Chapter 7 Terms

Two-tailed test of a hyypothesis

.Is one in which the alternative hypothesis does not specify departure from Ho in a particular direction and is written with the symbol =/=.

Three Regions of Graph

1. Mean in the middle 2. z critical values are the z-scores of alpha/2 3. Greater than +critical value or less than -critical value rejects the null hypothesis 4. In between the critical values is the acceptance region or accepts null hypothesis. Or fails to detect based upon insufficient evidence. 5. Note: Mu o is the hypothesized population mean.

Degree of confidence

1.00- alpha= 0.95 0.99 0.90 etc

ztest 90%

1.645 z test statistic

ztest 95%

1.96 z test statistic

ztest 99%

2.575 z test statistic

Decision rule

Basis for rejecting or accepting the null hypothesis based on the z score solved for.

Null Hypothesis

Denoted H0, represents the hypothesis that will be accepted unless the data provide convincing evidence that it is false. This usually represents the "status quo" or some claim about the population parameter that the researcher wants to test. The mean is equal (=) to a certain value. The true population is NOT significantly different from the original claim of the mean.. Ex: The True Population Mean is Equivalent to 0.724 The True Population Mean is NOT significantly different from 0.724

Alternative Research Hypothesis

Denoted Ha , represents the hypothesis that will be accepted only if the data provides convincing evidence of its truth. This usually represents the values of a population parameter for which the researcher wants to gather evidence to support. The mean is =\= to or < or > than the certain value. The true population IS significantly different from the original claim of the mean.. Ex: The True Population Mean is NOT Equivalent to 0.724 The True Population Mean IS significantly different from 0.724

True Positive

Ex: The Test shows Drug Use, and the Athlete IS really using

True Negative

Ex: The Test shows NO drug use, and the Athlete is truly "clean"

mu o or U0

Hypothesized Population Mean

Consequences of Type 1 and Type 2 Errors

If consequences of a Type I error are worse, make ALPHA SMALLER, say, from 0.05 to 0.01. If consequences of a Type II error are worse, make ALPHA LARGER, SAY, FROM 0.01 TO 0.05. The risk of Type I and Type II errors are Inversely related. Note: As Type 1 Error goes up, Risk of Type 2 Error goes down. As Type 1 Error goes down, Risk of Type 2 Error goes up.

Relation to Margin of Error

If the observed difference is xbar- mu is greater than the margin of error than the z test value will be greater than the critical value or in the reject null hypothesis area. If the observed difference is xbar-mu is less than the margin of error than the z-test value will be less than the critical value or in the accepted null hypothesis area.

Test Statistic

Is a sample statistic, computed from information provided in the sample, that the researcher uses ton decide between the null and alternative hypothesis. Tells us the number of standard errors that xbar actually falls away from mu or the hypothesized population mean (Calculated in the z-test formula).

Hypothesis

Is a statement about the numerical value of a population parameter.

One-tailed Test OF A HYPOTHESIS

Is one in whcih the alternate hypothesis is directional and includes the symbol < or >.

Observed Significance Level or p-value for a Specific Statistical Test

Is the probability (assuming Ho is true) of observing a value of the test statistic that is at least as contradictorty to the null hypothesis, and supportive of the alternate hypothesis, as the actual one computed from the sample data.

Rejection Region of a Statistical Test

Is the set of possible values of the test statistic for which the researcher will reject Ho in favor of Ha.

THE LEFT TAILED TEST Pvalue

LOOK AT STUDY GUIDE PICTURE Ho Ha<0.60 Pvalue= area from test statistic to the end of the region of rejection. Pvalue= 0.4984+0.5000.

TWO TAILED TEST Pvalue

LOOK AT STUDY GUIDE PICTURE Ho Ha=/=0.60 Pvalue= area from test statistic to the end of the nearest tail, then multiply it by two (for both tails). Pvalue= (0.0016*2)= 0.0032.

RIGHT TAILED TEST Pvalue

LOOK AT STUDY GUIDE PICTURE Ho Ha>0.60 Pvalue= area from test statistic to the end of the rejection region. Pvalue=0.500-0.4984= 0.0016.

