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**Thinking About Inference**

BPS chapter 15 © 2010 W.H. Freeman and Company

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Significance level A certain manufacturer of paints uses an additive to get the drying time for a specific paint to be 75 minutes. If there’s too much additive, the drying time could be longer than specified but too little additive will decrease the drying time. In testing the amount of additive, they use these hypotheses: H0: = 7 ml vs. Ha: 7 ml. Which of the following would be an implication of having a small ? Concluding that the mean amount of additive is different from 7 ml more often. Concluding that the mean amount of additive is not different from 7 ml more often.

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**Significance level (answer)**

A certain manufacturer of paints uses an additive to get the drying time for a specific paint to be 75 minutes. If there’s too much additive, the drying time could be longer than specified but too little additive will decrease the drying time. In testing the amount of additive, they use these hypotheses: H0: = 7 ml vs. Ha: 7 ml. Which of the following would be an implication of having a small ? Concluding that the mean amount of additive is different from 7 ml more often. Concluding that the mean amount of additive is not different from 7 ml more often.

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**Statistical significance**

A group of researchers wanted to know if there was a difference in average yearly income taxes paid between residents of two very large cities in the midwestern United States. The average for the first city was $6,505 and for the second city, it was $6,511. The difference provided a P-value of Were these results statistically significant? No, because a $6 difference is probably too small to really matter. No, because the P-value is small. Yes, because the P-value is small. Yes, because the difference of $6 is bigger than 0.

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**Statistical significance (answer)**

A group of researchers wanted to know if there was a difference in average yearly income taxes paid between residents of two very large cities in the midwestern United States. The average for the first city was $6,505 and for the second city, it was $6,511. The difference provided a P-value of Were these results statistically significant? No, because a $6 difference is probably too small to really matter. No, because the P-value is small. Yes, because the P-value is small. Yes, because the difference of $6 is bigger than 0.

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**Practical significance**

A group of researchers wanted to know if there was a difference in average yearly income taxes paid between residents of two very large cities in the midwestern United States. The average for the first city was $6,505 and for the second city, it was $6,511. The difference provided a P-value of Were these results practically significant? No, because a $6 difference is probably too small to really matter. No, because the P-value is small. Yes, because the P-value is small. Yes, because the difference of $6 is bigger than 0.

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**Practical significance (answer)**

A group of researchers wanted to know if there was a difference in average yearly income taxes paid between residents of two very large cities in the midwestern United States. The average for the first city was $6,505 and for the second city, it was $6,511. The difference provided a P-value of Were these results practically significant? No, because a $6 difference is probably too small to really matter. No, because the P-value is small. Yes, because the P-value is small. Yes, because the difference of $6 is bigger than 0.

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**Type I error Which of the following defines Type I error?**

Reject H0 when H0 is true. Reject H0 when H0 is false. Do not reject H0 when H0 is true. Do not reject H0 when H0 is false.

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**Type I error (answer) Which of the following defines Type I error?**

Reject H0 when H0 is true. Reject H0 when H0 is false. Do not reject H0 when H0 is true. Do not reject H0 when H0 is false.

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**Type II error Which of the following defines Type II error?**

Reject H0 when H0 is true. Reject H0 when H0 is false. Do not reject H0 when H0 is true. Do not reject H0 when H0 is false.

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**Type II error (answer) Which of the following defines Type II error?**

Reject H0 when H0 is true. Reject H0 when H0 is false. Do not reject H0 when H0 is true. Do not reject H0 when H0 is false.

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Type I and Type II error If we fail to reject the null hypothesis, we may have made A Type I error. A Type II error. Either a Type I or Type II error.

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**Type I and Type II error (answer)**

If we fail to reject the null hypothesis, we may have made A Type I error. A Type II error. Either a Type I or Type II error.

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Type I error Suppose that a regulatory agency will propose that Congress cut federal funding to a metropolitan area if its mean level of NOx is unsafe—that is, if it exceeds 5.0 ppt. The agency gathers sample NOx concentrations on 60 different days and calculates a test of significance to assess whether the mean level of NOx is greater than 5.0 ppt. What is a Type I error in context? Believing the mean level of NOx exceeds 5.0 ppt when it really does. Believing the mean level of NOx exceeds 5.0 ppt when it really does not. Believing the mean level of NOx is 5.0 ppt or less when it really is. Believing the mean level of NOx is 5.0 ppt or less when it really is not.

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Type I error (answer) Suppose that a regulatory agency will propose that Congress cut federal funding to a metropolitan area if its mean level of NOx is unsafe—that is, if it exceeds 5.0 ppt. The agency gathers sample NOx concentrations on 60 different days and calculates a test of significance to assess whether the mean level of NOx is greater than 5.0 ppt. What is a Type I error in context? Believing the mean level of NOx exceeds 5.0 ppt when it really does. Believing the mean level of NOx exceeds 5.0 ppt when it really does not. Believing the mean level of NOx is 5.0 ppt or less when it really is. Believing the mean level of NOx is 5.0 ppt or less when it really is not.

