 4 decimals)?

.6827

The port of South Louisiana, located along 54 miles of the Mississippi River between New Orleans and Baton Rouge, is the largest bulk cargo port in the world. The U.S. Army Corps of Engineers reports that the port handles a mean of 4.5 million tons of cargo per week. Assume that the number of tons of cargo handled per week is normally distributed with a standard deviation of .82 million tons. What is the probability that the port handles less than 5 million tons of cargo per week (to 4 decimals)?

.7290

A random variable is normally distributed with a mean of 50 and a standard deviation of 5. What is the probability that the random variable will assume a value between and (to 4 decimals)?

.9545

The port of South Louisiana, located along 54 miles of the Mississippi River between New Orleans and Baton Rouge, is the largest bulk cargo port in the world. The U.S. Army Corps of Engineers reports that the port handles a mean of 4.5 million tons of cargo per week. Assume that the number of tons of cargo handled per week is normally distributed with a standard deviation of .82 million tons. What is the probability that the port handles 3 or more million tons of cargo per week (to 4 decimals)?

.9663

This property holds for all continuous probability distributions and is the analog of the condition that the sum of the probabilities must equal ___ for a discrete probability function.

1

The time needed to complete a final examination in a particular college course is normally distributed with a mean of 80 minutes and a standard deviation of 10 minutes. Assume that the class has students and that the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time (to nearest whole number)?

10

Delta Airlines quotes a flight time of hours, minutes for its flights from Cincinnati to Tampa. Suppose we believe that actual flight times are uniformly distributed between hours and hours, minutes. What is the expected flight time, in minutes?

130

A person must score in the upper 2% of the population on an IQ test to qualify for membership in Mensa, the international high-IQ society. There are Mensa members in countries throughout the world (Mensa International website, January , ). If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what score must a person have to qualify for Mensa (to whole number)?

131

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night. Assume that room rates are normally distributed with a standard deviation of $55. What is the cost of the 20% most expensive hotel rooms in New York City? Round up to the next dollar.

251 or more

The port of South Louisiana, located along 54 miles of the Mississippi River between New Orleans and Baton Rouge, is the largest bulk cargo port in the world. The U.S. Army Corps of Engineers reports that the port handles a mean of 4.5 million tons of cargo per week. Assume that the number of tons of cargo handled per week is normally distributed with a standard deviation of .82 million tons. Assume that 85% of the time the port can handle the weekly cargo volume without extending operating hours. What is the number of tons of cargo per week that will require the port to extend its operating hours (to 2 decimals)?

5.35

Normal Probability Distribution Characteristics: (the basis for the empirical rule) _____ % of values of a normal random variable are within +/-1 standard deviation of its mean. _____ % of values of a normal random variable are within +/-2 standard deviations of its mean. _____ % of values of a normal random variable are within +/-3 standard deviations of its mean.

68.3; 95.4; 99.7

________ ____ __________, a French mathematician, published The Doctrine of Chances in 1733. He derived the normal distribution.

Abraham de Moivre

Excel has two functions for computing cumulative probabilities and x values for any normal distribution: NORM._______ is used to compute the cumulative probability given an x value. NORM.____ is used to compute the x value given a cumulative probability.

DIST; INV

Excel has two functions for computing probabilities and z values for a standard normal probability distribution. NORM.S._______ function computes the cumulative probability given a z value. NORM.S.______ function computes the z value given a cumulative probability. "S" in the function names reminds us that these functions relate to the standard normal probability distribution.

DIST; INV

Uniform Probability Distribution Expected Value of x:

E(x) = (a+ b)/2

____________ _____________ ___________ Function: 𝑓(𝑥)=1/(𝜎sqrt2𝜋)𝑒−(𝑥−𝜇)^2/2𝜎^2 Where μ = mean 𝜎= standard deviation 𝜋= 3.14159 e= 2.71828

Normal Probability Density

Uniform Probability Distribution Variance of x:

Var(x) = (b-a)^2/12

Assume z is a standard normal random variable. What is the value of z if the area between -z and z is .754? a. 1.16 b. .123 c. 2.16 d. .377

a. 1.16

The weight of football players is normally distributed with a mean of 200 pounds and a standard deviation of 25 pounds. What is the minimum weight of the middle 95% of the players? a. 151 b. 196 c. 249 d. None of these answers are correct.

a. 151

What type of function defines the probability distribution of ANY continuous random variable? a. Probability density function b. Uniform distribution function only c. Both the normal distribution function and the uniform distribution function d. Normal distribution function only

a. Probability density function

A standard normal distribution is a normal distribution with _____. a. a mean of 0 and a standard deviation of 1 b. any mean and a standard deviation of 1 c. a mean of 1 and a standard deviation of 0 d. any mean and any standard deviation

a. a mean of 0 and a standard deviation of 1

In a standard normal distribution, the range of values of z is from _____. a. minus infinity to infinity b. 0 to 1 c. -3.09 to 3.09 d. -1 to 1

a. minus infinity to infinity

For a standard normal distribution, a negative value of z indicates _____. a. the z is to the left of the mean b. a mistake has been made in computations, because z is always positive c. the z is to the right of the mean d. the area corresponding to the z is negative

a. the z is to the left of the mean

The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 and a standard deviation of 5,000 miles. What is the probability that a randomly selected tire will have a life of at least 30,000 miles? a. .0228 b. .9772 c. .4772 d. None of these answers are correct.

b. .9772

For any continuous random variable, the probability that the random variable takes on exactly a specific value is _____. a. any value between 0 and 1 b. 0 c. 1 d. .50

b. 0

Which of the following is NOT a characteristic of the normal probability distribution? a. The random variable assumes a value within plus or minus three standard deviations of its mean 99.72% of the time. b. The graph of the curve is the shape of a rectangle. c. The mean is equal to the median, which is also equal to the mode. d. The total area under the curve is always equal to 1.

b. The graph of the curve is the shape of a rectangle.