Two TAILED CRITICAL VALUE APPROACH WHERE Decision Rule actually CHANGES DEPENDing ON Ha

Look at Study Guide Ho=0.60 Ha=/= 0.60 +zalpha/2 and -zalpha/2 Decision Rule: Reject Null if Ztest>+Zalpha/2 or of Ztest<-Zalpha/2

Left TAILED CRITICAL VALUE APPROACH WHERE Decison Rule actually CHANGES DEPENDing ON Ha

Look at Study Guide Ho=0.60 Ha<0.60 To the left of -Zalpha.... Decision rule: Reject null hypothesis if Ztest<-Zalpha

Right TAILED CRITICAL VALUE APPROACH WHERE Decision Rule actually CHANGES DEPENDing ON Ha

Look at Study Guide Ho=0.60 Ha>0.60 To the right of Zalpha.... Decision rule: Reject null hypothesis if Ztest>Zalpha

Type 2 Error (False Negative)

Occurs if the researcher accepts the null hypothesis when, in fact, Ho is false. The probability of commiting a Type 2 error is denoted by B. Ex: in sports, a false negative a type 2 error which indicates NO drug use via test, when in reality, the athlete IS REALLY using.

Type 1 Error (False Positive)

Occurs if the researcher rejects the null hypothesis in favor of the alternative hypothesis when, in fact Ho, is true. The probability of committing a Type 1 Error is denoted by greek letter area. Ex: in sports, a false positive a type 1 error which indicates drug use, when in reality, the athlete is NOT really using

Z Test Statistic Non Numeric Proportions

One Tailed Test: Ho: p (hat)=po Ha: p (hat)<po or p(hat)>po Two Tailed Test: Ho: p(hat)=po Ha: p(hat)=/=po Test Statistic: where denominator is op= sqrt((po*qo)/n) and is the standard error of the proportion and qo=1-po p(hat)= sample proportion (xbar= point estimate) po=hypothesized population mean Numerator or p(hat)-po is the observed difference. (mu o= hypothesized mean) Conditions Required: 1. A random sample is selected from a binomial population. 2. The sample size n is large. (This condition will be satisfied if both npo> or equal to 15 and nqo> or equal to 15.) STEPS: Step Zero: Find the Sample Proportion of x/n (small over big) Step One: State the Research Question as 2 Competing Statements (hypotheses) Null: (Ho:)The true population proportion that Pays is NOT signifly MORE THAN 60% / po=0.60 Alternative: (Ha:) The true population proportion that Pays IS significantly MORE than 60%/ po>0.60 Step Two: Select Alpha (aka, level of significance) And draw the diagram with labeled regions (Alpha = 0.01) Step Three: Collect Data, Calculate Appropriate Test Statistic: Z test for Large Sample Proportion Use the z test formula Step Four: Reach the Statistical Decision: Does the Data lead us to Reject Null, or Accept the Null? Decision: Data leads us to Reject the Null Hypothesis Step Five: Interpret the finding, and State Potential Decision Error: The data led us to Reject Null, so we are led to support the Alternative Hypothesis which states: Alternative: (Ha:) The true population proportion that Pays IS significantly MORE than 60%

Interpretation of Pvalues:

Pvalue indicates the likelihood the Research Results are due to Chance Alone, if the Null were really True. If the Pvalue is small enough (smaller than Alpha), we say the likelihood it is due to chance Is so small, it is probably NOT due to chance.

