BNAD 277 Exam 2 Study Guide PDF

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BNAD 277 Exam 2 Study Guide Chapter 9: Hypothesis Tests 

Developing Null and Alternative Hypothesis Null Hypothesis: The tentative assumption (Denoted by H0) Alternative Hypothesis: The opposite of the null hypothesis (Denoted by Ha) Useful Questions:  What is the purpose of collecting the sample?  What conclusions are we hoping to make? o The Alternative Hypothesis as a Research Hypothesis  Begin with the alternative hypothesis and make it the conclusion that the researcher hopes to support  Example: Prove new fuel system increases cars mpg up from 24   

o H0: μ ≤ 24 o Ha: μ > 24 o The Null Hypothesis as an Assumption to Be Challenged 

Begin with the null hypothesis and make it the assumption the value of the population parameter is true  Example 1: The label on a soft drink states it contains at least 67.6 ounces (government agency checking if true)

o H0: μ ≥ 67.6 o Ha: μ < 67.6 

Example 2: The company wants to fill the bottles to 67.6 ounces (company does not want to over or under fill)

o H0: μ = 67.6 o Ha: μ ≠ 67.6 o Summary of Forms for Null and Alternative Hypotheses 

Three types of tests:  One-Tailed Tests o The Hypothesis is worse than Original (Soda is under filled)

 H0: μ ≥ μ0  Ha: μ < μ0 o The Hypothesis is better than the original (New mpg is higher)  H0: μ ≤ μ0  Ha: μ > μ0 

Two-Tailed Tests o The Hypothesis equals the assumption (Soda production filling correctly)

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H0: μ = μ0 Ha: μ ≠ μ0

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Tips  

H0: ALWAYS has (≤, ≥, =) Ha: OFTEN is what the test is attempting to establish i.e. the user is looking for evidence to support (μ < μ0, μ > μ0, μ ≠ μ0) Type I and Type II Errors o Type II: Accepting the Null hypothesis, but the Alternative hypothesis is true o Type I: Rejecting the Null hypothesis when it is true o Level of Significance: Probability of making a Type I error when the null hypothesis is true as an equality (Denoted as α) o Significance Tests: Applications of hypothesis testing that only control for the Type I error (Common hypothesis test)  Can only find Type I error, does not test for Type II  Say “Do not reject” not accept to avoid risk of making Type II error

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Population Mean: σ is Known o One-Tailed Test 

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Lower Tail Test:  H0: μ ≥ μ0  Ha: μ < μ0 Upper Tail Test:  H0: μ ≤ μ0  Ha: μ > μ0 Test Statistic: z

P Value Approach  Uses Test Statistic z to compute probability  P-Value: Provides a measure of evidence against the null hypothesis provided by the sample. Smaller P values indicate more evidence against the null  Rejection Rule Using p-Value: o Reject H0 if p-value ≤ α Critical Value Approach

Critical Value: Value of a test statistic that corresponds to an area of α or the largest value of the test statistic that will result in the rejection of the null  Rejection Rule Critical Value Approach o Reject H0 if z ≥ za  Summary  P-value and critical value approach always lead to same conclusion  P-value’s advantage is it tells us how significant the results are  Computation of p-Value for One-Tailed Tests o Computer the test statistic  Lower Tail  Use standard normal distribution to compute probability that z is less than or equal to the value of the test statistic (area in the lower tail)  Upper Tail  Use standard normal distribution to compute probability that z is greater than or equal to the value of the test statistic (area in the upper tail) o Two-Tailed Test  Form:  H0: μ = μ0  Ha: μ ≠ μ0  P-Value approach  1. Compute the Test Statistic z  2. If o Upper tail  compute probability that z is greater than or equal to the value of the test statistic (area in the upper tail) o Lower Tail  Use standard normal distribution to compute probability that z is less than or equal to the value of the test statistic (area in the lower tail)  3. Double the probability from step 2 to obtain the p-Value  Critical Value approach  Steps: o 1. Find Test Statistic z o 2. α/2 o 3. Find -zα/2 and zα/2  Two-Tailed Rejection Rule o Reject H0 if z ≤ -zα/2 or if z ≥ zα/2 o Summary and Practical Advice  Steps of Hypothesis Testing: (Sample of at least 30)  1. Develop the Null and Alternative hypothesis 

