Elements of Statistics II

Yailin1983

Main

Home >Homework Answsers >Psychology homework help

HOME WORK

see attachment

2 years ago
27.03.2024
5

Report issue

files (2)

Module4ESIIDiscussion.docx
ESIIModule4Class.docx

Module4ESIIDiscussion.docx

Module 4: Discussion Forum: Hypothesis Testing

Instructions

This space has been created for you to share the link of a newspaper or blog article that discusses the importance for companies of using hypothesis testing with your classmates.

1. Write a summary about the blog or article.

2. Write at least two paragraphs.

3. The paragraph must have at least 3 sentences.

4. Read your classmates blogs and select two of them to discuss.

5. Establish a dialogue with them.

General Instructions for the Discussion Forum

1. Post your answer as established by your instructor on the course calendar. Your comments must be written in your own words. You can offer examples and quotes to support your proposals. Citations of other authors must be properly documented (author's name, title, date, etc.).

2. Post your comments to the response of at least two (2) of your classmates on or before the day set by your instructor on the course calendar. Your reaction may be based on personal experiences, study material, or additional information obtained from the Online Library or others, and may include:

· Some understanding received from what is published that synthesizes the information and offers new perspectives or suggestions.

· The validation or rejection of the idea (supported by your experience or research).

· New information that broadens, adds or contrasts perspectives (based on reading and evidence).

3. Remember that your work must be original and must not contain material copied from books or the internet. You must respect the intellectual property of the authors and not commit plagiarism.

4. Examine the criteria used to evaluate your assignment to find out how to get the highest grade for your work. The assignments are graded or evaluated through rubrics or by the distribution of points.

5. Before submitting your entry, read your message several times. This will ensure that it contains the exact information you want to communicate.

Remember to review the academic expectations for your submission.

Submission Instructions:

· Submit your initial discussion post by 11:59 PM Eastern on Wednesday.

· Contribute a minimum of 350 words for your initial post. It should include at least 2 academic sources, formatted and cited in APA.

· Respond to at least two of your classmates' discussion posts by 11:59 PM Eastern on Sunday. Ask a question, and provide a different viewpoint.

ESIIModule4Class.docx

Module 4: Hypothesis Testing

Hypothesis Testing:

With Inferential Statistics we can draw conclusions on unknown population parameters based on the information contained in a sample (estimation and hypothesis testing).

Hypothesis testing is a standard procedure for testing a claim about a property of a population

Parameters are the characteristics or values of the population (examples: population mean “μ” and population standard deviation “σ”).

Statistics are the characteristics or values of the sample (examples: sample mean “x̄” and sample standard deviation “S”)

Hypothesis Testing:

Hypothesis: Is a claim or statement about a property of the population.

Hypothesis Test: (Test of Significance) Is a standard procedure for testing a claim about a property of a population.

Basics of Hypothesis Testing:

1. Given a claim, identify the Null Hypothesis �0 and the Alternative Hypothesis �1.

2. Given a significance level α, identify the critical values.

3. Given a claim and sample data, calculate the value of the test statistic.

4. State the conclusion (reject the Null Hypothesis of fail to reject the Null Hypothesis).

Decision	�0 is True	�0 is not True
Fail to reject �0	Right (Correct Decision)	Type II Error (β)
Reject �0	Type I Error ( α)	Right (Correct Decision)

Type I error is also called “the significance level”

Hypothesis Testing (testing a claim about the population mean μ when the population standard deviation σ is known)

�0: �=�0

�1: �≠�0

�0: �=�0

�1: �>�0

�0: �=�0

�1: �<�0

�>��/2��

�<−��/2

�>��

�<−��

(Triola, 2018)

Example:

A random sample of 49 students indicates these students had a mean grade of 73 points on a final exam, with a standard deviation of 8 points. With a significance level of 5%, is there sufficient evidence to support the claim that the population mean is:

A. equal to 70 points?

B. greater than 75 points?

C. less than 80 points?

Solution:

Given data:

Sample mean: x̄ = 73

Sample standard deviation: S = 8

Sample size: n = 49

Significance level o α = 0.05

Note: since the sample size is greater than 30, then the sample standard deviation is the best estimator of the population standard deviation σ

Part a) equal to 70 points?

