Business Statistics Chapter 7

which of the following is true regarding continuous and discrete random variables?

continuous random variables usually result from measuring

How would you describe the shape of a normal distribution?

it is "bell shaped" and symmetrical, with the center at μ

what are the upper and lower limits of the random variable for the normal distribution?

no limits. it is asymptotic in both directions

normal probability distribution

- It is bell-shaped and has a single peak at the center of the distribution. The arithmetic mean, median, and mode are equal and located in the center of the distribution. The total area under the curve is 1.00. Half the area under the normal curve is to the right of this center point, and the other half is to the left of it. -It is symmetrical about the mean. If we cut the normal curve vertically at the center value, the shapes of the curves will be mirror images. Also, the area of each half is 0.5 -It falls off smoothly in either direction from the central value. That is, the distribution is asymptotic: The curve gets closer and closer to the X-axis but never actually touches it. To put it another way, the tails of the curve extend indefinitely in both directions. -The location of a normal distribution is determined by the mean, μ. The dispersion or spread of the distribution is determined by the standard deviation, σ.

Suppose you have two normal distributions: one has mean = 16 and sd=2 and the other has mean=20 and sd=3. which of the following statements are true? select all that apply.

- the distribution with mean=20 is to the right of the other -the distribution with sd=2 is more peaked

What are the defining characteristics of the standard normal distribution? choose all that apply

- the sd= 1 -the mean is zero

Which of the following is an example of a continuous random variable?

Square footage of a house

a college entrance exam has scores that follow a normal distribution with a mean of 540 and standard deviation of 60. about 68 percent of students taking the exam will score within that range?

480 to 600 they are 1sd away from the mean -> 95%

z VALUE

The signed distance between a selected value, designated x, and the mean, μ, divided by the standard deviation, σ So, a z value is the distance from the mean, measured in units of the standard deviation.

where does the curve of the normal distribution touch the x-axis?

it never touches, but approaches asymptotically in both directions

Which of the following are true because the normal probability distribution is symmetric about the mean? select all that apply

the area on each side of the mean equal one-half the shape to the left of the mean is a mirror image of the shape to the right of the mean

what determines the width and location of the normal distribution?

the mean (μ) determines the location and the standard deviation (σ) determines the width. they are the only parts of the equation that are not fixed constants.

The average weight of a Honeycrisp apple is 110 grams and the standard deviation is 20 grams. what is the z-value for an apple that weighs 140 grams?

z = 1.5

a graduate school entrance exam has scores that are normally distributed with a mean of 560 and a standard deviation of 90. what percentage of examinees will score less than 400?

0.0375 z=(400-560)/90 =-1.78; find that on ztable and get 0.0375

a graduate school entrance exam has scores that are normally distributed with a mean of 560 and a standard deviation of 90. what percentage of examinees will score more than 700?

0.0594 z=(700-560)/90=1.56. z-table (0.5000-0.4406)

a graduate school entrance exam has scores that are normally distributed with a mean of 560 and a standard deviation of 90. what percentage of examinees will score between 400 and 500?

0.2139 z=(400-500)/90=1.78 z=(500-560)/90=0.67 use z-table (0.4625-0.2486)

scores on a graduate school entrance exam follow a normal distribution with a mean of 560 and a standard deviation of 90. what is the probability that a randomly chosen test taker will score between 490 and 560?

0.2823 z=(490-560)/90=0.78. use excel function normdist or a z-table (0.3133)

applicants for graduate school take a test that has scores which are normally distributed with a mean of 560 and a standard deviation of 90. what is the probability that a randomly chosen applicant will score between 550 and 650?

0.3851 z=(550-560)/90=0.11; z=(650-560)/90=1.00 use z-table with (0.3413+0.0438)

applicants for graduate school take a test that has scores which are normally distributed with a man of 560 and a standard deviation of 90. what is the probability that randomly chosen applicant will score between 500 and 600?

0.4186 z=(500-560)/90=-.67; z=(600-560)/90=0.44. z table to do (.2486+.17)

the lifetime of a particular brand of 60w light bulbs follows a normal distribution with a mean of 330 hours and a standard deviation of 11 hours. what is the probability that a randomly chosen 60w bulb will last between 325 and 345 hours?

0.5867 z=(325-330)/11=0.45; z=(345-330)/11=1.36 on z table use (0.1736+0.4131)

the lifetime of a particular type of AA batter follows a normal distribution with a mean of 200 and standard deviation of 10. about all (99.7%) of the batteries will last within what range?

170-230 3sd away from mean -> 99.7% 3sd=10*3 = 30 200-30=170 200+30=230

The empirical rule

68%, 95%, 99.7% if a random variable is normally distributed, then: 1. Approximately 68% of the observations will lie within plus and minus one standard deviation of the mean. 2. About 95% of the observations will lie within plus and minus two standard deviations of the mean. 3. Practically all, or 99.7% of the observations, will lie within plus and minus three standard deviations of the mean. We can use the Empirical Rule to answer these questions. 1. We can expect about 68% of the batteries to last between 17.8 and 20.2 hours, found by 19.0 ± 1(1.2) hours. 2. We can expect about 95% of the batteries to last between 16.6 and 21.4 hours, found by 19.0 ± 2(1.2) hours. 3. We can expect about 99%, or practically all, of the batteries to last between 15.4 and 22.6 hours, found by 19.0 ± 3(1.2) hours.

an i.q. test has scores that follow a normal distribution with a mean of 105 and standard deviation of 10. about 95% of persons taking the test will score whithin what range?

85-125

standard normal probability distribution

it is unique because it has a mean of 0 and a standard deviation of 1 Any normal probability distribution can be converted into a standard normal probability distribution by subtracting the mean from each observation and dividing this difference by the standard deviation. The results are called z values or z scores

which of the following is a result of the normal probability distribution being bell-shaped?

the mean, median, and mode are at the center

which of the following is symmetrical about the mean?

the normal curve

which of the following is NOT true regarding continuous random variables?

the number of successes must be less than or equal to the number of trials. this refers is characteristic of the binomial distribution, which is discrete

which one of the following is true?

the variance of the standard normal distribution is 1. since sd=1, sd^2 = 1

Which of the following is NOT true about z-values?

z-values are measured in the same units as x. z-values do not have units

The average height for American male is 71 inches and the standard deviation is 3 inches. What is the z-value for the height of 68 inches?

z= -1

STANDARD NORMAL VALUE

z=(x − μ) /σ where: x is the value of any particular observation or measurement. μ is the mean of the distribution. σ is the standard deviation of the distribution.

the average score on an IQ test is 110 and the standard deviation is 10 grams. what is the z-value for a score of 130?

z=2 (130-110)/10 z-u/sd