2. Specify the Level of Significance 3. Collect the sample data and compute the value of the Test Statistic  P-Value approach o 4. Use the value of the Test Statistic to compute the p-Value o 5. Reject the Null if p-Value is less than Level of Significance o 6. Interpret the statistical conclusion in the context of the application  Critical Value approach o 4. Use the Level of Significance to determine the Critical Value and the rejection rule o 5. Use the value of the Test Statistic and the rejection rule to determine whether to reject the Null o Interpret the statistical conclusion in the context of the application o Relationship between Interval  Estimation and Hypothesis  Confidence Interval of a population is the same as a Two-Tail Test Statistic +/- the sample mean  Testing  If the Confidence Interval contains the hypothesized value do not reject Population Mean: σ is Unknown o One-Tailed Test  Same as known (See above), but use T-Table and Degrees of Freedom (n-1) o Two-Tailed Test  Same as Known (See above), but use T-Table and Degrees of Freedom (n-1) o Tip:  Check for skewedness  If not normally distributed sample must be larger than 30 to be accurate o Summary and Practical Advice  

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Population Proportion o Same as Population mean except 

Use Standard Error of the to calculate Test Statistic

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P(^) = proportion of sample P0 = Hypothesized proportion

o Summary  

Proportions tests are similar to population mean np ≥ 5 and n(1-p) ≥ 5 for results to be accurate

Chapter 10: Inference about Means and Proportions with Two Population

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Inferences about the Difference Between Two Population Means: σ1 and σ2 Known o Independent Random Sample: Samples selected from two populations in such a way that the elements making up one sample are chosen independently of the element making up the other sample. o Interval Estimation of μ1 – μ2 

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Point Estimator of the Difference between Two Population Means  X1 – X2  X’s are sample mean age for the random samples Standard Error of X1 – X2

Interval Estimator  Point Estimator +/- Zα/2 x Standard Error o Hypothesis Test about μ1 – μ2  Three Forms of Tests  Lower Tail o H0: μ1 – μ2 ≥ D0 o Ha: μ1 – μ2 < D0  Upper Tail o H0: μ1 – μ2 ≤ D0 o Ha: μ1 – μ2 > D0  Two Tail o H0: μ1 – μ2 = D0 o Ha: μ1 – μ2 ≠ D0 *D0 denotes the hypothesized difference between μ1 – μ2  Test Statistic for Two Populations 

P-value approach  Use P-value approach to draw conclusion (See Chapter 9) o Practical Advice  Make sure your samples are big enough/normally distributed 

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Inferences about the Difference Between Two Population Means: Unknown o Interval Estimation of μ1 – μ2  

Same as above Known, but you have to use T-distribution and Degrees of Freedom Degrees of Freedom: t Distribution with two independent Random Samples

*Round Degrees of Freedom down

o Hypothesis Test about μ1 – μ2 

Same as above Known (see above), but use T-Table and Degrees of Freedom

o Practical Advice 

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Total of the two populations should be 20+ to be accurate

Inferences about the Difference Between Two Population Means: Matched Samples o Matched Samples: Samples in which each data value of one sample is matched with a corresponding data value of the other sample i.e. Same workers use two different methods, random if use method 1 or 2 first o Tip: only consider the difference column o D bar/X bar= Sum of the differences / population o Standard Deviation of Two Populations Matched Samples

o Test Statistic

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o P-Value approach  Use test statistic, T-Table, and Degrees of Freedom (n-1) o Interval Estimation  D bar +/- tα/2 x Test Statistic Inferences about the Difference Between Two Population Proportions o Interval Estimation of p1 – p2  Notation  P1: Proportion of population 1  P2: Proportion of population 2  P1 Bar: Proportion for random sample of population 1  P2 Bar: Proportion for random sample of population 2  Point Estimator of the Difference Between Two Population Proportions  P1 Bar – P2 Bar  Standard Error of Two Population Proportions

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Margin of Error for Two Population Proportions

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 zα/2 x Interval Estimator of the Difference Between Two Population Proportions

P1 Bar – P2 Bar +/- zα/2 x Hypothesis Tests about p1 – p2 Pooled estimator of p: An estimator of a population proportion obtained by computing weighted average of the point estimators obtained from two independent samples Standard Error of Proportion when p 1 = p2  Sq. Root [p(1-p)(1/n1 + 1/n2)] Pooled Estimator: USE WHEN p IS UNKNOWN  n1p1Bar + n2p2Bar / n1 + n2 Test Statistic for Hypothesis tests about Two Population Proportion

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o o o o o

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(P1 Bar – P2 Bar) /

Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions  

Goodness of Fit Test Goodness of Fit Test: A chi-square test that can be used to test that a probability distribution has a specific historical or theoretical probability distribution. This test was demonstrated for a multinomial probability distribution o Multinomial Probability Distribution  Multinomial Probability Distribution: A probability distribution where each outcome belongs to one of three or more categories. The multinomial probability distribution extends the binomial probability distribution from two to three or more outcomes per trail.  Notation  Pa=Probability of A  Pb= Probability of B  Pc= Probability of C, …….  Hypothesis  H0: The probabilities are as stated  Ha: The probabilities are not as stated  Test Statistic for Goodness of Fit