1) State the null and alternative hypothesis:

�0: μ = 70

�1: μ ≠ 70

(Two tails hypothesis testing)

Reject �0 if: Zc is < −��/2 or Zc is >��/2

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: = ��=x̄−��/�=73−708/49=38/7=2.63

3) State the rejection region for the statistic using α = 0.05

Zc is < −��/2 or Zc is > ��/2

α = 0.05

α /2 = 0.025

−��/2 = -1.96 and ��/2 = 1.96 (Using the normal distribution table)

Comparing the test statistics with the critical values:

2.63 is not < than -1.96

2.63 > 1.96 (Reject �0)

4) Conclusion: Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is different to 70 points.

Part b) greater than 75 points?

1) State the null and alternative hypothesis:

�0: μ = 75

�1: μ > 75

(One tail -right tail- hypothesis testing)

Reject �0 if: Zc is > ��

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: ��=x̄−��/�=73−758/49=−28/7=−1.41

3) State the rejection region for the statistic using α = 0.05

Zc is > Zα

α = 0.05

��=1.645 (Using the normal distribution table)

Comparing the test statistics with the critical value:

-1.41 < 1.645 (Fail to reject Ho)

4) Conclusion: Fail to reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is not greater than 75 points.

Part c) less than 80 points?

1) State the null and alternative hypothesis:

�0: μ = 80

�1: μ < 80

(one tail -left tail- hypothesis testing)

Reject �0 if: Zc is < -Zα

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: ��=x̄−��/�=73−808/49=−78/7=−4.92

3) State the rejection region for the statistic using α = 0.05

Zc is < -Zα

α = 0.05

��=−1.645 (Using the normal distribution table)

Comparing the test statistics with the critical value:

-4.92 < - 1.645 (Reject �0)

4) Conclusion: Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is less than 80 points.

Hypothesis Testing (testing a claim about the population mean μ when the population standard deviation σ is unknown)

�0: �=�0

�1: �≠�0

�0: �=�0

�1: �>�0

�0: �=�0

�1: �<�0

�>�(�/2;�−1)��

�<−�(�/2;�−1)

�>�(�;�−1)

�<−�(�;�−1)

(Triola, 2018)

Example:

A random sample of 16 students was selected and the mean grade was calculated to be 90 points. Since there is no information about the population standard deviation, the sample standard deviation was also calculated, and the value was 2 points. Assuming that the population grades follow a normal distribution, is there any statistical evidence that the grade population mean is less than 95 points (α = 5%)?

�0: μ = 95

�1: μ < 95

(One tail -left tail- hypothesis testing)

Reject �0 if: tc is < −��,(�−1)

2) State and compute an appropriate test statistic:

��=x̄−��/�=95−902/16=52/4=10

3) State the rejection region for the statistic using α = 0.05

tc is < −��,(�−1)

α = 0.05

n – 1 = 15

�0.05,(15)=−1.753 (Using the normal distribution table)

Comparing the test statistics with the critical value:

10 > - 1.753 (Fail to Reject �0)

4) Conclusion: Fail to Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is not less than 80 points.

References:

Textbooks (Suggested)

Rajaretnam, T. (2016). Statistics for social sciences. Sage Publications, Inc. ISBN-13: 9789351506560

Triola, M. F. (2018). Elementary statistics (13th ed.). Pearson. ISBN-13: 978-0134462455 https://librarylogin-carolina.uagm.edu/login?url=https://search.ebscohost.com/login.aspx?direct=true&db=e000xww&AN=1214457&site=ehost-live&ebv=EB&ppid=pp_Cover

https://youtu.be/-FtlH4svqx4

https://youtu.be/zJ8e_wAWUzE

ESIIModule4Class.docx

Module 4: Hypothesis Testing

Hypothesis Testing:

With Inferential Statistics we can draw conclusions on unknown population parameters based on the information contained in a sample (estimation and hypothesis testing).

Hypothesis testing is a standard procedure for testing a claim about a property of a population

Parameters are the characteristics or values of the population (examples: population mean “μ” and population standard deviation “σ”).

Statistics are the characteristics or values of the sample (examples: sample mean “x̄” and sample standard deviation “S”)

Hypothesis Testing:

Hypothesis: Is a claim or statement about a property of the population.

