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**Overview Definition Hypothesis**

in statistics, is a claim or statement about a property of a population page 366 of text Various examples are provided below definition box

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**Null Hypothesis: H0 Statement about value of population parameter**

Must contain condition of equality =, , or Test the Null Hypothesis directly Reject H0 or fail to reject H0 Give examples of different wording for and Š, such as ‘at least’, ‘at most’, ‘no more than’, etc.

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**Alternative Hypothesis: Ha**

Must be true if H0 is false , <, > ‘opposite’ of Null Give examples of different ways to word °,< and >, such as ‘is different from’, ‘fewer than’, ‘more than’, etc.

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**Note about Forming Your Own Claims (Hypotheses)**

If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis. By examining the flowchart for the Wording of the Final Conclusion, Figure 7-4, page 375, this requirement for support of a statement becomes clear.

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**HO The defendant Claim about a**

Legal Trial Hypothesis Test HO The defendant Claim about a is not guilty population parameter HA The defendant Opposing claim about a is guilty population parameter Result The evidence The statistic indicates a convinces the rejection of HO, and the jury to reject alternate hypothesis is the assumption accepted. of innocence. The verdict is guilty By examining the flowchart for the Wording of the Final Conclusion, Figure 7-4, page 375, this requirement for support of a statement becomes clear.

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Test Statistic a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis page 371 of text

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**For large samples, testing claims about population means**

Test Statistic a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis For large samples, testing claims about population means page of text Example on page 372 of text x - µx z = n

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Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis page 372 of text

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Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region

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Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region

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Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Regions

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**Significance Level denoted by **

the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. common choices are 0.05, 0.01, and 0.10 This is the same introduced in Section 6-2, where we defined the degree of confidence for a confidence interval to be the probability 1 -

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Critical Value Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis page 372 of text

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Critical Value Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis The critical value depends on the type of test being conducted. The text will start with critical values that are z scores. Later tests will have critical values that are t scores and X2 values. Example on page 373 of text. Critical Value ( z score )

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Critical Value Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis Reject H0 Fail to reject H0 The critical value separates the curve into areas where one would reject the null (the critical region), and where one would fail to reject the null (the rest of the curve). Critical Value ( z score )

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**Two-tailed,Right-tailed, Left-tailed Tests**

The tails in a distribution are the extreme regions bounded by critical values. page 373 of text

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Two-tailed Test H0: µ = 100 Ha: µ 100

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** is divided equally between the two tails of the critical**

Two-tailed Test H0: µ = 100 Ha: µ 100 is divided equally between the two tails of the critical region

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** is divided equally between the two tails of the critical**

Two-tailed Test H0: µ = 100 Ha: µ 100 is divided equally between the two tails of the critical region Means less than or greater than Analysis of what the symbol ° means which helps students to realize this is a two tailed test.

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** is divided equally between the two tails of the critical**

Two-tailed Test H0: µ = 100 Ha: µ 100 is divided equally between the two tails of the critical region Means less than or greater than Reject H0 Fail to reject H0 Reject H0 100 Values that differ significantly from 100

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Right-tailed Test H0: µ 100 Ha: µ > 100

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Right-tailed Test H0: µ 100 Ha: µ > 100 Points Right

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**Right-tailed Test H0: µ 100 Ha: µ > 100 Points Right Values that**

Fail to reject H0 Reject H0 Values that differ significantly from 100 100

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Left-tailed Test H0: µ 100 Ha: µ < 100

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Left-tailed Test H0: µ 100 Ha: µ < 100 Points Left

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**Left-tailed Test H0: µ 100 Ha: µ < 100 Points Left Values that**

Reject H0 Fail to reject H0 Values that differ significantly from 100 100

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**Conclusions in Hypothesis Testing**

always test the null hypothesis 1. Reject the H0 2. Fail to reject the H0 need to formulate correct wording of final conclusion page 374 of text. Examples at bottom of page and top of page 375

