Hypothesis Testing and Single Sample z test Practice Problems Note: unless indicated otherwise, use level of significance of .05 for all questions 1. A study was conducted to investigate whether the underperformance of students in high schools in the United States can be reduced. There has been recent interest in the issue of \"mindsets\" to improve academic performance. It has been shown that individuals with a \"growth mindset\" (i.e., a belief that students can grow their intelligence and that academic challenges are an opportunity for growth, rather than a sign that the student is not capable of learning) may do better in school than those with a \"fixed mindset\" (i.e., a belief that students are born with a fixed level of ability that cannot be changed). To examine this issue, a research study was conducted in which a group of 49 high school students were given a 45 minute \"growth mindset intervention\" in which they were taught that the brain can grow and reorganize as a result of hard work and academic challenges. At the end of the school year, the GPA of a sample of students who were given the \"growth mindset intervention\" was compared to the overall GPA of all students at the high school. It is known that the overall GPA of students at the high school is 2.40 with a SD of .25. Complete the following steps to conduct a statistical test that addresses if the \"growth mindset intervention\" leads to a difference in the GPA in high school students. a. What is the Independent Variable (IV) with its respective levels and what is the Dependent Variable (DV) in this experiment? b. Step 1: Write the null and research (alternative) hypotheses in symbols. c. Step 2: Assume that we draw all possible random samples of 49 students from the population of students at the high school. What would be the appropriate (i.e., compute) standard error (SE) for a distribution of samples of size 49? Give an interpretation in words of what this number means. d. Step 2 (continued): The critical value for the test statistic. e. Step 3: Find the obtained (i.e., computed) test statistic for a sample (n=49) with a mean of 2.50. Step 4: Make a statistical decision about the null. Will you reject or fail to reject the null based on your sample data? (Be sure to include appropriate test statistic and decision criteria). f. Step 4 (continued): Provide a substantive conclusion about the results from your sample data. (Be sure to include the research context and descriptive statistics). g. Continuing with this example, (i.e. the same population mean, standard deviation, etc.), calculate the effect size of the test anxiety on GPS (using Cohen's d) for the sample of students in the \"growth mindset intervention group\" who had a mean GPA of 2.50. How large of an effect size is this, small, medium, or large? h. The following table lists the possible decisions that can be made based on the research study above. Label which boxes would be correct decisions and which would be errors (Type I and Type II). Truth H0 True H0 False \"Growth Mindset Intervention\" does not affect GPA \"Growth Mindset Intervention\" does affect GPA Reject H0 \"Growth Mindset Intervention \" does affect GPA Statistical Decision i. j. Fail to reject H0 \"Growth Mindset Intervention \" does not affect GPA What would a Type I error mean in the context of this study? What would a Type II error mean in the context of this study? 2. The national average for students on the Science Attitudes Questionnaire is 10.0, with a standard deviation of 1.0. Creative Teaching Elementary School has created a new textbook which they use in their school because they believe it will increase student interest in science. A random sample of 25 students from Creative Teaching Elementary School are given the Science Attitudes Questionnaire and their mean score was 10.2. Go through all 4 steps of hypothesis testing to decide if the new textbook at Creative Teaching Elementary School significantly impacts students' attitudes about math. a. Step 1: Write out the hypotheses b. Step 2: Assume that we draw all possible random samples of 25 students from the population of students at the college. What would be the appropriate standard error (SE) for a distribution of samples of size 25? c. Step 2 (continued): What is the critical value for this test statistic? d. Step 3: Calculate the test statistic e. Step 4: Make a statistical decision about the null. Will you reject or fail to reject the null based on your sample data? (Be sure to include appropriate test statistic and decision criteria). f. Step 4 (continued): Provide a substantive conclusion about the results from your sample data. (Be sure to include the research context and descriptive statistics). g. The following table lists the possible decisions that can be made based on the research study above. Label which boxes would be correct decisions and which would be errors (Type I and Type II). Truth H0 True H0 False New Textbook at Creative Teaching Elementary School does not affect student attitudes about science New Textbook at Creative Teaching Elementary School does affect student attitudes about science Reject H0 New Textbook at Creative Teaching Elementary School does affect student attitudes about science Statistical Decision Fail to reject H0 New Textbook at Creative Teaching Elementary School does not affect student attitudes about science h. What would a Type I error mean in the context of this study? i. What would a Type II error mean in the context of this study