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1 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman C.M. Pascual S TATISTICS Chapter 8 Hypothesis Testing

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2 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Fundamentals of Hypothesis Testing 8-3 Testing a Claim about a Mean: Large Samples 8-4 Testing a Claim about a Mean: Small Samples 8-5 Testing a Claim about a Proportion 8-6 Testing a Claim about a Standard Deviation

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3 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 8-1 Overview Definition Hypothesis in statistics, is a claim or statement about a property of a population

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4 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.

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5 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 8-2 Fundamentals of Hypothesis Testing

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6 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 8-1 Central Limit Theorem

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7 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman µ x = 98.6 Figure 8-1 Central Limit Theorem The Expected Distribution of Sample Means Assuming that = 98.6 Likely sample means

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8 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman z = x = or z = 1.96 x = or µ x = 98.6 Figure 8-1 Central Limit Theorem The Expected Distribution of Sample Means Assuming that = 98.6 Likely sample means

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9 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 8-1 Central Limit Theorem The Expected Distribution of Sample Means Assuming that = 98.6 z = x = or z = 1.96 x = or Sample data: z = x = or µ x = 98.6 Likely sample means

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10 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Components of a Formal Hypothesis Test

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11 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Null Hypothesis: H 0 Statement about value of population parameter Must contain condition of equality =, , or Test the Null Hypothesis directly Reject H 0 or fail to reject H 0

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12 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Alternative Hypothesis: H 1 Must be true if H 0 is false , ‘opposite’ of Null

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13 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.

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14 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Note about Testing the Validity of Someone Else’s Claim Someone else’s claim may become the null hypothesis (because it contains equality), and it sometimes becomes the alternative hypothesis (because it does not contain equality).

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15 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Test Statistic a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis

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16 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 z =z = x - µ x n

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17 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis

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18 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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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19 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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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20 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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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21 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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

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22 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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

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23 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Critical Value ( z score ) 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

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24 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Critical Value ( z score ) Fail to reject H 0 Reject H 0 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

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25 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Two-tailed,Right-tailed, Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values.

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26 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Two-tailed Test H 0 : µ = 100 H 1 : µ 100

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27 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Two-tailed Test H 0 : µ = 100 H 1 : µ 100 is divided equally between the two tails of the critical region

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28 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Two-tailed Test H 0 : µ = 100 H 1 : µ 100 Means less than or greater than is divided equally between the two tails of the critical region

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29 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Two-tailed Test H 0 : µ = 100 H 1 : µ 100 Means less than or greater than 100 Values that differ significantly from 100 is divided equally between the two tails of the critical region Fail to reject H 0 Reject H 0

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30 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Right-tailed Test H 0 : µ 100 H 1 : µ > 100

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31 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Right-tailed Test H 0 : µ 100 H 1 : µ > 100 Points Right

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32 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Right-tailed Test H 0 : µ 100 H 1 : µ > 100 Values that differ significantly from Points Right Fail to reject H 0 Reject H 0

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33 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Left-tailed Test H 0 : µ 100 H 1 : µ < 100

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34 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Left-tailed Test H 0 : µ 100 H 1 : µ < 100 Points Left

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35 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Left-tailed Test H 0 : µ 100 H 1 : µ < Values that differ significantly from 100 Points Left Fail to reject H 0 Reject H 0

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36 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Conclusions in Hypothesis Testing always test the null hypothesis 1. Reject the H 0 2. Fail to reject the H 0 need to formulate correct wording of final conclusion See Figure 8-4

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37 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman FIGURE 8-4 Wording of Final Conclusion Does the original claim contain the condition of equality Do you reject H 0 ?. Yes (Original claim contains equality and becomes H 0 ) No (Original claim does not contain equality and becomes H 1 ) Yes (Reject H 0 ) “There is sufficient evidence to warrant rejection of the claim that... (original claim).” “There is not sufficient evidence to warrant rejection of the claim that... (original claim).” “The sample data supports the claim that... (original claim).” “There is not sufficient evidence to support the claim that... (original claim).” Do you reject H 0 ? Yes (Reject H 0 ) No (Fail to reject H 0 ) No (Fail to reject H 0 ) (This is the only case in which the original claim is rejected). (This is the only case in which the original claim is supported). Start

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38 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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)

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39 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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

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40 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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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41 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Table 8-2 Type I and Type II Errors True State of Nature We decide to reject the null hypothesis We fail to reject the null hypothesis The null hypothesis is true The null hypothesis is false Type I error (rejecting a true null hypothesis) Type II error (rejecting a false null hypothesis) Correct decision Correct decision Decision

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42 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.