Hypothesis Test: (Test of Significance) Is a standard procedure for testing a claim about a property of a population.

Basics of Hypothesis Testing:

1. Given a claim, identify the Null Hypothesis �0 and the Alternative Hypothesis �1.

2. Given a significance level α, identify the critical values.

3. Given a claim and sample data, calculate the value of the test statistic.

4. State the conclusion (reject the Null Hypothesis of fail to reject the Null Hypothesis).

Decision	�0 is True	�0 is not True
Fail to reject �0	Right (Correct Decision)	Type II Error (β)
Reject �0	Type I Error ( α)	Right (Correct Decision)

Type I error is also called “the significance level”

Hypothesis Testing (testing a claim about the population mean μ when the population standard deviation σ is known)

�0: �=�0

�1: �≠�0

�0: �=�0

�1: �>�0

�0: �=�0

�1: �<�0

�>��/2��

�<−��/2

�>��

�<−��

(Triola, 2018)

Example:

A. equal to 70 points?

B. greater than 75 points?

C. less than 80 points?

Solution:

Given data:

Sample mean: x̄ = 73

Sample standard deviation: S = 8

Sample size: n = 49

Significance level o α = 0.05

Note: since the sample size is greater than 30, then the sample standard deviation is the best estimator of the population standard deviation σ

Part a) equal to 70 points?

1) State the null and alternative hypothesis:

�0: μ = 70

�1: μ ≠ 70

(Two tails hypothesis testing)

Reject �0 if: Zc is < −��/2 or Zc is >��/2

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: = ��=x̄−��/�=73−708/49=38/7=2.63

3) State the rejection region for the statistic using α = 0.05

Zc is < −��/2 or Zc is > ��/2

α = 0.05

α /2 = 0.025

−��/2 = -1.96 and ��/2 = 1.96 (Using the normal distribution table)

Comparing the test statistics with the critical values:

2.63 is not < than -1.96

2.63 > 1.96 (Reject �0)

4) Conclusion: Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is different to 70 points.

Part b) greater than 75 points?

1) State the null and alternative hypothesis:

�0: μ = 75

�1: μ > 75

(One tail -right tail- hypothesis testing)

Reject �0 if: Zc is > ��

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: ��=x̄−��/�=73−758/49=−28/7=−1.41

3) State the rejection region for the statistic using α = 0.05

Zc is > Zα

α = 0.05

��=1.645 (Using the normal distribution table)

Comparing the test statistics with the critical value:

-1.41 < 1.645 (Fail to reject Ho)

4) Conclusion: Fail to reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is not greater than 75 points.

Part c) less than 80 points?

1) State the null and alternative hypothesis:

�0: μ = 80

�1: μ < 80

(one tail -left tail- hypothesis testing)

Reject �0 if: Zc is < -Zα

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: ��=x̄−��/�=73−808/49=−78/7=−4.92

3) State the rejection region for the statistic using α = 0.05

Zc is < -Zα

α = 0.05

��=−1.645 (Using the normal distribution table)

Comparing the test statistics with the critical value:

-4.92 < - 1.645 (Reject �0)

4) Conclusion: Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is less than 80 points.

Hypothesis Testing (testing a claim about the population mean μ when the population standard deviation σ is unknown)

�0: �=�0

�1: �≠�0

�0: �=�0

�1: �>�0

�0: �=�0

�1: �<�0

�>�(�/2;�−1)��

�<−�(�/2;�−1)

�>�(�;�−1)

�<−�(�;�−1)

(Triola, 2018)

Example:

�0: μ = 95

�1: μ < 95

(One tail -left tail- hypothesis testing)

Reject �0 if: tc is < −��,(�−1)

2) State and compute an appropriate test statistic:

��=x̄−��/�=95−902/16=52/4=10

3) State the rejection region for the statistic using α = 0.05

tc is < −��,(�−1)

α = 0.05

n – 1 = 15

�0.05,(15)=−1.753 (Using the normal distribution table)

Comparing the test statistics with the critical value:

10 > - 1.753 (Fail to Reject �0)

4) Conclusion: Fail to Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is not less than 80 points.

