Business Analytics Final SG

If A and B are independent events, P(A) = .2, and P(B) = .7, determine P(A∪B).

0.14

0!

1

The probability of failure, q, is 1 - ____.

1 - p

How Large for the Sample?

1. If the sample size is at least 30, then for most sampled populations, the sampling distribution of sample means is approximately normal 2. If the population is normal, then the sampling distribution of x bar is normal regardless of the sample size.

How to Find Q1, Q2, and Q3:

1. Order your data set from lowest to highest values 2. Find the median (Q2) 3. At Q2, split the ordered data set into two halves 4. Q1 is the median of the lower half of the data 5. Q3 is the median of the upper half of the data

Properties of f(x): f(x) is a continuous function such that...

1. f(x) > or equal to 0 for all x 2. The total area under the curve of f(x) is equal to 1. 3. Area under curve = total probability!

1st Quartile (Q1) is the ______ percentile 2nd Quartile (median) is the _____ percentile 3rd Quartile (Q3) is the _____ percentile

25th, 50th, 75th

If n = 15 and p = .4, then the standard deviation of the binomial distribution is _______.

3.6

What are the 3 important percentages in normal distributions?

68.26% 95.44% 99.73%

standard normal distribution

A normal distribution with a mean of 0 and a standard deviation of 1.

Population vs. sample

A population is a set of all elements about which we wish to draw conclusions from, while a sample is a subset of the elements of a population.

Let A, B and C be events and assume the following probabilities: P(A) = 0.2, P(B) = 0.3, P(C) = 0.5, P(A ∩ B) =0.06, P(A ∩ C) = 0, P(B ∩ C) = 0.5. Which two of the three events are mutually exclusive?

A, C

A newly married couple plans to have two children & would like to know all possible outcomes: What is the probability of at least one girl?

All possible outcomes: BB, BG, GB, GG Want to know probabilities (assuming all equal): P(BB) = P(BG) = P(GB) = P (GG) = ¼ = 1/n = 3/4

________ says that if the sample size is sufficiently large, then the sample means are approximately normally distributed.

Central Limit Theorem

Car mileage and temperature are examples of ________ random variables.

Continuous

Cross-sectional data vs. Time series data

Cross sectional data is collected at the same or approximate point in time, while time series data is collected over different time periods

Range

Largest measurement - smallest measurement

Chebyshev's Theorem

Let μ and σ be a population's mean and standard deviation, then for any value k >1.

Coefficient of Variation

Measures the size of the standard deviation relative to the size of the mean (Standard deviation/ mean) x 100%

Outliers

Outliers are measurements that are very different from other measurements

Addition

P(A ∪ B) = P(A) + P(B)

Probability of an event

P(E) = number of favorable outcomes/total number of possible outcomes

If x = the number of occurrences in a specified interval, then x is a _____ random variable

Poisson

Find Q1, Q2, and Q3: 2, 4, 6, 8, 12, 13, 16, 18, 20

Q1: 5 Q2: 12 Q3: 17

The interquartile range (IQR) is

Q3-Q1

Which two of the following variables is discrete? A. Shoe size B. Waist measurement C. Foot length D. Dress size

Shoe size & dress size

Variance

The average of the squared deviations of the different values of the random variable from the expected value.

Complement

The complement (A') of an event A is the set of all sample space outcomes not in A.

Conditional Probability

The probability of an event A, given that the event B has occurred, is called the condition probability of A given B: P(A|B) = P(A∩B) / P(B)

Central Limit Theorem

The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution. (sample size of at least 30)

T or F: In binomial distributions, Probability of success, p, is constant from trial to trial

True

T or F: Point Estimation is a form of statistical inference.

True

T or F: The highest point of the curve is over the mean

True

T or F: an area under a continuous probability distribution is a probability.

True

Skewed to the left

the left tail of the histogram is longer than the right tail.

Mode

the most frequently occurring score(s) in a distribution

Which of the following is not a discrete random variable? A) the number of times a light changes red in a 10-minute cycle B) the number of minutes required to run 1 mile C) the number of defects in a sample selected from a population of 100 products D) the number of criminals found in a five-mile radius of a neighborhood

the number of minutes required to run 1 mile

Frequency

the number of times a value, value range or event occurs in gathered data.

Skewed to the right

the right tail of the histogram is longer than the left tail

Sample space

the set of all possible outcomes of a probability experiment

Standard Deviation

the square root of the variance

Median

the value of the MIDDLE point of the ordered measurements

Bar chart

vertical/horizontal rectangle represents the frequency for each category. Only good for categorical/qualitative variables!!

A standard normal distribution has a mean of ________ and standard deviation of ________.

zero ; one

Which of the following is a categorical variable?

Whether a person has a traffic violation

Is sample mean quantitative?