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**Wording of Final Conclusion**

Start Does the original claim contain the condition of equality “There is sufficient evidence to warrant rejection of the claim that. . . (original claim).” (This is the only case in which the original claim is rejected). Yes (Original claim contains equality and becomes H0) Do you reject H0?. Yes (Reject H0) No (Fail to reject H0) “There is not sufficient evidence to warrant rejection of the claim that. . . (original claim).” No (Original claim does not contain equality and becomes Ha) (This is the only case in which the original claim is supported). page 375 of text Also found on Formula and Table insert provided with text. Many instructors allow students to use this flowchart when taking exams. The conclusion process indicted in this chart will be used with other statistical processes in other chapters. Discussion of the two results that support or reject conclusions, whereas the other two do not provide enough evidence for support or rejection. Do you reject H0? Yes (Reject H0) “The sample data supports the claim that (original claim).” No (Fail to reject H0) “There is not sufficient evidence to support the claim that. . . (original claim).”

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**Accept versus Fail to Reject**

some texts use “accept the null hypothesis we are not proving the null hypothesis sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect) page 374 of text The term ‘accept’ is somewhat misleading, implying incorrectly that the null has been proven. The phrase ‘fail to reject’ represents the result more correctly.

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**Power of a Hypothesis Test**

Definition Power of a Hypothesis Test is the probability (1 - ) of rejecting a false null hypothesis, which is computed by using a particular significance level and a particular value of the mean that is an alternative to the value assumed true in the null hypothesis.

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**for testing claims about population means**

Assumptions for testing claims about population means 1) The sample is a simple random sample. 2) The sample is large (n > 30). a) Central limit theorem applies b) Can use normal distribution 3) If is unknown, we can use sample standard deviation s as estimate for . page 381 of text

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**Traditional (or Classical) Method of Testing Hypotheses**

Goal Identify a sample result that is significantly different from the claimed value page 382 of text

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The traditional (or classical) method of hypothesis testing converts the relevant sample statistic into a test statistic which we compare to the critical value.

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**Hypotheses Testing 5 Step Process**

1. State the hypotheses 2. Decide on a model. 3. Determine the endpoints of the rejection region and state the decision rule. 4. Compute the test statistic 5. State the conclusion These are the steps in the flowchart of Figure 7-5 on page 383.

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**Test Statistic for Claims about µ when n > 30**

x - µx z = Test Statistic for Claims about µ when n < 30 x - µx t = n

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Decision Criterion Reject the null hypothesis if the test statistic is in the critical region Fail to reject the null hypothesis if the test statistic is not in the critical region

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**FIGURE 7-4 Wording of Final Conclusion**

Does the original claim contain the condition of equality Yes (Reject H0) “There is sufficient evidence to warrant rejection of the claim that. . . (original claim).” (This is the only case in which the original claim is rejected). Yes (Original claim contains equality and becomes H0) Do you reject H0?. No (Fail to reject H0) “There is not sufficient evidence to warrant rejection of the claim that. . . (original claim).” No (Original claim does not contain equality and becomes Ha) (This is the only case in which the original claim is supported). page 375 of text Also found on Formula and Table insert provided with text. Many instructors allow students to use this flowchart when taking exams. The conclusion process indicted in this chart will be used with other statistical processes in other chapters. Discussion of the two results that support or reject conclusions, whereas the other two do not provide enough evidence for support or rejection. Do you reject H0? Yes (Reject H0) “The sample data supports the claim that (original claim).” No (Fail to reject H0) “There is not sufficient evidence to support the claim that. . . (original claim).”

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Example: Given a data set of 106 healthy body temperatures, where the mean was 98.2o and s = 0.62o , at the 0.05 significance level, test the claim that the mean body temperature of all healthy adults is equal to 98.6o. This examples starts at the bottom of page 383 and concludes on page Remind students that the first time through the entire Hypothesis Testing procedure will seem ‘intense’. However, after many practices (doing some exercises provided in the text), the process will begin to seem a little more ‘natural’.