References:

Textbooks (Suggested)

Rajaretnam, T. (2016). Statistics for social sciences. Sage Publications, Inc. ISBN-13: 9789351506560

https://youtu.be/-FtlH4svqx4

https://youtu.be/zJ8e_wAWUzE

Module4ESIIDiscussion.docx

Module 4: Discussion Forum: Hypothesis Testing

Instructions

This space has been created for you to share the link of a newspaper or blog article that discusses the importance for companies of using hypothesis testing with your classmates.

1. Write a summary about the blog or article.

2. Write at least two paragraphs.

3. The paragraph must have at least 3 sentences.

4. Read your classmates blogs and select two of them to discuss.

5. Establish a dialogue with them.

General Instructions for the Discussion Forum

· Some understanding received from what is published that synthesizes the information and offers new perspectives or suggestions.

· The validation or rejection of the idea (supported by your experience or research).

· New information that broadens, adds or contrasts perspectives (based on reading and evidence).

3. Remember that your work must be original and must not contain material copied from books or the internet. You must respect the intellectual property of the authors and not commit plagiarism.

5. Before submitting your entry, read your message several times. This will ensure that it contains the exact information you want to communicate.

Remember to review the academic expectations for your submission.

Submission Instructions:

· Submit your initial discussion post by 11:59 PM Eastern on Wednesday.

· Contribute a minimum of 350 words for your initial post. It should include at least 2 academic sources, formatted and cited in APA.

· Respond to at least two of your classmates' discussion posts by 11:59 PM Eastern on Sunday. Ask a question, and provide a different viewpoint.

ESIIModule4Class.docx

Module 4: Hypothesis Testing

Hypothesis Testing:

With Inferential Statistics we can draw conclusions on unknown population parameters based on the information contained in a sample (estimation and hypothesis testing).

Hypothesis testing is a standard procedure for testing a claim about a property of a population

Parameters are the characteristics or values of the population (examples: population mean “μ” and population standard deviation “σ”).

Statistics are the characteristics or values of the sample (examples: sample mean “x̄” and sample standard deviation “S”)

Hypothesis Testing:

Hypothesis: Is a claim or statement about a property of the population.

Hypothesis Test: (Test of Significance) Is a standard procedure for testing a claim about a property of a population.

Basics of Hypothesis Testing:

1. Given a claim, identify the Null Hypothesis �0 and the Alternative Hypothesis �1.

2. Given a significance level α, identify the critical values.

3. Given a claim and sample data, calculate the value of the test statistic.

4. State the conclusion (reject the Null Hypothesis of fail to reject the Null Hypothesis).

Decision	�0 is True	�0 is not True
Fail to reject �0	Right (Correct Decision)	Type II Error (β)
Reject �0	Type I Error ( α)	Right (Correct Decision)

Type I error is also called “the significance level”

Hypothesis Testing (testing a claim about the population mean μ when the population standard deviation σ is known)

�0: �=�0

�1: �≠�0

�0: �=�0

�1: �>�0

�0: �=�0

�1: �<�0

�>��/2��

�<−��/2

�>��

�<−��

(Triola, 2018)

Example:

A. equal to 70 points?

B. greater than 75 points?

C. less than 80 points?

Solution:

Given data:

Sample mean: x̄ = 73

Sample standard deviation: S = 8

Sample size: n = 49

Significance level o α = 0.05

Note: since the sample size is greater than 30, then the sample standard deviation is the best estimator of the population standard deviation σ

Part a) equal to 70 points?

1) State the null and alternative hypothesis:

�0: μ = 70

�1: μ ≠ 70

(Two tails hypothesis testing)

Reject �0 if: Zc is < −��/2 or Zc is >��/2

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: = ��=x̄−��/�=73−708/49=38/7=2.63

3) State the rejection region for the statistic using α = 0.05

Zc is < −��/2 or Zc is > ��/2

α = 0.05

α /2 = 0.025

−��/2 = -1.96 and ��/2 = 1.96 (Using the normal distribution table)

Comparing the test statistics with the critical values:

2.63 is not < than -1.96

2.63 > 1.96 (Reject �0)

4) Conclusion: Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is different to 70 points.

Part b) greater than 75 points?

