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ECN 221 - EXAM 3
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Key Concepts:
Terms in this set (17)
Confidence Interval
If we have no particular idea about the value of the population parameter, and merely want some information, the ______ should be used.
Hypothesis Test
If we want to make a comparison with a. given value for a population parameter, then a ____ should be used.
null hypothesis H0
- represents the status quo
- believed to be true unless there is overwhelming evidence to the contrary
alternative hypothesis H1
- represents the opposite of the null hypothesis
- is believed to be true if the null hypothesis is found to be false
- this is usually what we want to test/find out
- In my research I am trying to "prove" a certain hypothesis
Type I error
occurs when the null hypothesis is rejected when it's true, also known as a false positive
Type II error
occurs when we fail to reject the null hypothesis when it is not true, also known as a false negative.
One-tail hypothesis
alternative hypothesis is stated as either < or >
Two-tail hypothesis
alternative hypothesis is expressed as u≠u0.
choose a significance level for a for the hypothesis test and compare the critical value to the test statistic zx.
The Role a Plays in the Hypothesis Testing
Changing the a changes the critical z-score in the hypothesis test, which in turn, changes the rejection region in the sampling equilibrium
Hypothesis Testing for the Population Mean when σ is Known
Step 1: Identify the null and alternative hypotheses
Step 2: Set a value for the significance level, a
Step 3: Determine the critical value
- σ is known so use a z-score; the critical z-score identifies rejection region for this one tail test
- since this is a one-tail test the entire area for a = 0.05 is placed on the right side (upper tail) of the sampling distribution
Step 4: Calculate the appropriate test statistic
Formula for the z-test statistic for a hypothesis test when σ is known
Step 5: Compare the z-test statistic zx with the critical z-score za
- for the one-tail upper test, reject the null hypothesis if zx > za
Step 5a: (Optional) Compare the sample mean x bar with the critical sample mean xa
Step 6: State the conclusion
Hypothesis Testing for the Population Mean when σ is Unknown t-distribution, degrees of freedom
Substitute the population standard deviation with the sample standard deviation, s, in place of σ
Use the student's t-distribution rather than the normal distribution
- assume that the population of interest follows the normal probability distribution
Step 1: Identify the null and the alternative hypothesis
Step 2: Set a value for the significance level, a
Step 3: Determine the appropriate critical value
- σ is unknown so use a t-score; n = 100 so use 99 degrees of freedom
- Since this is a one-tail test and the alternative hypothesis states that you expect and increase in the average the entire area for a = 0.05 is placed on the right side (upper tail) of the sampling distribution
Step 4: Calculate the appropriate test statistic
Formula for the t-test statistic for a hypothesis test for the population (when σ is unknown):
Do not reject the null hypothesis if tx < ta
Step 5a: (Optional) Compare the sample mean x bar with the critical sample mean xa
- The critical sample mean, xa, is the sample mean that marks the boundary of the rejection region
Formula for the critical sample mean for a hypothesis test for the population mean (when σ is unknown):
We need to find a score that brackets the critical t-score in the n-1=100-1=99 row
The p-value will be between the two probabilities shown in the corresponding column headings
Hypothesis Testing for the Proportion of a Population
Proportion data follow the binomial distribution, which can be approximated by the normal distribution if nπ ≥ 5 and n(1 - π) ≥ 5
Where:
π = The probability of a success in the population
n = The sample size
p-values
The probability value (p-value) is the probability of observing a sample mean ast least as extreme as the one selected for the hypothesis test, assuming the null hypothesis is true.
The p-value is sometimes referred to as the observed level of significance ot the weight of evidence
Provides a third approach to deciding whether or not to reject the null hypothesis
The smaller the p-value, the greater the weight of evidence we have for rejection thenull hypothesis H0
Hypothesis Testing to Compare the Difference Between Two Means with Independent Samples, σ1 and σ2 Known
Goal: examine the difference between two means is the result of subtracting the sampling distribution for the mean of one population from the sampling distrubtion for the mean of a second population
When samples are independent of one another, the results you observe when sampling from one population have no impact on the results you observe when sampling from the second population
We are facing the usual rules with the CLT:
- If the sample size is less than 30, than this hypothesis test requires that the population follow the normal distribution
- If our sample size is greater than or equal to 30, we know from the Central Limit Theorem that the sampling distribution follows the normal distribution so there are no restrictions on the population distribution
Step 1: Identify the null and alternative hypotheses
Step 2: Set a value for a significance level
Step 3: Calculate the appropriate test statistic
- Calculate for the standard error first
Step 4: Determine the appropriate critical value
Step 5: Compare the z-test statistic zx with the critical z-score a a/2
- For the two-tail test, reject the null hypothesis if |zx| > |z a/2|
Step 6: Calculate the p-value
Step 7: State the conclusion
Hypothesis Testing with Dependent Samples
Step 0: Calculate the matched-pair differences: d
Let population 1 = Company evaluation before social media campaign (x1)
Let population 2 = Company evaluation after social media (x2)
Step 1: Calculate the mean and the standard deviation of the matched pair differences
- Mean:
- Standard Deviation of matched pair differences:
Hypothesis Testing to Compare Two Population Proportions with Independent Samples
p-values, confidence intervals
F-distribution
- The ANOVA test-statistic always follows the F-distribution
- The F-distribution is right-skewed and the reject region is in the right tail
- The F-distribution has two types of degrees of freedom, D1, for SSb and D2 for SSW
One-way Anova Randomized Block ANOVA
- With the new method we still have to calculate the SST and the SSB and the equation for these two don't change
- But we will partition the SSW into the sum of squares block (SSBL) and the sum of squares error (SSE) to add the information about days of the week
- SST = SSB + SSBL +SSE
Simple Regression Analysis
- Simple regression analysis is used to determine a straight line that best fits a series of ordered pairs (x,y)
o This technique is known as simple variable regression because we are using only one independent variable
o Formula for the equation describing a straight line through ordered pairs
The Least Squares Method
The _______identifies the linear equation that best fits a set of ordered pairs
- used to find the values for a (the y-intercept) and b (the slope of the line)
The resulting best fit line is called the regression line
Goal: minimize the total squared error between the values of y and y hat
The elast squares method will minimize the sum of squares error (SSE)
The Coefficient of Determination R2 Hypothesis Testing for Slope and for Model Confidence Intervals
- The portion of the total variation in the independent variable (y) that is explained by its relationship witht the independent variable (x)
- R2 varies from 0% to 100%
- Higher values of R squared are more desirable than the lower ones because we would like to be able to explain as much of the variation in the dependent variable as possible
- For the simple regression R squared is actually the swaure of the correlation coefficient we calculated in Chapter 3
- R squared = ssr / sst
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