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Example: Given a data set of 106 healthy body temperatures, where the mean was 98.2o and s = 0.62o , at the 0.05 significance level, test the claim that the mean body temperature of all healthy adults is equal to 98.6o. Steps: 1) State the hypotheses H0 : = 98.6o Ha : 98.6o 2) Determine the model Setting up the claim, null, and alternative are critical to the rest of the process. If any of these are identified incorrectly, the final result of the procedure will be in great jeopardy. Two tail Z test, n > 30

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** = 0.05 /2 = 0.025 (two tailed test) z = - 1.96 1.96**

3) Determine the Rejection Region = 0.05 /2 = (two tailed test) 0.4750 0.4750 0.025 0.025 This is a two-tailed test because the alternative indicated ‘is not equal to’. See Section 7-2, page The critical values are determined from Table A-2, looking up the area of in the body of the table. (Review Section 5-4, page 249 of text. z =

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**z = = = - 6.64 x - µ 98.2 - 98.6 4) Compute the test statistic n**

0.62 n 106 Care must be taken when using a calculator to find the z value. If students wish to compute the value all in one step on the calculator, parentheses will need to be placed around the numerator and the denominator.

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**5) State the Conclusion z = - 6.64 REJECT H0 z = - 1.96 µ = 98.6**

Sample data: x = 98.2o or z = Reject H0: µ = 98.6 Reject H0: µ = 98.6 Fail to Reject H0: µ = 98.6 Place the test statistic z score (-6.64) correctly in place on the z score number line. This correct placement puts it far down into the left critical region. z = µ = 98.6 or z = 0 z = 1.96 z = There is sufficient evidence to warrant rejection of claim that the mean body temperatures of healthy adults is equal to 98.6o. REJECT H0

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**for testing claims about population means**

Assumptions for testing claims about population means 1) The sample is a simple random sample. 2) The sample is small (n 30). 3) The value of the population standard deviation is unknown. 4) The sample values come from a population with a distribution that is approximately normal. page 399 of text

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**Test Statistic for a Student t-distribution**

x -µx t = s n Critical Values Found in Table A-3 Degrees of freedom (df) = n -1 Critical t values to the left of the mean are negative page 400 of text Most common mistake made with this procedure is to not use Table A-3 to find the critical values. Some use Table A-2 incorrectly.

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**Important Properties of the Student t Distribution**

1. The Student t distribution is different for different sample sizes (see Figure 6-5 in Section 6-3). 2. The Student t distribution has the same general bell shape as the normal distribution; its wider shape reflects the greater variability that is expected with small samples. 3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a = 1). 5. As the sample size n gets larger, the Student t distribution get closer to the normal distribution. For values of n > 30, the differences are so small that we can use the critical z values instead of developing a much larger table of critical t values. (The values in the bottom row of Table A-3 are equal to the corresponding critical z values from the normal distributions.) page 400 of text

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**the population essentially**

Figure Choosing between the Normal and Student t-Distributions when Testing a Claim about a Population Mean µ Start Use normal distribution with x - µx Is n > 30 ? Yes Z / n (If is unknown use s instead.) No Is the distribution of the population essentially normal ? (Use a histogram.) No Use nonparametric methods, which don’t require a normal distribution. Yes Use normal distribution with page 401 of text Example of the small sample procedure is on page 402 of text Is known ? x - µx Z / n No (This case is rare.) Use the Student t distribution with x - µx t s/ n

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The larger Student t critical value shows that with a small sample, the sample evidence must be more extreme before we consider the difference is significant. bottom of page 403 of text

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**A company manufacturing rockets claims to use an **

average of 5500 lbs of rocket fuel for the first 15 seconds of operation. A sample of 6 engines are fired and the mean fuel consumption is 5690 lbs with a sample standard deviation of 250 lbs. Is the claim justified at the 5% level of significance? 2.571 -2.571 1. HO: µ = HA: µ 5500 2. Two tail t test, n < 30, unknown population standard deviation 1.862 3. t critical for 5% for a two tail test with 5 d.f. is 2.571 Fail to reject HO, there is no evidence at the .05 level that the average fuel consumption is different from µ = 5500 lbs

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**P-Value Method Table A-3 includes only selected values of **

Specific P-values usually cannot be found Use Table to identify limits that contain the P-value Some calculators and computer programs will find exact P-values page 404 of text Example on this page too.