1) State the null and alternative hypothesis:

�0: μ = 75

�1: μ > 75

(One tail -right tail- hypothesis testing)

Reject �0 if: Zc is > ��

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: ��=x̄−��/�=73−758/49=−28/7=−1.41

3) State the rejection region for the statistic using α = 0.05

Zc is > Zα

α = 0.05

��=1.645 (Using the normal distribution table)

Comparing the test statistics with the critical value:

-1.41 < 1.645 (Fail to reject Ho)

4) Conclusion: Fail to reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is not greater than 75 points.

Part c) less than 80 points?

1) State the null and alternative hypothesis:

�0: μ = 80

�1: μ < 80

(one tail -left tail- hypothesis testing)

Reject �0 if: Zc is < -Zα

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: ��=x̄−��/�=73−808/49=−78/7=−4.92

3) State the rejection region for the statistic using α = 0.05

Zc is < -Zα

α = 0.05

��=−1.645 (Using the normal distribution table)

Comparing the test statistics with the critical value:

-4.92 < - 1.645 (Reject �0)

4) Conclusion: Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is less than 80 points.

Hypothesis Testing (testing a claim about the population mean μ when the population standard deviation σ is unknown)

�0: �=�0

�1: �≠�0

�0: �=�0

�1: �>�0

�0: �=�0

�1: �<�0

�>�(�/2;�−1)��

�<−�(�/2;�−1)

�>�(�;�−1)

�<−�(�;�−1)

(Triola, 2018)

Example:

�0: μ = 95

�1: μ < 95

(One tail -left tail- hypothesis testing)

Reject �0 if: tc is < −��,(�−1)

2) State and compute an appropriate test statistic:

��=x̄−��/�=95−902/16=52/4=10

3) State the rejection region for the statistic using α = 0.05

tc is < −��,(�−1)

α = 0.05

n – 1 = 15

�0.05,(15)=−1.753 (Using the normal distribution table)

Comparing the test statistics with the critical value:

10 > - 1.753 (Fail to Reject �0)

4) Conclusion: Fail to Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is not less than 80 points.

References:

Textbooks (Suggested)

Rajaretnam, T. (2016). Statistics for social sciences. Sage Publications, Inc. ISBN-13: 9789351506560

https://youtu.be/-FtlH4svqx4

https://youtu.be/zJ8e_wAWUzE

Module4ESIIDiscussion.docx

Module 4: Discussion Forum: Hypothesis Testing

Instructions

This space has been created for you to share the link of a newspaper or blog article that discusses the importance for companies of using hypothesis testing with your classmates.

1. Write a summary about the blog or article.

2. Write at least two paragraphs.

3. The paragraph must have at least 3 sentences.

4. Read your classmates blogs and select two of them to discuss.

5. Establish a dialogue with them.

General Instructions for the Discussion Forum

· Some understanding received from what is published that synthesizes the information and offers new perspectives or suggestions.

· The validation or rejection of the idea (supported by your experience or research).

· New information that broadens, adds or contrasts perspectives (based on reading and evidence).

3. Remember that your work must be original and must not contain material copied from books or the internet. You must respect the intellectual property of the authors and not commit plagiarism.

5. Before submitting your entry, read your message several times. This will ensure that it contains the exact information you want to communicate.

Remember to review the academic expectations for your submission.

Submission Instructions:

· Submit your initial discussion post by 11:59 PM Eastern on Wednesday.

· Contribute a minimum of 350 words for your initial post. It should include at least 2 academic sources, formatted and cited in APA.

· Respond to at least two of your classmates' discussion posts by 11:59 PM Eastern on Sunday. Ask a question, and provide a different viewpoint.

ESIIModule4Class.docx

Module 4: Hypothesis Testing

Hypothesis Testing:

With Inferential Statistics we can draw conclusions on unknown population parameters based on the information contained in a sample (estimation and hypothesis testing).

Hypothesis testing is a standard procedure for testing a claim about a property of a population

Parameters are the characteristics or values of the population (examples: population mean “μ” and population standard deviation “σ”).

Statistics are the characteristics or values of the sample (examples: sample mean “x̄” and sample standard deviation “S”)

Hypothesis Testing:

Hypothesis: Is a claim or statement about a property of the population.

Hypothesis Test: (Test of Significance) Is a standard procedure for testing a claim about a property of a population.

