Stats

How would you solve (k-factorial)? |4| |2|

4!/2!(4-2)! = 4!/2!2! = 4x3x2!/2!2! one of the "2!" would cancel, because there is one on both the top and bottom on the division line, leaving you with: 4x3/2 = 12/2 = 6

Assume the variable under consideration has a density curve. The percentage of all possible observations of the variable that lie to the right of 5 equals the area under its density curve to right of _____, expressed as a percentage.

5

The area under a particular normal curve between 10 and 15 is 0.9147. A normally distributed variable has the same mean and standard deviation as the parameters for this normal curve. What percentage of all possible observations of the variable lie between 10 and 15?

91.47% For a normally distributed variable, the percentage of all possible observations that lie within any specified range equals the corresponding area under its associated normal curve, expressed as a percentage. The percentage is 100 times the area between 10 and 15 that lies under the corresponding normal curve.

Median: a. The middle value of a data set in its ordered list b. The lowest value of a data set in its ordered list

A

Qualitative variable

A non numerically valued variable- can't do arithmetic with

Quantitative variable

A numerically valued variable- can do arithmetic with

Discrete variable

A quantitative variable whose possible values can be listed- usually involved a count of something

Continuous variable

A quantitative variable whose possible values form some interval of numbers- usually involves a measurement of something

What is a density curve?

A smooth curve that identifies the shape of distribution

Find the z-scores that separate the middle 42% of the distribution from the area in the tails of the SND.

Answer: -0.55 and 0.55 Determine left tail: (1-0.42)/2 = 0.29 Z-score of the closest area is -0.55 Determine right tail: 1 - 0.29 = 0.71 Z-score of the closest area is 0.55

Find the value of z0.34

Answer: 0.41 1 - 0.34 = 0.66 Area closest on the table is 0.6591 Corresponding z-score is 0.41

Assume that adults have IQ scores that are normally distributed with a mean of u = 100 and a standard deviation o = 15. Determine the quartiles for adults IQs.

Answer: 25% of adults have an IQ below 89.95 50% of adults have an IQ below 100 75% of adults have an IQ below 110.05 Area associated w/ first quartile is 0.25, find corresponding z-score. z = -0.67 x-value is determined by x = u + z x(multi.) o x = 100+(-0.67)x15 = 89.95 Area associated w/ second quartile is 0.5, find corresponding z-score. z = 0.00 x = 100+(0.00)x15 = 100 Area associated w/ third quartile is 0.75 z = 0.67 x = 100+(0.67)x15 = 110.05

A variable is normally distributed with mean 18 and standard deviation 5. Find the percentage of all possible values of the variable that lie between 7 and 11.

Answer: 6.69% z = (x-u)/o z = (7-18)/5 = -2.2 z = (111-18)/5 = -1.4 Use a standard normal table and... Area to the left of -2.2 = 0.0139 Area to the left of -1.4 = 0.0808 Subtract the two areas 0.0808 - 0.0139 = 0.0669 Multiply by 100%

Assume that adults have IQ scores that are normally distributed with a mean of u = 100 and a standard deviation o = 15. Find the percentage of adults that have an IQ between 84 and 116.

Answer: 71.54% z = (x-u)/o z = (84-100)/15 = -1.07 z = (116-100)/15 = 1.07 Area-left z = -1.07 = 0.1423 Area-left z = 1.07 = 0.8577 = 0.8577 - 0.1423 = 0.7152

Assume that adults have IQ scores that are normally distributed with a mean of u = 100 and a standard deviation o = 15. Find the percentage of adults that have an IQ exceeding 121.

Answer: 8.08% z = (x-u)/o z = (121-100)/15 = 1.4 Area- left = 0.9192 Area- right = 1-0.9192 = 0.0808 Remember, it asks for area EXCEEDING. Exceeding = right

Assume that adults have IQ scores that are normally distributed with a mean of u = 100 and a standard deviation o = 15. Obtain the 95th percentile for the IQ of adults

Answer: 95% of adults have an IQ below 124.68 Find the z-score such that the area to the left of it is 0.95. z = 1.645 x = 100+(1.645)x15 = 124.68

Determine the area under the SNC that lies between z = -1.77 and z = 0.

