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8-1 BASICS OF HYPOTHESIS TESTING

Some examples of typical hypotheses (or claims):

• µ < 98.60F “The mean body temperature of humans is less than 98.60

F ”

• p > 0.5 “The proportion of consumers not comfortable with drone deliveries is greater than 0.5.”

• σ =15 “ The population of college students has IQ scores with a standard deviation equal to 15”

The term null is used to indicate no change or no effect or no difference. We conduct the hypothesis test by assuming the parameter is equal to some specified value so that we can work with a single distribution having a specific value

H

0 ALWAYS CONTAINS THE EQUALITY!!!!!!

Note: if the original claim contain equality, then write it as Null hypothesis, otherwise, it is

a Alternative hypothesis.

Examples: Claim: There is less than 10% of smokers on campus. Claim in symbolic form: p < 0.10. Note that this does not contain equality, it is an alternative hypothesis. The opposite must be a null hypothesis. ��: � = 0.10 �: � < 0.10

DEFINITION In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a procedure for testing a claim about a property of a population.

DEFINITIONS

The null hypothesis (denoted by H0) statement that the value of a population

parameter (such as proportion, mean, or standard deviation) is equal to some claimed value.

The alternative hypothesis (denoted by H1 or Ha or HA) is a statement that the

parameter has a value that somehow differs from the null hypothesis. For the methods of this chapter, the symbolic form of the alternative hypothesis must use one of these symbols <, >, or ≠

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Example: Write the null hypothesis and the alternative hypothesis. a. Claim: the mean height of men is greater than 174 cm.

b. Claim: at least 90% of students use a smartphone.

c. Claim: The mean pulse rate (in bpm) of adult male is equal to 69 bpm.

d. Claim: The standard deviation of pulse rate of adult male is less than 11.3 bpm.

Note: If you are conducting a study and want to use a hypothesis test to support your claim, your claim must be worded so that it becomes the alternative hypothesis. You can never support a claim that a parameter is equal to a specific value.

Select the Significance Level α

The significance level α is the same α introduced in section 7-1, where we defined “critical value.” Common choices for α are 0.05, 0.01 and 0.10.

DEFINITION The significance level α for a hypothesis test is the probability value used as the cutoff for determining when the sample evidence constitutes significant evidence against the null hypothesis. By its nature, the significance level α is the probability of mistakenly rejecting the null hypothesis when it is true.

Significance level α = P(rejecting H0when H0

is true)

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Identify the statistic Relevant to the Test and Determine Its Sampling

Distribution (such as normal, t, or χ 2 )

Table 8-2 lists parameters along with the corresponding sampling distributions.

Example: The claim p > 0.5 is a claim about the population proportion p, so use the normal distribution, provided that the requirements are satisfied. (With n=1009, p =0.5, and q = 0.5.

np ≥ 5 and nq ≥ 5 are both true.)

Find the Value of the Test Statistic

Example: Find the test statistic.

DEFINITION: The test statistic is a value used in making a decision about the null hypothesis. If

it is found by converting the sample statistic ( such as ) to a score (such

as z, t, or χ 2 ) with the assumption that the null hypothesis is true.

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Find the critical values for two-tailed, left-tailed, right-tailed

Depending on the claim being tested, the critical region could be in the two extreme tails, it could be in the left tail, or it could be in the right tail.

• Two-tailed test: The critical region is in the two extreme regions (tails) under the curve

• Left-tailed test: The critical region is in the extreme left region (tail) under the curve

• Right-tailed test: The critical region is in the extreme right region (tail) under the curve

DEFINITION: The critical region (or rejection region) is the area corresponding to all values of the test statistic that cause us to reject the null hypothesis

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Note: Critical values depend on the null hypothesis, and the sampling distribution, and the significance level α .

Example: Assume normal distribution. Find critical values.

a. Using a significance level of � = 0.05 and it is a right tailed test. Find critical value.

b. Using a significance level of � = 0.01 and it is a left-tailed test. Find critical value. c. Using a significance level of � = 0.05 and it is a two-tailed test. Find critical values.

d. Using a significance level of � = 0.10 and �: � < 0.75. Find critical value.

