Section 8-2 Basics of Hypothesis Testing

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Section 8-2 Basics of Hypothesis

Testing


Rare Event Rule for Inferential Statistics

If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct.


Components of aFormal Hypothesis

Test


Null Hypothesis:

• The null hypothesis (denoted by ) is a statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value.

• We test the null hypothesis directly.

• Either reject or fail to reject .

0H

0H

0H 0H


Alternative Hypothesis:

• The alternative hypothesis (denoted by ) is the statement that the parameter has a value that somehow differs from the null hypothesis.

• The symbolic form of the alternative hypothesis must use one of these symbols: .

1H

1 a AH or H or H

, ,


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.


Note about Identifying and

Figure 8-2

1H0H


Example:

a) The mean annual income of employees who took a statistics course is greater than $60,000.

b) The proportion of people aged 18 to 25 who currently use illicit drugs is equal to 0.20.

c) The standard deviation of human body temperatures is equal to 0.62 degrees F.

d) The majority of college students have credit cards.

e) The standard deviaton of duration times of the Old Faithful geyser is less than 40 sec.

f) The mean weight of airline passengers (including carry-on bags) is at most 195 lbs.


The test statistic is a value used in making a decision about the null hypothesis, and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true.

Test Statistic


Test Statistic - Formulas

Test statistic for proportion

Test statistic for standard deviation

p̂ pz

pq

n

Test statistic for mean

or x x

z ts

n n

22

2

1n s


Critical Region

The critical region (or rejection region) is the set of all values of the test statistic that cause us to reject the null hypothesis. For example, see the red-shaded region in the previous figure.


Significance Level

The significance level (denoted by ) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. This is the same introduced in Section 7-2. Common choices for are 0.05, 0.01, and 0.10.


P-ValueThe P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true.

Critical region in the left tail:

Critical region in the right tail:

Critical region in two tails:

P-value = area to the left of the test statistic

P-value = area to the right of the test statistic

P-value = twice the area in the tail beyond the test statistic


Caution

Don’t confuse a P-value with a proportion p.Know this distinction:

P-value = probability of getting a test statistic at least as extreme as the one representing sample data

p = population proportion


P-value method:Using the significance level :

If P-value , reject .

If P-value , fail to reject .

Decision Criterion

0H

0H


Decision CriterionFind the P-value and determine the decision about the

null hypothesis if the test is conducted at the 0.05 level of significance.

a) H1: < 95 and z = -1.89

b) H1: p is not equal to 0.6 and z = 1.62

c) H1: > 41 and z = 2.05


Wording of Final Conclusion

Figure 8-7


Caution

Never conclude a hypothesis test with a statement of “reject the null hypothesis” or “fail to reject the null hypothesis.” Always make sense of the conclusion with a statement that uses simple nontechnical wording that addresses the original claim.


Example

State the final conclusion.

a) Original claim: The percentage of blue M&Ms is greater than 5%.

Decision: Fail to reject the null hypothesis.

a) Original claim: The percentage of Americans who know their credit score is equal to 20%.

Decision: Fail to reject the null hypothesis.

a) Original claim: The percentage of on-time U.S. airline flights is less than 75%.

Decision: Reject the null hypothesis.


Type I and Type II Errors


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.


Hypothesis Test –P-Value Method

• State the null and alternative hypotheses in symbolic form.• Select the level of significance based on the seriousness of a type 1 error. The values of 0.05 and 0.01 are most common.• Identify the statistic that is relevant to this test and determine its sampling distribution (such as normal z, t, or chi-square.)• Find the test statistic and find the P-value. Draw a graph and show the test statistic and P-value. Technology may be used for this step.• Reject H0 if the P-value is less than or equal to the level of significance, , or Fail to reject H0 if the P-value is greater than .• Restate the previous decision in simple non-technical terms, and address the original claim.

Section 8-2 Basics of Hypothesis Testing

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