PSY230_M3 Assignment_Spring24 a

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Arizona State University *

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Statistics

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Apr 3, 2024

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PSY230_Bienemann Spring 2024 M3 Assignment (30 points) Things to Review Before Starting the Assignment: 1. Complete the guided Practice Problems first if you have not already done so. It will help you become proficient in answering the questions in this assignment. 2. Check out the module, Assignment Guides , which contains tutorials on how to show the path/process work, essential procedures in Word and Excel, and APA formatting examples. How to Successfully Complete this Assignment: 1. Use this Word document to fill in the answers to the questions. You must type out a clear answer to each question, even if the answer is also contained in the Excel file submitted to show work . 2. Download the Excel file for this assignment and use that data set to answer questions throughout the assignment. 3. All calculation processes must be included to receive full credit. You can use the (1) standard calculation method or (2) alternative excel method (see ‘Assignment Guides’ module > ‘Showing Math Work’). Note: To avoid losing any unnecessary points, be sure to take note of all point deductions and totals found in red underneath each question to ensure you meet the required criteria. Q1. Hypothesis Testing with a Z test (10 points total) A research institute examined the IQs (intelligence quotients) of Americans and reported a mean ( μ ) of 115 points and a standard deviation ( σ ) of 15 points. But I wonder if the statistics in my city, Chicago, would be different from the national average. Specifically, I have a hunch that individuals in Chicago might have higher IQs compared to the national average. To test this idea, I randomly picked 15 friends of mine living in Chicago and asked them their individual IQs. The data set can be found on sheet Q1 in the M3_Data Excel file. I would like to perform a Z test to see if the IQs in Chicago (represented by the sample of 15 individuals) are significantly higher than the national average IQ. The significance level for my Z test was set at α = .05. The hypotheses should be directional because I am predicting a specific direction of difference, and the test should be one-tailed. Q1a. What is the dependent variable in this study? [1 point] The dependent variable in this study is the IQ’s. Q1b. What should be my null and alternative hypotheses? State each hypothesis using both words and statistical symbol notation. [2 points] 1 for each hypothesis: 0.5 for written, 0.5 for notation Written Symbolic Symbol bank for copy & pasting:  =      Null The average IQ of people in Chicago is not going to be higher than the H 0 :  IQ American  IQ Chicago All course content and materials, including this assignment and all other items published within the Canvas course shell, are copyrighted materials and students may not share outside the class, upload to online websites not approved by the instructor, sell, or distribute course content or notes taken during the conduct of the course (see ACD 304–06 , “Commercial Note Taking Services” and ABOR Policy 5-308 F.14 for more information). Students must refrain from uploading to any course shell, discussion board, or website used by the course instructor or other course forum, material that is not the student’s original work, unless the students first comply with all applicable copyright laws; faculty members reserve the right to delete materials on the grounds of suspected copyright infringement.

PSY230_Bienemann Spring 2024 national average. Alternative The Average IQ of people In Chicago Is higher than the national average. H 1 :  IQ American  IQ Chicago Q1c. Calculate the sample mean. Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] 0.5 if the process is correct but the result was calculated incorrectly 𝑀 = ∑ 𝑋 N 133+92+100+114+122+134+94+112+109+148+129+92+154+120+144 15 M= 119.8 Q1d. Calculate standard error using the population standard deviation σ given in the research scenario ( SE , which is the standard deviation of the sampling distribution) Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] deduct 0.5 if the process is correct but the result was calculated incorrectly σ m = σ √ N = 15 √ 15 =3.87 Q1e. Calculate the Z statistic (which indicates where our sample mean is located on the sampling distribution) Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] deduct 0.5 if the process is correct but the result was calculated incorrectly Z = M − ❑ m 2 a ❑ Z = 119.8 − 115 3.87 ❑ = 1.24 Q1f. Determine the critical Z value(s). Explain how you find the answer. [1 point] 0.5 for the Z value, 0.5 for the rationale We have a one-tail test And the critical Z value that is on the right side, since we are hypothesizing that the individual IQ in Chicago is higher than the national, α = .05 So that means that I will be on the lookout for 5% in the tail area. So that means that I will be using 5.05% since alpha equals .05. we’re looking for 5%. P( Z > Z crt) =0.05 where Z is the standard normal variable but P( Z > 1.64) = 0.05 the corresponding Z value is 1.64. So the critical Z value is 1.64 All course content and materials, including this assignment and all other items published within the Canvas course shell, are copyrighted materials and students may not share outside the class, upload to online websites not approved by the instructor, sell, or distribute course content or notes taken during the conduct of the course (see ACD 304–06 , “Commercial Note Taking Services” and ABOR Policy 5-308 F.14 for more information). Students must refrain from uploading to any course shell, discussion board, or website used by the course instructor or other course forum, material that is not the student’s original work, unless the students first comply with all applicable copyright laws; faculty members reserve the right to delete materials on the grounds of suspected copyright infringement.

