BNAD 276 Exam #4
The wider a confidence interval...
-the less precise the confidence interval is -precision is directly linked with the width of the confidence interval
What does the numerator in the Sharpe Ratio measure?
-measures the extra reward that investors receive for the added risk taken
How to calculate the average residual (deviation) score:
( ∑x ) / ( N ) = Avg Residual Score
Intersection of two events:
(Event A and Event B)
The higher the positive correlation between assets return in a portfolio the....
-Greater the risk in the portfolio
Type 11 error
Beta
What would be the Null hypothesis in a experiment done by a fisher trying to see if there is a difference in fish caught with type of bait used to catch the fish?
Null Hypothesis: There is no difference in type of bait used, and number of fish caught
What do u calculate when two populations are assumed to have the same variance
Pooled estimate of the common variance
A ____________ probability is calculated by drawing on personal and subjective judgment
Subjective Probability: is calculated by drawing on personal and subjective judgment.
If the population standard deviation is unknown, it can be estimated using what...?
"s"
Coefficient of regression:
"b"
Correlation Coefficient
- "r" -stands for how consistent the data is
Null hypothesis consists of these signs
- = - ≤ - ≥
If we reject the Null hypothesis when it is actually false we have committed.......what type of error is this?
- no error
Example of variance not being accounted for...
- subtract r^2(0.64) from 1 = 1 - 0.64 = .36 = 36% -relationship between moms and daughters heights r=0.8 = 1 - .64 = 0.36 = 36%
Positively skewed distribution..
- the Mode < Median < Mean -50%ile is the Mean of the distribution
Alternative hypothesis consists of these signs
- ≠ - < - >
A Negatively skewed distribution...
-50 %ile is equal to the Median -the Mean < Median < Mode
Law of Large Numbers:
-According to a famous Law of Large Numbers, the empirical probability approaches the classical probability if the experiment is run a very large number of times
Concerning mean difference with Matched-pairs sampling requires 2 conditions...?
-Both X1 and X2 are normally distributed -The sample size (n) has to be greater or equal to 30
Statements about P-value and Critical-value approaching to hypothesis testing..
-Both approaches/uses the same decision rule concerning when to reject Ho.
By saying Complement of A, what does this mean pertaining to probability...?
-Complement of A means probability of "not A"
What gives variability info?
-Correlation "r" gives variability info -Residuals gives variability info
Which of the 7 methodologies are most often graphed using scatter plots?
-Correlation methodologies -Simple and Muiltiple regression
A person who is Risk-Averse...
-Demands a reward for risk taking
What if your observed Correlation coefficient ( r ) is 0... what can u conclude..?
-Do not reject the Null hypothesis, there is no correlation between variables as the null predicted
Objective Probabilities:
-Empirical Probability: a relative frequency of occurrence -Classical Probability: logical analysis
A risk-neutral person has the choice to receive $1,000 if he does nothing or he can receive $5,000 if he can make a free throw.. what would he do?
-Go for the free throws because the prize value $5000 exceeds the risk-free cash prize of $1000 -Risk neutral people base decisions solely on the basis of expected values
What hypothesis tests may be performed?
-Left tailed test -Right tailed test -Two tailed test
Tdf distribution...?
-Like the z distribution, tdf distributions are bell-shaped around 0 with tails that keep getting closer to the horizontal axis but never touches it -Has slightly broader tails than the z distribution -As degrees of freedom increases, the tdf distribution becomes similar to the z distribtion -If degrees of freedom is infinite, then z & tdf distributions would be identical (exactly the same)
Higher the positive correlation of assets in a portfolio...
-Means a greater risk of the portfolio
The p-value is calculated assuming the...
-Null hypothesis is true
Which of the 7 methodologies are most often graphed using line or bar graphs?
-One way ANOVA -Two way ANOVA
When do we find a point estimate?
-Population variances have to be assumed equal to use a pooled estimate
Sharpe Ratio: "Reward-to-Varability ratio"
-Sharpe ratio is a ratio calculated by dividing the difference of the mean return from the risk-free rate by the asset's deviation -the greater the Sharpe ratio, the better the investment compensates its investors for risk
Which of the 7 methodologies are most often graphed using bar graphs?