Basics of Hypothesis Testing:

1. Given a claim, identify the Null Hypothesis �0 and the Alternative Hypothesis �1.

2. Given a significance level α, identify the critical values.

3. Given a claim and sample data, calculate the value of the test statistic.

4. State the conclusion (reject the Null Hypothesis of fail to reject the Null Hypothesis).

Decision	�0 is True	�0 is not True
Fail to reject �0	Right (Correct Decision)	Type II Error (β)
Reject �0	Type I Error ( α)	Right (Correct Decision)

Type I error is also called “the significance level”

Hypothesis Testing (testing a claim about the population mean μ when the population standard deviation σ is known)

�0: �=�0

�1: �≠�0

�0: �=�0

�1: �>�0

�0: �=�0

�1: �<�0

�>��/2��

�<−��/2

�>��

�<−��

(Triola, 2018)

Example:

A. equal to 70 points?

B. greater than 75 points?

C. less than 80 points?

Solution:

Given data:

Sample mean: x̄ = 73

Sample standard deviation: S = 8

Sample size: n = 49

Significance level o α = 0.05

Note: since the sample size is greater than 30, then the sample standard deviation is the best estimator of the population standard deviation σ

Part a) equal to 70 points?

1) State the null and alternative hypothesis:

�0: μ = 70

�1: μ ≠ 70

(Two tails hypothesis testing)

Reject �0 if: Zc is < −��/2 or Zc is >��/2

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: = ��=x̄−��/�=73−708/49=38/7=2.63

3) State the rejection region for the statistic using α = 0.05

Zc is < −��/2 or Zc is > ��/2

α = 0.05

α /2 = 0.025

−��/2 = -1.96 and ��/2 = 1.96 (Using the normal distribution table)

Comparing the test statistics with the critical values:

2.63 is not < than -1.96

2.63 > 1.96 (Reject �0)

4) Conclusion: Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is different to 70 points.

Part b) greater than 75 points?

1) State the null and alternative hypothesis:

�0: μ = 75

�1: μ > 75

(One tail -right tail- hypothesis testing)

Reject �0 if: Zc is > ��

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: ��=x̄−��/�=73−758/49=−28/7=−1.41

3) State the rejection region for the statistic using α = 0.05

Zc is > Zα

α = 0.05

��=1.645 (Using the normal distribution table)

Comparing the test statistics with the critical value:

-1.41 < 1.645 (Fail to reject Ho)

4) Conclusion: Fail to reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is not greater than 75 points.

Part c) less than 80 points?

1) State the null and alternative hypothesis:

�0: μ = 80

�1: μ < 80

(one tail -left tail- hypothesis testing)

Reject �0 if: Zc is < -Zα

2) State and compute an appropriate test statistic:

Per the Central Limit Theorem: ��=x̄−��/�=73−808/49=−78/7=−4.92

3) State the rejection region for the statistic using α = 0.05

Zc is < -Zα

α = 0.05

��=−1.645 (Using the normal distribution table)

Comparing the test statistics with the critical value:

-4.92 < - 1.645 (Reject �0)

4) Conclusion: Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is less than 80 points.

Hypothesis Testing (testing a claim about the population mean μ when the population standard deviation σ is unknown)

�0: �=�0

�1: �≠�0

�0: �=�0

�1: �>�0

�0: �=�0

�1: �<�0

�>�(�/2;�−1)��

�<−�(�/2;�−1)

�>�(�;�−1)

�<−�(�;�−1)

(Triola, 2018)

Example:

�0: μ = 95

�1: μ < 95

(One tail -left tail- hypothesis testing)

Reject �0 if: tc is < −��,(�−1)

2) State and compute an appropriate test statistic:

��=x̄−��/�=95−902/16=52/4=10

3) State the rejection region for the statistic using α = 0.05

tc is < −��,(�−1)

α = 0.05

n – 1 = 15

�0.05,(15)=−1.753 (Using the normal distribution table)

Comparing the test statistics with the critical value:

10 > - 1.753 (Fail to Reject �0)

4) Conclusion: Fail to Reject the null hypothesis, meaning that with a significance level of 5%, there is statistical evidence that the population mean is not less than 80 points.

References:

Textbooks (Suggested)

Rajaretnam, T. (2016). Statistics for social sciences. Sage Publications, Inc. ISBN-13: 9789351506560

https://youtu.be/-FtlH4svqx4

https://youtu.be/zJ8e_wAWUzE

other Questions(10)