PSY230_Bienemann Spring 2024 Q1g. Compare the Z statistic with the appropriate critical Z value and then draw a conclusion about the result of the hypothesis test. - Is the Z statistic more extreme than the critical Z? [0.5 point] - Do you “reject” or “fail to reject” the null hypothesis? [0.5 point] The Z statistic is less extreme and lower than the critical Z so we fail to reject the null hypothesis. Q1h. Write a sentence to answer the research question (you can use the wording from the hypotheses or explain it in another way) [1 point] The average IQ of the 15 individuals from Chicago is not higher than the national average IQ of Americans Q1i. Calculate the standardized effect size Round the result to the hundredth (2 nd place to the right of the decimal). [1 point total] deduct 0.5 if the process is correct but the result was calculated incorrectly m-µ/ σ= 119.8-115/15=0.32 Q2. Hypothesis Testing with a Z Test (12.5 points) According to the CDC report, the mean life expectancy in the US population is currently (µ) 85.8 years with a standard deviation ( σ ) of 15.9. A researcher examined the age of death of 80 people recently recorded in several Florida hospitals and calculated the mean to be 89.4 years old. He runs a two-tailed Z test with α = .05 to see if Florida has a significantly different life expectancy compared to the US population. Because the researcher is not predicting a direction, the hypotheses should be non- directional, and the test should be two-tailed. *Find a blueprint for your work on sheet Q2 in the M3_Data Excel file Q2a. What is the dependent variable in this research question? [.5 point] The dependent variable is the life expectancy Q2b. What should the null and alternative hypotheses? State each hypothesis using both words and statistical symbol notation. [2 points] 1 for each hypothesis: 0.5 for written, 0.5 for notation Written Symbolic Symbol bank for copy & pasting:  =      Null The life expectancy mean in Florida will not be different from the expectancy of H 0 :  =85.8 All course content and materials, including this assignment and all other items published within the Canvas course shell, are copyrighted materials and students may not share outside the class, upload to online websites not approved by the instructor, sell, or distribute course content or notes taken during the conduct of the course (see ACD 304–06 , “Commercial Note Taking Services” and ABOR Policy 5-308 F.14 for more information). Students must refrain from uploading to any course shell, discussion board, or website used by the course instructor or other course forum, material that is not the student’s original work, unless the students first comply with all applicable copyright laws; faculty members reserve the right to delete materials on the grounds of suspected copyright infringement.