Steps for Hypothesis Testing

Step One: State Two competing, mutually exclusive Statements (outcomes), only one of which will be supported by the data. 1. Null Hypotheses: 2. Alternative Hypothesis: Step Two: Select the Level of Statistical Significance (aka, Select Alpha Level) Step 3: Collect data, calculate Sample Mean, etc., and Calculate the appropriate Test Statistic, here, Numeric Data, Large "n" So we use Z test statistic INTERPRET Z TEST STATISTIC: Z TEST indicates the number of Std. Errors that Xbar actually fell away from Hypothesized Mean,Mu Step 4: Statistical Decision: Data leads us to REJECT THE NULL HYPOTHESIS, B/C? XBAR FELL MORE THAN 1.96 STD ERRORS ABOVE MU STEP 5: This means, the True Population Mean is significantly GREATER than 85, b/c Xbar fell so far above Mu. Whenever the Data leads us to Reject the Null Hypothesis, it means the Observed Difference IS Statistically Significant. Had Ztest fallen within the Margin of Error, it would not have been statistically significant

DATA DECISION CHART

Study and write out. Make sure you know and understand. It is in notes and study guide.

T Test Statistic Numeric Means

T Test Statistic Mu o or U0 is the hypothesized population mean n< 30 where x bar-mu or numerator is observed difference Where denominator is standard error of the mean OR S- SQRT(N)

Standard Error of the Mean

The SDEV for a distribution of Means, is not the Standard Deviation, but the STANDARD ERROR OF THE MEAN, sdev/SQRT(N)

Z or T critical

The critical value that indicates the number of standard errors xbar must fall away from the mean or mu, to reject the null. The critical values of indicate the number of standard errors xbar must fall away to be considered significantly different from mu.

P(Type 1 Error)

The result is alpha or "alpha error".

Alpha

Usually given to draw the curve and is known as the level of statistical significance.

Making Sense of Z test Statistic

What does Z test statistic tell us? Z test statistic indicates the number of Std. Errors that Phat (0.6504) falls away from the Hypothesized Population Proportion (Po) (0.60). Here, Z test is +2.94, which indicates that Phat Falls 2.94 Std. Errors ABOVE the Hypothesized Population Proportion (Po).

Rejecting the Null Vs. Accepting the Null

When data leads us to Reject the Null, we say, There was sufficient evidence for us to Reject the Null Hypothesis Accepting the Null: When data leads us to Accept the Null, we say, There was Insufficient evidence for us to Reject the Null Hypothesis Aka, Fail to Reject the Null, due to insufficient evidence of a significant difference

Observed difference

XbAr- mu

Z Test Statistic Numeric Means

Z Test Statistic Mu o or U0 is the hypothesized population mean n> or equal to 30 where x bar-mu or numerator is observed difference Where denominator is standard error of the mean OR PHETA- SQRT(N)

Pvalue

is the area under the curve from the Test Statistic to the end of the Region of Rejection

Alpha

is the area under the curve in the Region(s) of rejection (area beyond Critical Value(s))

What a Statistically Significant Difference Means:

• Every time the data leads us to Reject Null, there is a Statistically Significant Diff (here, Phat & Po) • A Statistically Significant Difference, the Observed Diff is LARGER than the margin of error • A Statistically Significant Difference, is a research finding that is NOT due to chance. What a Statistically Significant Difference Means: If a finding is NOT due to chance, it is likely to be repeated, if the study were repeated. ○Statistically Significant results are reliable, stable, likely to have same outcome time and time again when the study is replicated

Prompt: "If" a decision error were made (which we do not ever know, since we would need the entire population data to know this), what kind of an error "could" have been made?

• If the data leads us to Accept the Null, the only possible decision error is an Acceptance Error, Type II. -And, what is the "risk" of committing a Type I Error? Alpha = P(Type I Error)= 0.01 • If data leads us to Reject the Null, the only possible decision error is a rejection error, Type I Error ○ And what is the "risk" of making a Type II error? (you will not be asked this, but it is = Beta).

What is the relationship between Statistical Significance and Pvalues?

□ Any time data leads us to Reject Null, the results are Statistically Significant □ If data leads us to Accept Null, then the results are NOT Statistically significant □ A statistically significant finding, is one that is NOT DUE TO CHANCE

Application of Pvalues: Using Pvalues in Statistical Decision Rule

○ Decision Rule using Pvalues: Reject the Null Hypothesis if Pvalue < Alpha This decision rule applies to any hypothesis: same answer for all.... -Right Tailed Test - Left Tailed Test - Or Two Tailed