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7 Elementary Statistics Hypothesis Testing

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Introduction to Hypothesis Testing Section 7.1

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A statistical hypothesis is a claim about a population. Alternative hypothesis H a The claim, that if accepted, will replace the status quo Null hypothesis H 0 The status quo or assumed belief

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Write the claim about the population. Then, write its complement. A hospital claims its ambulance response time is less than 10 minutes. Writing Hypotheses claim assumed status quo

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Begin by assuming the null hypothesis is true. Hypothesis Test Strategy Collect data from a random sample taken from the population and calculate the necessary sample statistics. If the sample statistic has a low probability of being drawn from a population in which the null hypothesis is true, you will reject H 0. (As a consequence, you will support the alternative hypothesis.) If the probability is not low enough, fail to reject H 0.

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A type I error: Null hypothesis is actually true but the decision is to reject it. Level of significance, Maximum probability of committing a type I error. Actual Truth of H 0 Errors and Level of Significance H 0 True H 0 False Do not reject H 0 Reject H 0 Correct Decision Correct Decision Type II Error Type I Error Decision

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Right-tail test Two-tail test Left-tail test Types of Hypothesis Tests H a is more probable

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1. Write the null and alternative hypothesis. 2. State the level of significance. 3. Identify the sampling distribution. Write H 0 and H a as mathematical statements. This is the maximum probability of rejecting the null hypothesis when it is actually true. (Making a type I error.) The sampling distribution is the distribution for the test statistic assuming that the condition in H 0 is true and that the experiment is repeated an infinite number of times. Steps in a Hypothesis Test

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4. Find the test statistic and standardize it. Perform the calculations to standardize your sample statistic. 5. Calculate the P-value for the test statistic. This is the probability of obtaining your test statistic or one that is more extreme from the sampling distribution.

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If the Z score is greater than Z c, reject H 0. If the Z score is less than Z c, fail to reject H 0. 7. Make your decision. 8. Interpret your decision. Since the claim is the alternative hypothesis, you will eit her support the claim or determine there is not enough evidence to support the claim. 6. Find the critical values.

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Sampling distribution for The rejection region is the range of values for which the null hypothesis is not probable. It is always in the direction of the alternative hypothesis. Its area is equal to . A critical value separates the rejection region from the non-rejection region. Rejection Regions Rejection Region Critical Value z c zzczc

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The critical value z c separates the rejection region from the non-rejection region. The area of the rejection region is . Find z c for a left-tail test with =.01. Find z c for a right-tail test with =.05. Find –z c and z c for a two-tail test with =.01. -z c = –2.33 z c = 2.575 z c = 1.645 Critical Values -z c zczc Rejection region Rejection region zczc -z c Rejection region Rejection region –z c = –2.575

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Hypothesis Testing for the Mean (n 30) Section 7.2

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The z-Test for a Mean The z-test is a statistical test for a population mean. The z-test can be used: (1) if the population is normal and s is known or (2) when the sample size, n, is at least 30. The test statistic is the sample mean and the standardized test statistic is z. When n 30, use s in place of.

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A health group claims the mean sodium content in one serving of a cereal is greater than 230 mg. You work for a national health service and are asked to test this claim. You find that a random sample of 52 servings has a mean sodium content of 232 mg and a standard deviation of 10 mg. At = 0.05, do you have enough evidence to accept the group’s claim? 1. Write the null and alternative hypothesis. 2. State the level of significance. = 0.05 3. Determine the sampling distribution. Since the sample size is at least 30, the sampling distribution is normal. The z-Test for a Mean (P-value)

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4. Find the test statistic and standardize it. 5. Calculate the P-value for the test statistic. Since this is a right-tail test, the P-value is the area found to the right of z = 1.44 in the normal distribution. From the table P = 1 – 0.9251 n = 52 s = 10 Test statistic z = 1.44 Area in right tail P = 0.0749.

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7. Make your decision. 8. Interpret your decision. Rejection region 6. Find the critical values. Since H a contains the > symbol, this is a right-tail test. z = 1.44 < Z c = 1.645 thus does not fall in the rejection region, so fail to reject H 0 There is not enough evidence to accept the group’s claim that there is greater than 230 mg of sodium in one serving of its cereal. 1.645 Z c = = 0.05 so find 0.0500 in the z table which corresponds to a Z c of 1.645

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Hypothesis Testing for the Mean (n < 30) Section 7.3

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Find the critical value t 0 for a left-tailed test given = 0.01 and n = 18. Find the critical values –t 0 and t 0 for a two-tailed test given d.f. = 18 – 1 = 17 t 0 t 0 = –2.567 d.f. = 11 – 1 = 10 –t 0 = –2.228 and t 0 = 2.228 The t Sampling Distribution = 0.05 and n = 11. Area in left tail t 0

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A university says the mean number of classroom hours per week for full-time faculty is 11.0. A random sample of the number of classroom hours for full-time faculty for one week is listed below. You work for a student organization and are asked to test this claim. At = 0.01, do you have enough evidence to reject the university’s claim? 11.8 8.6 12.6 7.9 6.4 10.4 13.6 9.1 1. Write the null and alternative hypothesis 2. State the level of significance = 0.01 3. Determine the sampling distribution Since the sample size is 8, the sampling distribution is a t-distribution with 8 – 1 = 7 d.f. Testing –Small Sample

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t = –1.08 does not fall in the rejection region, so fail to reject H 0 at = 0.01 n = 8 = 10.050 s = 2.485 7. Make your decision. 6. Find the test statistic and standardize it 8. Interpret your decision. There is not enough evidence to reject the university’s claim that faculty spend a mean of 11 classroom hours. 5. Find the rejection region. Since H a contains the ≠ symbol, this is a two-tail test. 4. Find the critical values. –3.4993.499 t0t0 –t0–t0

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