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43 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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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44 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Steps in Hypothesis Testing 1.State the null and alternative hypothesis; 2.Select the level of significance; 3.Determine the critical value and the rejection region/s; 4.State the decision rule; 5.Compute the test statistics; and 6.Make a decision, whether to reject or not to reject the null hypothesis.

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45 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 1 A manufacturer claims that the average lifetime of his lightbulbs is 3 years or 36 months. The stabdard deviation is 8 months. Fifty (50) bulbs are selected, and the average lifetime is found to be 32 months. Should the manufacturer’s statement be rejected at

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46 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 1 Solution: Step 1. State the hypothesis: –Ho: µ = 36 months –Ha : µ 36 months Step 2. Level of significance Step 3. Determine critical values and rejection region

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47 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 1 Solution: Step 3. Determine critical values and rejection region –Z = +/ (from Appendix B of z values) Step 4. State the decision rule –Reject the null hypothesis if Zc > or Zc =

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48 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 1 Solution: Step 5. Compute the test statistic. Z c = (32-36)/ (8/(50) 0.5 = zc =zc = x - µ x n

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49 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 1 Solution: Step 6. Make a decision. Z c = is less than Z = And it falls in the rejection region in the left tail. Therefore, reject H o and conclude that the average lifetime of lightbulbs is not equal to 36 months.

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50 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 1 Solution: Step 6. Make a decision. Z c = is less than Z = And it falls in the rejection region in the left tail. Therefore, reject H o and conclude that the average lifetime of lightbulbs is not equal to 36 months.

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51 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 2 A test on car braking reaction times for men between 18 to 36 years old have produced a mean and standard deviation of second and second, respectively. When 40 male drivers of this age group were randomly selected and tested for their breaking reaction times, a mean of second came out. At the = 0.10, test the claim of the driving instructor that his graduates had faster reaction times. Z c = is less than Z = And it falls in the rejection region in the left tail. Therefore, reject H o and conclude that the average lifetime of lightbulbs is not equal to 36 months.

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52 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 2 Solution: Step 1. State the hypothesis: –Ho: µ = second –Ha: µ < second Step 2. Level of significance

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53 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 2 Solution: Step 3. Determine critical values and rejection region Z = - 1/.28 (from Appendix B of z values) Step 4. State the decision rule –Reject the null hypothesis if Zc <

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54 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 2 Solution: Step 5. Compute the test statistic. Z c = ( )/ (0.123/(40) 0.5 = zc =zc = x - µ x n

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55 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 2 Solution: Step 6. –Since the test statistics falls within the non-critical region, do not reject Ho. There is enough evidence to support the instructor’s claim.; accept Ho..

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56 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Test on Small Sample Mean The t-test is a statistical test for the mean of a population and is used when the population is normally distributed, σ is unknown, and n < 30. The formula for the t-test with degrees of freedom are d.f. = n – 1 is t =t = x - µ s n

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57 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 3 In order to increase customer service, a muffler repair shop claim its mechanics can replace a muffler in 12 minutes. A time management specialist selected 6 repair jobs and found their mean time to be 11.6 minutes. The standard deviation of the sample was 2.1 minutes. A = 0.025, is there enough evidence to conclude that the mean time in changing a muffler is less than 12 minutes?

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58 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 3 Solution 1.State the hypothesis: 1.Ho: µ = 12 2.Ha: µ < 12 2.Step 2. Level of significance 3.Step 3. Since and d.f. = 6 – 1 = 5, then at Appendix C at t-value =

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59 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 1 Solution 4. Step 4. Reject Ho if t c < Compute for the test statistic t =t = x - µ s

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60 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example 3 t = (11.6 – 12)/(2.1(6) 0.5 = Step 6. Since the critical value fall within the non-critical region, do not reject Ho. Accept Ho.

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61 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Sit Work (Submit after class) 1.A diet clinic states that there is an average loss of 24 pounds for those who stay on the program for 20 weeks. The standard deviation is 5 pounds. The clinic tries a new diet, reducing salt intake to see whether that strategy will produce a greater weight loss. A group of 40 volunteers loses an average of 16.3 pounds each over 20 weeks. Should the clinic change the new diet? Use = 0.05

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62 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Assignment (Submit next meeting) 2. A recent survey stated that household received an average 37 telephone calls per month. To test the claim, a researcher surveyed 29 households and found that the average number of calls was The standard deviation of the sample was 6. At = 0.05, can the claim be substantiated?

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