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Type II error Suppose that a regulatory agency will propose that Congress cut federal funding to a metropolitan area if its mean level of NOx is unsafe—that is, if it exceeds 5.0 ppt. The agency gathers sample NOx concentrations on 60 different days and calculates a test of significance to assess whether the mean level of NOx is greater than 5.0 ppt. Which of the following best describes the implications of a Type II error? Cutting federal funding when in fact the level of NOx is greater than 5.0 ppt. Cutting federal funding when in fact the level of NOx is equal to 5.0 ppt or less. Providing federal funding when in fact the level of NOx is greater than 5.0 ppt. Providing federal funding when in fact the level of NOx is equal to 5.0 ppt or less.

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Type II error (answer) Suppose that a regulatory agency will propose that Congress cut federal funding to a metropolitan area if its mean level of NOx is unsafe—that is, if it exceeds 5.0 ppt. The agency gathers sample NOx concentrations on 60 different days and calculates a test of significance to assess whether the mean level of NOx is greater than 5.0 ppt. Which of the following best describes the implications of a Type II error? Cutting federal funding when in fact the level of NOx is greater than 5.0 ppt. Cutting federal funding when in fact the level of NOx is equal to 5.0 ppt or less. Providing federal funding when in fact the level of NOx is greater than 5.0 ppt. Providing federal funding when in fact the level of NOx is equal to 5.0 ppt or less.

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Type I and Type II error Suppose that a regulatory agency will propose that Congress cut federal funding to a metropolitan area if its mean level of NOx is unsafe—that is, if it exceeds 5.0 ppt. The agency gathers sample NOx concentrations on 60 different days and calculates a test of significance to assess whether the mean level of NOx is greater than 5.0 ppt. Suppose the agency concludes that the mean level of NOx exceeds 5.0 ppt. Which of the following is true? Neither a Type I error nor a Type II error could have been committed. We definitely did not make a Type I error, but a Type II error may have been committed. We definitely did not make a Type II error, but a Type I error may have been committed. We may have made both a Type I error and a Type II error.

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**Type I and Type II error (answer)**

Suppose that a regulatory agency will propose that Congress cut federal funding to a metropolitan area if its mean level of NOx is unsafe—that is, if it exceeds 5.0 ppt. The agency gathers sample NOx concentrations on 60 different days and calculates a test of significance to assess whether the mean level of NOx is greater than 5.0 ppt. Suppose the agency concludes that the mean level of NOx exceeds 5.0 ppt. Which of the following is true? Neither a Type I error nor a Type II error could have been committed. We definitely did not make a Type I error, but a Type II error may have been committed. We definitely did not make a Type II error, but a Type I error may have been committed. We may have made both a Type I error and a Type II error.

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**Significance level The significance level is A Type I error.**

A Type II error. The probability of a Type I error. The probability of a Type II error. The power of a test.

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**Significance level (answer)**

The significance level is A Type I error. A Type II error. The probability of a Type I error. The probability of a Type II error. The power of a test.

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** is A Type I error. A Type II error.**

The probability of a Type I error. The probability of a Type II error. The power of a test.

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** (answer) is A Type I error. A Type II error.**

The probability of a Type I error. The probability of a Type II error. The power of a test.

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**1 - 1 - is A Type I error. A Type II error.**

The probability of a Type I error. The probability of a Type II error. The power of a test.

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**1 - (answer) 1 - is A Type I error. A Type II error.**

The probability of a Type I error. The probability of a Type II error. The power of a test.

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Type II error Suppose we set our significance level to be = To decrease the probability of committing a Type II error, we can Increase our sample size. Decrease our sample size.

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Type II error (answer) Suppose we set our significance level to be = To decrease the probability of committing a Type II error, we can Increase our sample size. Decrease our sample size.

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**Significance level and power**

True or False: As the significance level for a test is decreased, the power is increased. True False

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**Significance level and power (answer)**

True or False: As the significance level for a test is decreased, the power is increased. True False

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Significance level True or False: As the significance level for a test is decreased, the probability of making a Type I error is increased. True False

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**Significance level (answer)**

True or False: As the significance level for a test is decreased, the probability of making a Type I error is increased. True False

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Significance level True or False: As the significance level for a test is decreased, the probability of making a Type II error is increased. True False

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**Significance level (answer)**

True or False: As the significance level for a test is decreased, the probability of making a Type II error is increased. True False

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**Significance level and power**

True or False: In a significance test with = 0.05, if n is increased, then the power of the test increases. True False

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**Significance level and power (answer)**

True or False: In a significance test with = 0.05, if n is increased, then the power of the test increases. True False

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**Significance level and power**

True or False: For small n, is approximately equal to 1 - . True False

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**Significance level and power (answer)**

True or False: For small n, is approximately equal to 1 - . True False

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**Significance level and power**

True or False: For large n (i.e., n > 30), is approximately equal to 1 - . True False

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**Significance level and power (answer)**

True or False: For large n (i.e., n > 30), is approximately equal to 1 - . True False

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Significance level True or False: In a significance test with = 0.05, if n is increased, then increases. True False

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**Significance level (answer)**

True or False: In a significance test with = 0.05, if n is increased, then increases. True False

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Type I and Type II Errors. For type I and type II errors, we must know the null and alternate hypotheses. H 0 : µ = 40 The mean of the population is 40.

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