The standard deviation of a standard normal distribution _____. a. can be any positive value b. is always equal to 1 c. can be any value d. is always equal to 0

b. is always equal to 1

For a continuous random variable x, the probability density function f(x) represents _____. a. the area under the curve at x b. the height of the function at x c. both the probability at a given value of x and the area under the curve at x d. the probability at a given value of x

b. the height of the function at x

The highest point of a normal curve occurs at _____. a. approximately three standard deviations to the right of the mean b. the mean c. one standard deviation to the right of the mean d. two standard deviations to the right of the mean

b. the mean

The assembly time for a product is uniformly distributed between 6 and 10 minutes. The probability of assembling the product between 7 and 9 minutes is _____. a. 1 b. 0 c. .50 d. .20

c. .50

For the standard normal probability distribution, the area to the left of the mean is _____. a. -.5 b. 1 c. 0.5 d. any value between 0 and 1

c. 0.5

Which of the following is NOT a characteristic of the normal probability distribution? a. The distribution is symmetrical. b. The mean of the distribution can be negative, zero, or positive. c. The standard deviation must be 1. d. The mean, median, and mode are equal.

c. The standard deviation must be 1.

For a uniform probability density function, the height of the function _____. a. is different for various values of x b. cannot be larger than1 c. is the same for each value of x d. decreases as x increases

c. is the same for each value of x

Larger values of the standard deviation result in a normal curve that is _____. a. narrower and more peaked b. shifted to the right c. wider and flatter d. shifted to the left

c. wider and flatter

A _____________ ___________ ____________ can assume any value in an interval on the real line or in a collection of intervals.

continuous random variable

The probability density function for a uniform distribution ranging between 2 and 6 is _____. a. undefined b. any positive value c. 4 d. .25

d. .25

The assembly time for a product is uniformly distributed between 6 and 10 minutes. The probability of assembling the product in 7 minutes or more is _____. a. 0 b. 1 c. .25 d. .75

d. .75

Given that z is a standard normal random variable, what is the value of z if the area to the right of z is .1401? a. 2.16 b. -1.08 c. .1401 d. 1.08

d. 1.08

Given that z is a standard normal random variable, what is the value of z if the area to the left of z is .9382? a. 2.1 b. 1.8 c. 1.77 d. 1.54

d. 1.54

The weight of items produced by a machine is normally distributed with a mean of 8 ounces and a standard deviation of 2 ounces. What percentage of items will weigh at least 11.7 ounces? a. 46.78% b. 96.78% c. 53.22% d. 3.22%

d. 3.22%

Assume z is a standard normal random variable. Then P(1.41 < z < 2.85) equals _____. a. .8285 b. .4772 c. .3413 d. None of these answers are correct.

d. None of these answers are correct.

If the mean of a normal distribution is negative, _____. a. the variance must also be negative b. the standard deviation must also be negative c. a mistake has been made in the computations, because the mean of a normal distribution cannot be negative d. None of these answers are correct.

d. None of these answers are correct.

The mean of a standard normal probability distribution _____. a. can be any value b. is always equal to 1 c. can be any value as long as it is positive d. None of these answers are correct.

d. None of these answers are correct.

The mean, median, and mode have the same value for which of the following probability distributions? a. exponential b. uniform c. Poisson d. normal

d. normal

For a __________ ___________ variable, the probability function f(x) provides the probability that the random variable assumes a particular value.

discrete random variable

Normal Probability Distribution Characteristics: The __________ can be any numerical value: negative, zero, or positive.

mean

Normal Probability Distribution Characteristics: The highest point on the normal curve is at the __________, which is also the median and mode.

mean

Normal Probability Distribution Characteristics: The entire family of normal probability distributions is defined by its ______ μ and its ____________ ____________ σ.

mean; standard deviation

The __________ _______________ _____________ is the most important distribution for describing a continuous random variable. It is widely used in statistical inference. It has been used in a wide variety of applications including: Heights of people Test scores Rainfall amounts Scientific measurements

normal probability distribution

The difference is that the _____________ _____________ function does not directly provide probabilities. However, the area under the graph of f(x) corresponding to a given interval does provide the probability that the continuous random variable x assumes a value in that interval.

probability density

The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the _________ ________________ _________ between x1 and x2.

probability density function

Normal Probability Distribution Characteristics: The ___________ ____________ determines the width of the curve: larger values result in wider, flatter curves.

standard deviation

Converting to Standard Normal Distribution: z= (𝑥−𝜇)/𝜎 We can think of z as a measure of the number of _________ ___________ x is from μ.

standard deviations

Normal Probability Distribution Characteristics: A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a __________ __________ ____________ distribution.

standard normal probability

t or f: It is not possible to talk about the probability of the random variable assuming a particular value (one single value). Instead, we talk about the probability of the random variable assuming a value within a given interval.

true

t or f: What we have in the real world is NOT a standard normal distributions, but just normal distributions.

true

Normal Probability Distribution Characteristics: Probabilities for the normal random variable are given by areas _________ the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right).

under

The ____________ ___________ _________ function is: f (x) = 1/(b-a) for a<=x<=b = 0 elsewhere where: a= smallest value the variable can assume b= largest value the variable can assume

uniform probability density

A random variable is ______________ distributed whenever the probability is proportional to the interval's length.

uniformly

Normal Probability Distribution Characteristics: The letter ____ is used to designate the standard normal random variable.

z

Normal Probability Distribution Characteristics: The distribution is symmetric; its skewness measure is __________.

zero