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**P-Value Method of Testing Hypotheses**

very similar to traditional method key difference is the way in which we decide to reject the null hypothesis approach finds the probability (P-value) of getting a result and rejects the null hypothesis if that probability is very low page 387 of text

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**P-Value Method of Testing Hypotheses**

Definition P-Value (or probability value) the probability that the test statistic is as far or farther from if the null hypothesis is true The attained significance level of a hypothesis test is the P value of its test statistic

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**P-Value Method of Testing Hypotheses**

Guidelines for rejecting HO based on the P value: If P < , then reject HO and accept HA If P > , then reserve judgement about HO

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**P-value Interpretation Small P-values (such as 0.05 or lower)**

Unusual sample results. Significant difference from the null hypothesis Large P-values (such as above 0.05) Sample results are not unusual. Not a significant difference from the null hypothesis

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**Figure 7-8 Finding P-Values**

Start Left-tailed What type of test ? Right-tailed Two-tailed Is the test statistic to the right or left of center ? Left Right P-value = area to the left of the test statistic page 390 of text P-value = twice the area to the left of the test statistic P-value = twice the area to the right of the test statistic P-value = area to the right of the test statistic P-value P-value is twice this area P-value is twice this area P-value Test statistic Test statistic Test statistic Test statistic

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**Testing Claims with Confidence Intervals**

A confidence interval estimate of a population parameter contains the likely values of that parameter. We should therefore reject a claim that the population parameter has a value that is not included in the confidence interval. page 392 of text

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**Testing Claims with Confidence Intervals**

Claim: mean body temperature = 98.6º, where n = 106, x = 98.2º and s = 0.62º 95% confidence interval of 106 body temperature data (that is, 95% of samples would contain true value µ ) 98.08º < µ < 98.32º 98.6º is not in this interval Therefore it is very unlikely that µ = 98.6º Thus we reject claim µ = 98.6º

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**Underlying Rationale of Hypotheses Testing**

If, under a given observed assumption, the probability of getting the sample is exceptionally small, we conclude that the assumption is probably not correct. When testing a claim, we make an assumption (null hypothesis) that contains equality. We then compare the assumption and the sample results and we form one of the following conclusions: page 393 of text

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**Underlying Rationale of Hypotheses Testing**

If the sample results can easily occur when the assumption (null hypothesis) is true, we attribute the relatively small discrepancy between the assumption and the sample results to chance. If the sample results cannot easily occur when that assumption (null hypothesis) is true, we explain the relatively large discrepancy between the assumption and the sample by concluding that the assumption is not true.

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Type I Error The mistake of rejecting the null hypothesis when it is true. (alpha) is used to represent the probability of a type I error Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6 Example on page 375 of text

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Type II Error the mistake of failing to reject the null hypothesis when it is false. ß (beta) is used to represent the probability of a type II error Example: Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6

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**Type I and Type II Errors**

True State of Nature The null hypothesis is true The null hypothesis is false Type I error (rejecting a true null hypothesis) We decide to reject the null hypothesis Correct decision Decision Type II error (rejecting a false null hypothesis) page 376 of text We fail to reject the null hypothesis Correct decision

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**Controlling Type I and Type II Errors**

For any fixed , an increase in the sample size n will cause a decrease in For any fixed sample size n , a decrease in will cause an increase in . Conversely, an increase in will cause a decrease in . To decrease both and , increase the sample size. page 377 in text

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