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Stat212 Statistical II Fall 2023 Midterm 2 Name:  For ALL hypothesis testing questions, please write down the null and alternative hypothesis  Please show your work to earn full credit 1. (4 points) Since 2000, the median income of U.S. individuals has risen about 1% (adjusted for inflation). However, over the same period, the median income for the four educational categories (high school dropouts, high school graduates, some college education, Bachelor’s degree or higher) have all decreased. What could be the possible reason for this? ( ) A. There are now more college graduates (who get higher pay) than 2000, and the income for college graduates have fallen at a larger rate than that from other educational groups B. There are now more college graduates (who get higher pay) than 2000, and the income for college graduates have fallen at a smaller rate than that from other educational groups C. There are now less college graduates (who get higher pay) than 2000, and the income for college graduates have fallen at a larger rate than that from other educational groups D. There are now less college graduates (who get higher pay) than 2000, and the income for college graduates have fallen at a smaller rate than that from other educational groups What is the statistical terminology for this phenomenon? 2. (2 points) In one way ANOVA with 6 groups to compare, what is the number of pairwise comparisons? ( ) A. 6 B. 15 C. 21 D. 30 3. (6 points) Suppose that a researcher is interested to know if stress during pregnancy varies according to age and educational levels. He randomly selected 48 pregnant women to conduct a study. Suppose there are 4 age groups (20-, 20-30, 30-40, 40+) and 3 education groups (less than college degree, college degree, and graduate degree). Choose the correct type of the design ( ) A. This is a one factor completely randomized design. B. This is a one factor randomized block design. C. This is a two factor design. If you choose B, what is the block variable? How many treatment groups here? If you choose C, how many treatment groups here? Furthermore if this is a balanced completely factorial design (same number of observations is every age by education group), how many pregnant women in each of the treatment group?

4. (4 points) Which statement is correct for figure 1? ( ) Which statement is correct for figure 2? ( ) Figure 1 Figure 2 A. There is main effect at “Condition” level, there is no main effect at “Group” level, and there is interaction B. There is no main effect at “Condition” level, there is main effect at “Group” level, and there is interaction C. There is main effect at “Condition” level, there is no main effect at “Group” level, and there is no interaction D. There is no main effect at “Condition” level, there is main effect at “Group” level, and there is no interaction 5. (14 points) The researchers want to investigate the effect of housing system of eggs on the mean of characteristics of eggs. There are four housing systems: BARN, CAGE, FREE and ORGANIC. The shell thickness of 6 eggs from each housing systems (total 24 eggs) are recorded. The average thickness of the FREE housing system is 5.02, and the average thickness of the ORGANIC housing system is 4.82. One way ANOVA table generates a mean square error of 0.09. Construct the 95% Fisher’s LSD and Tukey’s HSD CI of ࠵? ிோாா − ࠵? ைோீ஺ேூ஼

6. (20 points) If in the previous example, the researchers also determine the weight class (medium and large) for each egg. 3 out 6 eggs in each housing system are medium and the other 3 are large. Below is part of the ANOVA table Source DF SS MS F-Stat HOUSING 2.4233 WEIGHTCLASS 0.06 Interaction 0.9233 Error N.A. Total 4.2133 N.A. N.A. A. Fill in the ANOVA table (total 13 missing values) B. Are there main effects here? Do hypothesis testing at alpha=0.05. Does your conclusion consistent with question 5? C. Is there evidence of interaction in the model? Do hypothesis testing at alpha=0.05. If there is interaction, give an example of possible interaction. If no interaction, explain what “no interaction” means

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7. (15 points) Susan Sound predicts that students will learn most effectively with a constant background sound, as opposed to an unpredictable sound. She randomly divides 15 students into 2 groups of 7 and 8 each. All students study a passage of text for 30 minutes. Those in group 1 study with background sound at a constant volume in the background. Those in group 2 study with noise that changes volume periodically. After studying, all students take a 10 point multiple choice test over the material. Their scores have the following test statistics. Group Sample size Mean Variance Constant sound 7 6 4.86 Random sound 8 4 2.86 A. Construct the F statistic ANOVA table using data. Stat the hypotheses for ANOVA and use significance level 0.05 to test your hypotheses. What is the range of p-value of the test? B. Explain whether you use can conclusion above to support Susan’s prediction or not? Explain why and how.

8. (15 points) Suppose an investigator wishes to see if boys and girls at a local school respond differently to an attitudinal question regarding the educational value of extracurricular activities. A. Describe a way of data collection such that this could be a test of homogeneity B. Describe a way of data collection such that this could be a test of independency C. If the answer to the question has three levels: A = very valuable, U = uncertain, and D = little value, and the corresponding numbers of students with each answer are listed below A U D Total Boys 60 10 30 100 Girls 40 10 50 100 Total 100 20 80 200 State the null hypothesis and test it using alpha=0.01

9. (10 points) A researcher wants to test whether the university GPA is normally distributed. The GPA is partitioned into four groups A, B, C, D, which are defined as A. Below one standard deviations of the mean B. Within one standard deviation below the mean C. Within one standard deviation above the mean D. Above one standard deviations of the mean A random sample of 105 students are asked to find out their GPA groups, and it turns out that there are 9, 60, 17, 19 students in the 4 groups respectively. Is there sufficient evidence that the GPA is normally distributed? (keep two decimal places for probabilities involved here) 10. (10 points) The accompanying data refers to the concentration of the radioactive isotope strontium-90 in milk samples obtained from 5 randomly selected dairies in each of the four different regions. Region 1 6.4 ( ) 5.8 ( 3 ) 6.5 ( ) 7.7 ( 10 ) 6.1 ( ) Region 2 7.1 ( 9 ) 9.9 ( 14 ) 11.2 ( 17 ) 10.5 ( 16 ) 8.8 ( 12 ) Region 3 5.7 ( 2 ) 5.9 ( 4 ) 8.2 ( 11 ) 6.5 ( ) 5.1 ( 1 ) Region 4 9.5 ( 13 ) 12.1 ( 19 ) 10.3 ( 15 ) 12.4 ( 20 ) 11.7 ( 18 ) Fill in the ranks in the parentheses when needed (ranks 1-4, 9-20 are filled inside already). Test at level 0.01 to see whether true average strontium-90 concentration differs for at least two of the regions

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