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PSY230_Bienemann Spring 2024 the US population. Alternative The life expectancy mean in Florida will be significantly different from the expectancy of the U S population H 1 :  >85.8 Q2c. Calculate standard error ( SE , which is the standard deviation of the sampling distribution) Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] 0.5 if the process is correct but the result was calculated incorrectly σ/ √ N =15.9/ √ 80 =1.78 Q2d. Calculate the Z statistic (which indicates where our sample mean is located on the sampling distribution) Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] 0.5 if the process is correct but the result was calculated incorrectly Z = (M-  )/SE=(89.4-85.8)/1.78=2.02 Q2e. Determine the critical Z value(s). Explain how you find the answer. [1 point] 0.5 for the Z value, 0.5 for the rationale, deduct 0.5 if only one Z value is mentioned Since this is a two-tailed test and α = .05 if you split 5% into two tails you get 2.5% so this gives a Z value of 1.96 and because it is two-tailed we would get a positive and a negative, so it is both -1.96 and 1.96. Q2f. Compare the Z statistic with the appropriate critical Z value and then draw a conclusion about the result of the hypothesis test. What is the answer to the research question now? - Do you “reject” or “fail to reject” the null hypothesis? [0.5 point] - What is the answer to the research question? [0.5 point] Because our calculated statistic is more extreme than the critical value, we end up rejecting the null hypothesis i.e. the life expectancy in Florida is in fact, significantly different from the US population. Q2g. Calculate the standardized effect size. Round the result to the hundredth (2 nd place to the right of the decimal). [1 point total] deduct 0.5 if the process is correct but the result was calculated incorrectly Raw=  Florida-  US=89.4-85.8=3.6 Standard(  Florida-  US)/ σ= (3.6)/ 15.9=0.23 Q3. Hypothesis testing using a Z test (7.5 points) A professor has been teaching introductory statistics for many years and the final exam performance has been very consistent from class to class and the scores have been normally distributed. Overall, the whole data base (i.e., population) of final exam scores has a mean ( μ ) of 24.75 points (out of a maximum All course content and materials, including this assignment and all other items published within the Canvas course shell, are copyrighted materials and students may not share outside the class, upload to online websites not approved by the instructor, sell, or distribute course content or notes taken during the conduct of the course (see ACD 304–06 , “Commercial Note Taking Services” and ABOR Policy 5-308 F.14 for more information). Students must refrain from uploading to any course shell, discussion board, or website used by the course instructor or other course forum, material that is not the student’s original work, unless the students first comply with all applicable copyright laws; faculty members reserve the right to delete materials on the grounds of suspected copyright infringement.

PSY230_Bienemann Spring 2024 of 30 points) and a standard deviation ( σ ) of 1.75 points. The professor would like to revise the course design to see if student performance on the final exam could be improved. There were 14 students who took the new course as a trial run in a recent academic year, and their average final exam score was 25.8. The professor would like to run a hypothesis test to see if this sample of students in the recent trial course performed significantly better than the past population. In other words, the hypothesis was a comparison between the population taking the course with the new design (represented by the sample of 14 students) with the population taking the course with the old design. The professor is predicting an increase of final exam score with the new design, so the hypotheses should be directional, and the test should be one-tailed. The significance level is set at α = .01. *Find a blueprint for your work on sheet Q3 in the M3_Data Excel file Q3a. Identify the dependent variable for this study [0.5 point] The dependent variable is the old design of the exam and the new design of the exam scores. Q3b. State the null hypothesis and alternative hypothesis using both words and symbol notation. [2 points] 1 for each hypothesis: 0.5 for written, 0.5 for notation Written Symbolic Symbol bank for copy & pasting:  =      Null The new design of the final exam population scores will not be significantly higher than the old design of the exam scores H 0 :  NEW  OLD Alternative The new design of the final exam population scores will be significantly higher than the old design of the exam scores H 1 :  NEW   OLD Q3c. Calculate standard error (SE, which is the standard deviation of the sampling distribution) Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] 0.5 if the process is correct but the result was calculated incorrectly σ/ √ N =1.75/ √ 14 =0.47 Q3d. Calculate the Z statistic (which indicates where our sample mean is located on the sampling distribution) Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] 0.5 if the process is correct but the result was calculated incorrectly Z = (M-  )/SE = (25.8-24.75)/0.47=2.23 All course content and materials, including this assignment and all other items published within the Canvas course shell, are copyrighted materials and students may not share outside the class, upload to online websites not approved by the instructor, sell, or distribute course content or notes taken during the conduct of the course (see ACD 304–06 , “Commercial Note Taking Services” and ABOR Policy 5-308 F.14 for more information). Students must refrain from uploading to any course shell, discussion board, or website used by the course instructor or other course forum, material that is not the student’s original work, unless the students first comply with all applicable copyright laws; faculty members reserve the right to delete materials on the grounds of suspected copyright infringement.