DEFINITION In a hypothesis test, the critical value(s) separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to rejection of the null hypothesis.

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e. Using a significance level of � = 0.01 and �: � � 0.54. Find critical values.

P-Value Method

To find the P-value, first find the area beyond the test statistic, then use the procedure given in Figure 8-3. That procedure can be summarized as follows.

• Critical region in the left tail: P-value = area to the left of the test statistic.

• Critical region in the right tail: P-value = area to the right of the test statistic.

• Critical region in two tails: P-value = twice the area in the tail beyond the test statistic

DEFINITION In a hypothesis test, the P-value is the probability of getting a value of the test statistic that is at least as extreme as the test statistic obtained from the sample data, assuming that the null hypothesis is true.

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Example: Find p-values

a. The test statistic of � = 2.75 is obtained when testing a claim that � � 0.15. Find the p-value.

b. The test statistic of � = �2.05 is obtained when testing a claim that � < 0.15. Find the p-value.

c. The test statistic of � = 3.05 is obtained when testing a claim that � � 0.15. Find the p-value.

d. The test statistic of � = 4.23 is obtained when testing a claim that � �

�. Find the p-

value.

Make a Decision to Either Reject H0

or fail to reject H0

Decision criterion for the P-Value Method:

• If P-value ≤ α , reject H0. (“If the P is low, the null must go.”)

• If P-value ≥ α , fail to reject H0

Decision Criteria for the Critical Value Method:

• If the test statistic is in the critical region, reject H0

• If the test statistic is not in the critical region, fail to reject H0

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Write Conclusion Using Simple and Nontechnical Terms For wording of the final conclusion use the following (stick to the script). This is also in the formula sheet you can use during an exam.

Example: Make decision about the null hypothesis and state the conclusion (Critical value method)

a. � = 0.05 original claim is �: � � 0.67. Test statistic: z = 2.05

b. � = 0.05 original claim is �: � < 0.07. Test statistic: z = - 4.05

c. � = 0.01 original claim is �: � � 0.67. Test statistic: z =2.78

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Example: Make decision about the null hypothesis and state the conclusion (P value method)

a. Significance level: � = 0.05 original claim is �: � � 0.67. P-value: 0.1328

b. Significance level: � = 0.01 original claim is �: � < 0.75. P-value: 0.0257

c. Significance level: � = 0.02 original claim is �: � � 0.67. P-value: 0.0045

Accept of Fail to Reject? We should say that we “fail to reject the null hypothesis” instead of saying that we “accept the null hypothesis.”. The term accept is misleading, because it implies incorrectly that the null hypothesis has been proved, but we can never prove a null hypothesis. The phrase fail to reject says more correctly that the available evidence isn’t strong enough to warrant rejection of the null hypothesis. Final Conclusion: “there is not sufficient evidence to warrant rejection of the claim of no difference between 0.5 and the population proportion.” A better statement would be this: “Until stronger evidence is obtained, continue to assume that the population proportion is equal to 0.5.”

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Confidence Intervals for Hypothesis Tests A confidence interval estimate of a population parameter contains the likely values of that parameter. If a confidence interval does not include a claimed value of a population parameter, reject that claim. For two-tailed hypothesis tests, construct a confidence interval with a confidence level of 1−α , but for a one-tailed hypothesis test with significance level α ,

construct a confidence interval with a confidence level of 1− 2α . .

Type I and Type II Errors When testing a null hypothesis, we arrive at a conclusion of rejecting or failing to reject it. Our conclusions are sometimes correct and sometimes wrong (even if we apply all procedures correctly). Table 8-4 below includes two different types of errors and we distinguish between them by calling the type I and type Ii errors as described here:

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Example: Write the description of type 1 error and type 2 error.

The proportion of people who write with their left hand is equal to 0.1 Type 1 error: In reality p = 0.1, but we reject the claim that p = 0.1 Type 2 error: In reality p is not 0.1, but we fail to reject the claim that p = 0.1

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