YES

Normal distribution

a bell-shaped curve, describing the spread of a characteristic throughout a population

Percent Frequency

a display of data that indicates the percentage of observations for each data point or grouping of data points.

Non-Probability Sampling

a method of selecting units from a population using a subjective (i.e. non-random) method.

Population parameter

a number calculated from all the population measurements that describe some aspect of the population

Sample statistic

a number calculated using the sample measurements that describe some aspect of the sample

Histogram

a picture of the frequency distribution. QUANTITATIVE Step 1: Find # of classes (2^k > N) Step 2: Find class length ((largest value-smallest value) / # of classes) Step 3: Form non overlapping classes of equal width Step 4: Graph histogram

sampling distribution of the sample mean

a probability distribution of all possible sample means of a given sample size

Probability Distribution

a table, graph, or formula that gives the probability associated with each possible value that the variable can assume

Ordinal

a variable where there is a meaningful ordering, or ranking, of the categories EX: teaching effectiveness, class (freshman/junior/etc)

Nominative

a variable where there is no meaningful ordering, or ranking, of the categories. EX: gender, car color

Interval

all of the characteristics of ordinal plus measurements are on numerical scale with an arbitrary zero point, can only meaningfully compare values by the interval between them EX: temperature

Ratio

all the characteristics of interval plus measurements are on a numerical scale with a meaningful zero point, values can be compared by their interval and ratios (most quantitative variables) EX: earnings, profit, loss, age, height, distance

Multiple choice questions

allow more than two responses, usually analyzed with averages

All probability distributions are characterized by _________ and a ________.

an expected value (mean) variance (standard deviation squared)

Variable

any characteristic of an element EX: model design, lot type, list price, etc

subjective method

assessment based on experience, expertise or intuition

Relative frequency method

assigning probabilities based on experimentation or historical data (ex: estimate the probability that a randomly selected consumer prefers Coca-Cola to all other soft drinks.)

Mean

average

Intersection

belong to both A and B.

Union

belong to either A or B or both.

If x is the total number of successes in n trials of a binomial experiment, then x is a ______ random variable.

binomial

In the uniform distribution, _____ is the smallest value and ____ is the largest.

c ; d

Convenience sampling

choosing individuals who are easiest to reach

Pie chart

circle divided into slices where the size of each slice represents its relative frequency or percent frequency.

Contingency table

classify data on two dimensions

Dichotomous questions

clearly stated, easy to answer/analyze, limited info, "yes or no"

What are the two types of random variable?

discrete (countable) and continuous (decimals)

Multistage cluster sampling

divide population into clusters and then randomly select clusters to sample.

Stratified random sampling

divide population into non-overlapping groups (strata) and then select a random sample from each strata.

Data

facts and figures from which conclusions can be drawn

Empirical Rule

if a population has a mean (mu) and a standard deviation and is described by a normal curve then, 68.2% of the population measurements lie within one standard deviation of the mean. 95.44% lie within two standard deviations of the mean 99.73% lie within three standard deviations of the mean

Voluntary sampling

individuals are self-selected by responding to an incentive

The ________ of two events A and B is the event that consists of the sample space outcomes belonging to both event A and event B.

intersection

Qualitative (categorical)

labels or names used to identify an attribute of each element EX: name of customer, email address, address of house

Even if the population of individual items is not normal, the sampling distribution is approximately normal if the sample is ______ enough (Central Limit Theorem)

large

Systematic sampling

list population, select random starting point, sample each nth element.

Relationship among mean, median, and mode

mean > median > mode

The Central Limit Theorem states that as the sample size increases, the distribution of the sample ________ approaches the normal distribution.

means

Probability

measure of the chance that an experimental outcome will occur when an experiment is carried out.

If a population distribution is skewed to the right, then, given a random sample from that population, one would expect that the

median would be less than the mean/average

Open ended questions

most honest and complete information, cannot be readily summarized

If two events are independent, we can ________ their probabilities to determine the intersection probability.

multiply

If the population of individual items is normal, then the population of all sample means is also ___________.

normal

Sample Size

number of elements

Quantitative

numbers represent qualities EX: weight of package, shipping costs, package size

What is a random variable?

quantitative value that represents the outcomes from an experiment.

Judgment sampling

samples in which a person who is extremely knowledgeable about the population selects population elements he or she feels are most representative.

Probability Sampling

sampling where we know the chance that each element in the population will be included in the sample (cluster, systematic, stratified)

Dot Plots

see the overall pattern of the data by grouping the data into classes. Best for small to moderately sized data distributions.

Event

set of sample space outcomes

Relative Frequency

summarizes the proportion of items in each class. For each class, divide the frequency of the class by the total # of observations. Multiply by 100 to obtain the percent frequency

Data Set

the data that is collected for a particular study

Outcomes

the experimental outcomes in the sample space