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Steps o List Hypothesized Probability and Observed Frequency (fi)

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o Population sample x Hypothesized probability = Expected Frequency (ei) o Observed Frequency – Expected Frequency = Difference (fi – ei) o Squared Difference (fi – ei)2 o Squared Difference Divided by Expected frequency ((fi – ei)2/ei) P-Value Approach  Use Chi-Table  Find Test Statistic (X2)  Find Degrees of Freedom (k – 1) Critical Value Approach  Use Chi-Table  Find Level of Significance  Find Degrees of Freedom  Use found Critical Value to create Rejection Rule o Rejection Rules  Reject Ho if X2 ≥ Critical Value Summary  1. State null and alternative  2. Select sample and record observed frequencies  3. Assume the null is true and determine the expected frequencies  4. Compute Test Statistic  5. Rejection Rule: (Upper Tail) o P-Value approach: Reject H0 if p-value ≤ α o Critical Value approach: Reject H0 if X2 ≥ X2α

Test of Independence Test of Independence: A chi-square test that can be used to test for the independence of two random variables. If the hypothesis of independence is rejected, it can be concluded that the random variables are associated or dependent o Hypothesis  H0: Independent  Ha: Not Independent o Steps  Summarize data for each variable separately  Example: o V1: Percent Male & Percent Female (Male or Female / Total Sample)

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o V2:Percent Percent Prefer Light, Dark, Regular (Light or Dark or Regular / Total Sample) Find Expected Frequency of each (eij)  (Row I total)(Column j total) / Sample Size = Expected Find Chi-Square Statistic (X2)

Steps o Observed – Expected = Difference o Square Difference o Divide Squared Difference by Expected Frequency o Add All the answers up  Find Degrees of Freedom  (# rows – 1)(# columns – 1)  Draw Conclusion  P-value approach o Use Chi-Square Test Statistic and Degrees of Freedom to Find P-Value o Use P-Value rejection rule to draw conclusion  Critical Value Approach o Use Level of Significance and Degrees of Freedom to find the Critical Value o Use Critical Value rejection rule to draw conclusion o Summary  1. State Null and Alternative  2. Select sample and record observed frequencies in a table with rows and columns  3. Assume the null is true and compute expected Frequencies  4. Compute Test Statistic and Degrees of Freedom  5. Draw Conclusion  P-value approach o Reject null if p is less than or equal to level of significance  Critical Value approach o Reject null if Test Statistic is greater than or equal to Critical Value Testing for Equality of Three or More Population Proportions 

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Notation  P1= population proportion of population 1  P2= population proportion of population 2

 So on so forth  Pk= population proportion for population k  Hypothesis  H0: p1 = p2 = …..  Ha: Not all proportions are equal  Steps to find Test Statistic  Sum of Each Row / Total Population  Percent Found x First Row Observed Frequency= Expected Frequency for first row  Column Total – Expected Frequencies of row 1 column = Expected Frequencies of Row 2  Observed – Expected = Difference  Difference2  Difference2 / Expected Frequency  Add all answers up = Test Statistic  Find Degrees of Freedom  K–1  Draw Conclusions  P-value approach o Use Chi-Table, DoF, and Test Stat to find rough Level of Significance o Use P-value rejection rule to draw conclusion  Critical Value Approach o Use Chi-Table, Level of Significance to find Critical Value o Use Critical Value Rejection Rule to draw conclusion  Summary of Steps  1.State the null and alternative  2. Select samples from each population and record observed frequencies in table with 2 rows and k columns  3. Assume null is true and computed expected frequencies  4. Find Test Statistic and Degrees of Freedom  5. Draw conclusion using rejection rules o P-value approach o Critical value approach o A Multiple Comparison Procedure o Marascuilo Procedure: A multiple comparison procedure that can be used to test for a significant difference between pairs of population proportions. This test can be helpful in identifying differences between pairs of population proportions whenever the hypothesis of equal population proportions has been rejected  Used to identify where differences in population proportions exist  Steps:  1.Compute Sample proportions (row/column total / total column)

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2.Compute Absolute value of pairwise difference between sample proportions for each pair of populations (proportion 1 / Proportion 2) do each pair 3.Calculate Critical Value for each pair of proportions

4.Compare Absolute Values found in Step 2 to Critical Value o If Absolute Value exceeds Critical Value it is Significant o If Absolute Value does not exceed Critical Value the difference is not significant...