-T-tests -One way Anova -Two way Anova, only sometimes, most often uses line graph
When determining the value of t alpha,degrees of freedom you need what 2 things...?
-The Sample size ( n ) -Alpha = ( 1 - Confidence interval ) -Degrees of Freedom = ( n - 1 )
How does the variance of a distribution effects the likelihood of rejecting the null hypothesis?
-The larger the variance, the harder it is to reject the null hypothesis than a smaller variance (effect size has to be bigger)
Confidence Interval are less precise when...
-The wider the distribution, the less precise the estimate is -When variability (standard deviation) gets greater, the less precise the estimate -If the sample size of the population ( n ) is to small, then your estimate will be less precise
One-tailed Test:
-Use when trying to determine if your prediction is more/less than the mean -Like a right tailed test, can only be rejected on one side of the mean Ex. Ho: u ≤ uo verus. Ha: u > uo
For the Regression Line to be best at predicting y values from x values we would hope the standard error of the estimate to be big or small?
-We would want the standard error of the estimate to be as small as possible for the best predictions -We want our error (residuals) to be small to get the most precise results
The smaller the population size ( n ) the...
-Wider the Confidence interval
The larger the population standard deviation the ___________ the Confidence interval...?
-Wider the confidence interval
What variable are you predicting?
-You are predicting whatever the y-axis variable is
In most applications, the hypothesized difference between 2 populations means is...
-Zero
What type of correlation has the smallest y-intercept...?
-a Positive correlation ( r = 1 ) has the smallest y-intercept
Probability Tree:
-a graphical representation of the various possible sequences of an experiment -illustrates possible outcomes & their associated probabilities
Margin of error:
-accounts for the standard error of the estimator and the desired confidence level of the interval
Chi-Square:
-allows hypothesis testing for nominal data by counting how many in each category
How to decrease Variability:
-an increase in sample size -improvement of reliability of assesment tools and careful data collection techniques -
With control charts, as sample size increases, does your control chart get wider or narrower?
-as sample size increases, the control chart gets narrower
Standard Deviation:
-can never a negative number -is the square root of the variance
Left-tailed test:
-can only reject the null to the left of the hypothesized value of the population proportion Ex. Ho: U ≥ Uo verus Ha: U < Uo
Bernoulli Process:
-consists of n independent & identical trials of an experiment such that each trial... -only 2 possible outcomes, success or failure -the probabilities of success & failure remain the same from trial to trial
Correlation methodologies:
-correlation uses two quantitative variables that must be interval/ratio numeric scale -uses scatter plot
Sample Space:
-denoted "S", of an experiment that include all possible outcomes Example: Letter grades S=(A,B,C,D,F) -the event of passing grades (A,B,C,D) is a subset of S -the simple event of "failing grade" (F) is also a subset of S
If r = 1 then what can be concluded?
-dependent variable can be perfectly predicted by the independent variable -all of dependent variation can be accounted for by the independent variable -coefficient of determination is 100% -the standard error of the estimate is 0 because of the perfect correlation
Subjective Probabilities:
-draws on personal and subjective judgement
A pooled estimate of the common variance between 2 populations is used when...?
-find the pooled estimate when both populations are assumed to have the same population variance
Linear Regression assumptions...?