Area = (Atoleft z = 0) - (Atoleft z = -1.77) = 0.5000 - 0.0384 = 0.4616

Determine the area under the SNC that lies between z = -1.29 and z = 1.75

Area = (Atoleft z = 1.75) - (Atoleft z = -1.29) = 0.9599 - 0.0985 = 0.8614

Mean: a. The product of the observations divided by the number of observations b. Arithmetic average of a data set

B

Mode: a. The least frequently occurring value in a data set b. The most frequently occurring value in a data set

B

A variable is approximately normally distributed. If you draw a histogram of the distribution of the variable, roughly what shape will it have?

Bell shape

With which normal distribution is the standard normal curve associated? a. The normal distribution with a mean of x and a standard deviation of s b. The normal distribution with a mean of 1 and a standard deviation of 0 c. The normal distribution with a mean of 0 and a standard deviation of 1 d. The normal distribution with a mean of u and a standard deviation of o

C

Find the z-score such that the area under the standard normal curve to the left is 0.31.

Corresponding z-score is -0.50 The z-score must be less than zero because 0.31 is less than 0.5. Refer to a standard normal distribution table and look in the body of the table for an area closest to 0.31. The closest area is 0.3085 Find the z-score of the closest area, that is your answer

Single-value, limit, or cutpoint? The fuel tank capacity of all new SUV models, to the nearest tenth of a gallon.

Cutpoint grouping

Potential outliers are: a. Observations that lie in the middle of the lower limit b. Observations that lie outside of the upper limit c. Observations that lie beneath the lower and upper limits d. Observations that lie below the lower limit or above the upper limit

D

Characteristic of the graph of a probability density function: The total area under the graph of the equation over all possible values of the random variable must ______.

Equal 1, fam.

T/F A density curve is always on or above the vertical axis. Also, is this a property of a density curve or not?

False- A density curve is always on or above the horizontal axis This is a property of a density curve

T/F A frequency distribution of qualitative is a listing of the distinct values and their relatives frequencies. It is useful because it provides a table of the values of the observations and (relatively) how often they occur. It is also useful for comparing two data sets.

False- It is a listing of the distinct values and their frequencies. It is useful because it provides a table of the values of the observations and how often they occur.

The phrase "all possible observations of the variable exceed 21" corresponds to what are under the density curve?

The area to the right of 21

A curve has an area of 0.615 to the left of 65 and an area of 0.485 to the right of 65. Could this curve be a density curve for some variable? Why?

The curve could not be a density curve because the total are under the curve is greater than 1.

The second quartile:

The median of the entire ordered data set

The third quartile:

The median of the part of the entire ordered data set that lies at or above the actual median of the data set

The first quartile:

The median of the part of the entire ordered data set that lies at or below the actual median of the data set

Five-number summary is...

The minimum of the data set, Q1, Q2, Q3, and the maximum of the data set

Determine whether the random variable is continuous or discrete. State the possible values: The amount of snowfall.

The random variable is continuous, possible values being s is less than or equal to 0.

Determine whether the random variable is continuous or discrete. State the possible values: The number of people with blood type A in a random sample of 41 people.

The random variable is discrete, possible values being x = 0, 1, 2,... 41.

Which normal distribution has a wider spread: one with a mean of 8 and standard deviation of 9 or one with a mean of 9 and a standard deviation of 8? Why?

The wider spread would lay within the mean of 8 and standard deviation of 9. Because the larger the standard deviation, the flatter and more spread out the distribution is to determine which normal distribution has the wider spread.

Assume that the variable under consideration has a density curve. It is given that 28.6% of all possible observations of the variable less than 18. (a) Determine the area under the density curve that lies to the left of 18. (b) Determine the area under the density curve that lies to the right of 18.