PSY230_Bienemann Spring 2024 Q3e. Determine the critical value for Z. Explain how you come up with the answer. [1 point: 0.5 for the answer and 0.5 for the rationale] Since it is one-tailed and since our α =.01 I am looking for 1% on the right side so the Z value is 2.33. Q3f. Compare the Z statistic with the appropriate critical Z value and then draw a conclusion about the result of the hypothesis test. What is the answer to the research question? - Do you “reject” or “fail to reject” the null hypothesis? [0.5 point] - What is the answer to the research question? [0.5 point] Z statistic is 2.23 and the Z value is 2.33 The Z statistic is less extreme than our critical value, so we fail to reject the null hypothesis i.e. the new design of the introductory exam scores are not significantly higher than the old design. Q3g. Calculate the standardized effect size for this test. [0.5 point, must show the process to earn the credit] Is this considered a small, medium, or large effect according to the Cohen’s d standards? [0.5 point] Raw=  NEW-  OLD=25.8-24.75=1.05 Standard=(  NEW-  OLD)/ σ= (1.05)/1.75=0.60 It is a medium effect. Answer the following questions based on this alternative scenario: Because the previous sample is very small, it likely does not represent a full class adequately. So, the researcher decides to continue to collect data for another semester of the new course design until the sample becomes 30. The average new test score remains 25.8, the same as the previous scenario. All other aspects of the study remain unchanged. Q3h. What is the standard error with this sample? Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] 0.5 if the process is correct but the result was calculated incorrectly σ/ √ N =1.75/ √ 30 =0.32 Q3i. What is the Z statistic with this sample? Round the result to the hundredth (2 nd place to the right of the decimal). [1 point] 0.5 if the process is correct but the result was calculated incorrectly Z = (M-  )/SE = (25.8-24.75)/0.32 = 3.28 Q3j. Compare the Z statistic with the appropriate critical Z value and then draw a conclusion about the result of the hypothesis test. What is the answer to the research question now? - Do you “reject” or “fail to reject” the null hypothesis? [0.5 point] - What is the answer to the research question? [0.5 point] All course content and materials, including this assignment and all other items published within the Canvas course shell, are copyrighted materials and students may not share outside the class, upload to online websites not approved by the instructor, sell, or distribute course content or notes taken during the conduct of the course (see ACD 304–06 , “Commercial Note Taking Services” and ABOR Policy 5-308 F.14 for more information). Students must refrain from uploading to any course shell, discussion board, or website used by the course instructor or other course forum, material that is not the student’s original work, unless the students first comply with all applicable copyright laws; faculty members reserve the right to delete materials on the grounds of suspected copyright infringement.

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PSY230_Bienemann Spring 2024 The critical Z value is 2.33 and the Z statistic is 3.28 because the Z statistic is higher than the critical Z value. We reject the Null Hypothesis i.e. the mean of the exam scores for the new design is significantly higher than the old design. Q3k. Calculate the standardized effect size. [1 point] 0.5 if the process is correct but the result was calculated incorrectly Standard=(  NEW-  OLD)/ σ= (1.05)/1.75=0.60 Q3l. Based on the hypothesis test results with the different sample sizes, how did the increase in sample size impact the test results in terms of the Z statistic and the effect size? [1 point: 0.5 for impact on Z statistic; 0.5 for impact on the effect size] The effect size had no change, but the increased sample size Did change the test results in terms of the Z statistic because it did make the Z statistic more extreme than the critical value. So, we ended up rejecting the null hypothesis. All course content and materials, including this assignment and all other items published within the Canvas course shell, are copyrighted materials and students may not share outside the class, upload to online websites not approved by the instructor, sell, or distribute course content or notes taken during the conduct of the course (see ACD 304–06 , “Commercial Note Taking Services” and ABOR Policy 5-308 F.14 for more information). Students must refrain from uploading to any course shell, discussion board, or website used by the course instructor or other course forum, material that is not the student’s original work, unless the students first comply with all applicable copyright laws; faculty members reserve the right to delete materials on the grounds of suspected copyright infringement.