-for each value of x, there is a group of y values -these y values are normally distributed -the means of these normal distribution of y values tend to lie on the straight line of the regression line -the standard deviations of these normal distributions are equal
Empirical Rule:
-given a sample mean and a sample standard deviation that is symmetric and bell-shaped 1 & -1 standard deviations = 34% 2 & -2 S.D. = 95% 3 & -3 S.D. = 99%
Exhaustive:
-if all possible outcomes of a random experiment are included in the events Ex. "earning a metal" or "not earning a metal" is an exhaustive experiment since these are the only outcomes
Mutually Exclusive:
-if they do not share any common outcome of a random experiment Ex. "earning a metal" or "not earning a metal" in a single olympic event is mutually exclusive
one-tailed test
-involves a null hypothesis that can only be rejected on one side of the hypothesized value -can only reject the null hypothesis if the population mean is greater than Uo Ex. Ho: U ≤ Uo verus Ha: U > Uo
Probability:
-is a numeric value that measure the likelihood that an uncertain event occurs -value of your probability is between 0 (impossible events) and 1 (definite events)
Event:
-is a subset of the sample space
Experiment:
-is a trial that results in one of several uncertain outcomes
Residual Scores
-is calculated by taking the difference between the expected value (y') and the actual value (y) is the residual score -same thing as a deviation score
Chance variation
-is cause by a random number of events that are part of the production process
Binomial random variable:
-is defined as the # of successes achieved in the n trials of a Bernoulli process -possible outcomes are 0 , 1 , .... n
Two-tailed Test:
-is defined when ≠ is in the alternative hypothesis -Use a two-tail test when trying to see if your hypothesis differs from the mean or not -Increases your critical value Ex. Ho = u verus Ha ≠ u
Coefficient of determination
-is the name for r squared -is just the square of "r" -is always positive
Poisson Distribution
-is used when the number of occurrences over a given interval of a given time or space
Matched-pairs experiment
-is when the samples are paired or matched in one way -we watch for a natural pairing between one observation in the first sample and one observation in the second sample Ex. "Before" and "After" studies Ex. A pairing of observations Ex. Dependent Sampling
What happens to our critical statistic when moving from a 2-tailed test to a 1-tailed test...?
-it is not possible to reject the null (find significant difference) if the results of the 1-tailed test is in the unpredictable direction -that is the gamble made when deciding to use a 1-tailed test
When comparing two population means, their hypothesized difference is...
-may assume any value
Discrete Probability Distribution:
-may be viewed at as a table, algebraically, or graphically Ex. a experimenter rolling a dice, all numbers have a probability of 1/6
For matched paired sampling, the parameter of interest is considered...?
-mean difference
Numerator of the Sharpe Ratio measures what..?
-measures the extra reward that investors receive for the added risk taken
When calculating the regression line we try to..?
-minimize distance between predicted Ys and actual (data) Y points (length of green line=standard error of regression line) -because of the negative and positive values canceling each other out, we have to square those distance ( deviations ) -we are trying to minimize the "sum of squares of the vertical distances between predicted and actual data points
two-tailed test
-null can get rejected on both sides of the hypothesized value of the population proportion -includes population mean and population proportion -if the null is rejected in a 2-tailed test, this suggests that the true parameter does not equal the hypothesized value -defined when the alternative hypothesis has a "≠" Ex. Ho: u=Uo versus. Ha: P ≠ Po
One-way Anova: "analysis of variance"
-one way Anova compares more than two means, there is 1 IV and 1 DV -bar or line graph typically
When p is not less than the alpha..?
-p is not significant -we do not have support for our alternative hypothesis Ex. ( p > 0.05 )
Contrast point estimates with confidence intervals
-point estimates are a more specific estimate, but less likely to be exactly right -confidence intervals provide a range of scores, but is more likely to include the population parameter
If the sample data provides significance evidence that the Null is incorrect then we...
-reject the Null hypothesis
Right-tailed test:
-rejection of the null occurs on the right side of the hypothesized mean Ex. Ho: U ≤ Uo verus Ha: U > Uo
Classical (Uniform) approach:
-same probability for every possible outcome -for example a flipping a coin or rolling a dice, each outcome has a equal probability because the variables are random
What do u calculate as the point estimate of an unknown population
-sample variance
Y-intercept tells us what..?
-says that each x value should at least be the y-intercept value or greater
Binomial Distribution:
-shows the probabilities associated with the possible values of X (binomial random variable)
Example of variance being accounted for...
-square the correlation coefficient ("r") -relationship between mom and daughters height r=0.8 Total variance being accounted for: (0.8)^2= 0.64 = 64%
Multiplication Rule:
-states that the probability that A & B both occur is equal to the probability that A occurs given that B has occurred times the probability that B occurs For independent events: the probability that A & B will both occur is the product of the 2 probabilities multiplied together
In most applications, the hypothesized differences between 2 population means is what...
-sum of hypothesized differences of 2 different means is usually 1
T-distributions are used when we know what..?