This problem is asking for the AREA specifically, not the percentage, so: Divide the percentage of all possible observations of the variable that are less than 18 by 100% 28.6%/100% = 0.286 (a) Therefore, the area under the density curve that lies to the left of 18 is 0.286. (b) The total area under the density curve equals 1. The total area under the density curve is equal to area to the left of 18 plus the area to the right of 18. The area to the left of 18 is 0.286. So just subtract this value from 1 to find the area to right of 18. 1 - 0.286 = 0.714

Determine the area under the standard normal curve that lies between z = -0.17 and z = 0.17

To solve this, simply subtract the area to the left of z = -0.17 from the area to the left of z= 0.17. Area = (Atoleft z = 0.17) - (Atoleft z = -0.17) = 0.5675 - 0.4325 = 0.1350

k-factorial is denoted by k! and looks like...

k! = k(k-1)...2x1. Examples: 3! = 3x2x1 = 6 4! = 4x3x2x1 = 24 6! = 6x5x4x3x2x1 = 720

If you were asked to obtain the standardized version, z of x, what formula would you use?

z = (normally distributed variable or "x" - mean)/standard deviation AKA z = (x-u)/o

Use a standard normal distribution table to obtain the z-score that has an area of 0.67 to its right.

z = -0.44 Subtract 0.67 from 1 to find the are to the left of the z-score 1 - 0.67 = 0.33 Now, look at the body of the SNDT for the area closest to 0.33. The corresponding z-score if your answer. *If two area entries are equally close to the one desired, take the mean of the two corresponding z-scores as your answer.

The _______ is preferred when the mean is used as a measure of center.

Standard deviation

Single-value grouping:

Suitable for discrete data in which there are only small number of distinct values

With limit grouping, the "middle" of a class is the _____ average of the two ______ of the class; it is called the _____.

-average -class limits -class mark

With cutpoint grouping, the "middle" of a class is the _____ of the two _____ of the class; it is called the ______.

-average -cutpoints -class midpoint

According to a standard normal table, the area under the standard normal curve to the left of 2.07 is 0.9808. Determine the area under the SNC to the left of -2.07.

To fine the area to the right of z = -2.07, use the given area to the left of z = 2.07 and the fact that the total area under the standard normal curve is equal to 1. 1 - 0.9808 = 0.0192 Therefore, the area under the SNC to the left of z = -2.07 is 0.0192.

T/F IQR gives the range of the middle 50% of the observations

True

Characteristic of the graph of a probability density function: The height of the graph of the equation must be ________ for all possible values of the random variable.

Greater than or equal to 0.

What does the empirical rule day? For any data set ______ distribution the following is true. Approximately ____ of the observations lie within one standard deviation to either side of the mean. Approximately _____ of the observations lie within two standard deviations to either side of the mean. Approximately _____ of the observations lie within three standard deviations to either side of the mean.

Having a roughly bell-shaped 68% 95% 99.7%

A normal curve associated with a normal distribution is close to the ______ axis outside the range from u +/- ____ to u +/- _____.

Horizontal The formula is u - 3o to u + 3o

The ______ is preferred when the median is used as a measure of center.

Interquartile range

Single-value, limit, or cutpoint? The gas mileages, rounded to the nearest number of miles per gallon, of all new car models.

Limit grouping

Three most important measures of center

Mean, median, and mode

Distributions of variables such as height, test scores, and sale prices of houses in a given area all have roughly the shape of a ______. Also, how is a ______ shaped?

Normal curve A normal curve is bell shaped.

How to calculate situation P(A or B)

P(A)+P(B)-P(A&B)

Cutpoint grouping:

Particularly useful when the data set are continuous and are expressed with decimals

Lower limit (potential outliers) formula:

Q1-(1.5xIQR)- any observation(s) in the data set that fall below the number given by this formula are PI's

Upper limit (potential outliers) formula:

Q3+(1.5xIQR)- any observation(s) in the data set that fall above the number given by this formula are PI's

Interquartile range formula

Q3-Q1

Single-value, limit, or cutpoint? The number of automobiles per family.

Single-value grouping

T/F For a variable with a density curve, the percentage of all possible observations of the variable that lie within any specified range equals (at least approximately) the corresponding area under the density curve, expressed as a percentage.

True

T/F The corresponding mean and standard deviation of a normal curve are known as the "parameters".

True

T/F The mean of a normal distribution has no effect on its shape

True- the shape of a NB is determined completely by its standard deviation.

Limit grouping:

Useful when the data are expressed as whole numbers and there are too many distinct values to employ single-value grouping