-t-scores are used when we know the sample standard deviations
Binomial Probability Distribution:
-the expected value E(x)= u =np -the variance is np(1-np) -the standard deviation is √(standard deviation squared)
The greater the diversification of a portfolio, then..?
-the lower the correlation among the assets return -this minimizing the risk
By adding more participants ( n ).. then how does that effect everything else in distribution..?
-the means would be the same -variability goes down, which makes it more difficult to reject the Null hypothesis -makes it easier to reject the Null hypothesis
Independent events:
-the occurrence of one event does not effect the probability of the occurrence of the other event
Dependent events:
-the occurrence of one event is related to the probability of the occurrence of the other event
Conditional Probability
-the probability of an event given that another event has already occurred -the probability of A given B" -in general is greater than unconditional probability
Conditional Probabilities:
-the probability of an event given that another event has already occurred -"the probability of A given B" -in general, conditional is greater than unconditional probabilities
Unconditional Probabilities:
-the probability of an event without any restrictions
Mutually exclusive events A & B:
-the probability of their intersection is zero P(A union B) -the probability of the union can be found by simply adding A + B
r^2 is what...?
-the proportion of the total variance in one variable that is predictable by its relationship with the other variable -we lose the directionality of the relationship with r^2
Discrete
-the random variable assumes a countable number of distinct values
Continuous
-the random variable is characterized by (infinetly) uncountable values within any interval
Regression Coefficient
-the slope
The smaller the p-value..
-the smaller the p-value the stronger the evidence for rejecting the null hypothesis
Assignable variation
-the type of variation that is caused by specific events or factors that can usually be identified or eliminated
If the observed z falls beyond the critical z in the distribution curve..
-then we reject the null hypothesis -we have a significant result - p < alpha Ex. ( p < 0.05 ) -then we have support for our alternative hypothesis
Two-way Anova:
-two-way Anova usually compares more than 2 means, there is 2 IV's and 1 DV -may see a bar graph, but most likely a line graph is the standard
Control Chart:
-used to monitor the behavior of a production process -a plot of statistical information over time
Simple and Multiple Regression:
-uses correlation to predict values on one variable based on values for the other variable -uses scatterplot
Regression does what to predict values of different variables?
-uses the predictor variable (independent variable) to make predictions about the predicted variable (dependent variable)
Would using a regression line, or guessing the mean of the y variable, be better for minimizing error?
-using a regression line minimizes error and gives more accurate predictions
Factors that effect an confidence interval
-varability -sample size ( n ) -confidence level
Union of two events (Event A or Event B):
-when thinking about the union of two mutually exclusive events you need to be careful because you add the P(A) & P(B) together, you count the P(A and B) twice, so you have to subtract P(A union B) to avoid over-stating the probability
Type 1 error:
-when u reject the null, and the null turns out to be true -"false alarm" -related to the alpha
Type 11 error:
-when you don't reject the null, and the null turns out to be false -worse of an error than error type 1 -related to the beta
What are the 3 propositions of the Central Limit Theorem?
1. If sample size ( n ) is large enough, the mean of the sampling distribution will approach the mean of the population 2. If sample size ( n ) is large enough, the sampling distributions of means will be approximately normal, regardless of the shape of the population 3. The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size -As n increases standard error of the mean (SEM) decreases
What are the 3 steps used when formulating the competing hypothesis...?
1. identify the relevant population parameter of interest 2. Determine whether 1 or 2 tailed test 3. Include some form of equality sign in the null hypothesis & use the alternative hypothesis to establish a claim
How to calculate a Confidence Interval..?
= (Point estimate) ± (Margin of Error)
Lower Control Limit
= Expected value - (3 * standard deviations)
How to calculate a residual score:
= ∑ (Y' - Y) -It's the sum of the diffference between actual and expected y values
Standard error of the estimate (line):
= √ ( ∑(Y-Y')^2 ) / ( n-2 ) -a measure of the average amount of predictive error -the average of residuals (amount that Y' scores differ from Y scores) -a mean of the lengths of the green lines
Upper Control Limit
=Expected value + (3 * standard deviations)
Type 1 error
Alpha
The conclusion of a hypothesis that are drawn from the p-value approach versus the critical-value approach are....
Always the same
A classical probability is based on logical analysis rather than on observation or personal judgment.
Classical Probability: is based on logical analysis rather than on observation or personal judgment.
How to calculate Confidence Interval:
Confidence Interval = Point estimate ± Margin of the Error
How to calculate confidence level..
Confidence level = 100( 1 - alpha)%
Bernoulli Process
Consists of n independent & identical trials of an experiment such that each trial: -there are only 2 outcomes, success or failure -the probabilities of success and failure remain the same from trial to trial
An __________ probability is calculated as a relative frequency of occurrence.
Empirical Probability: is calculated as a relative frequency of occurrence.
Right-tailed test:
Ho: U ≤ Uo verus Ha: U > Uo
Left-tailed test:
Ho: U ≥ Uo verus Ha: U < Uo
If you want to reject the Null hypothesis what do u want?
If trying to reject the Null, then we want our -z or t score to be as big as possible and -we want our p-value to be as little as possible
In a hypothesis test, Uo and Po are hypothesized values of what?
In a hypothesis test, Uo and Po are hypothesized values of what? -the population mean and the population proportion
When conducting a T-test, the numerator/denominator of our test statistics (z & t) estimates what...?
Numerator: estimates the difference between groups (between means) Denominator: estimates the within group variability (by using the standard error of the mean)
Total Probability Rule expresses the probability of an event, A, in terms of probabilities of the intersection of A with any mutually exclusive and exhaustive events. The total probability rule based on two events, B and Bc, is..
P(A) = P(A ∩ B) + P(A ∩ Bc), or equivalently, P(A) = P(A | B)P(B) + P(A | Bc)P(Bc).
Confidence Interval:
Provides a range of values that with a certain level of confidence, contain the population parameter of interest
Objective Probability can be explained as...?
Since empirical and classical probabilities generally do not vary from person to person, they are often grouped as Objective Probabilities.
20% of participants received the placebo effect with SSRI medication out of 230 people... what is the standard deviation?
Standard Deviation = √(230 x .20 x .80) = 6.1
What is the measure that indicates how precise a prediction might be or how inaccurate a prediction is..?
Standard error of estimate: -is a measure of accuracy -cannot be negative -based on squared deviations, mean of the length between the green lines (y' and y) -measure of the variability of the dots around the regression line (average of each data point from the regression line, like standard deviation
How to find the average amount by which actual scores deviate on either side of the predicted score:
Step 1. Find error for each value (just the residuals) Ex. (Y-Y') Step 2. Add up all the residuals: ∑ (Y-Y')=0 -Then square the deviations: ∑ (Y-Y')^2 Step 3. Add square root & divide by the degrees of freedom (n-2) Ex. √ ∑(Y-Y')^2 / (n-2)
The Probability of all simple event must equal..?
Sum of all simple events should always equal 1.0
T-test (2 Means):
T-tests -always compares two means, there is one IV and one DV -usually a Bar Graph
The standard deviation of a portfolio return is a measure of what..?
The standard deviation of a portfolio return is a measure of what..? -the variability which is synonymous with risk
Binomial random variable:
is defined as the number of successes achieved in the n amount of trials when 2 Possible Outcomes: p= probability of success 1 - p = q = probability of failure -each time the trial is repeated, the probabilities of success and failure remain the same
How to tell if a distribution is significant finding?
p-value is less than < 0.05 alpha
If r squared ( coefficient of determination ) is 0.8 then what % of variance can be explained? -what is the % of variance that can not be explained?
r^2= 0.8 80% is explained by variance (accounted for) 20% is the % of variance that is not explained or account for 1 - 0.8 = 0.2 = 20%
The X-bar ( mean of a random variable ) follows the normal distribution when the population is normally distributed and when the sample size is equal or greater than ...?
sample size of 30 or great
Right-tailed test illustrates what..?
that the p-value = P(Z ≥ z)
The larger the population mean...?
the wider the confidence interval
Confidence Intervals
using distributions to estimate means
Z-distributions are used when we know what...?
z-score is used when we know the population standard deviation
