Homework 1

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Feb 20, 2024

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Rachel Ludeke January 28, 2017 Stats II Homework I 11.10 a) We check the normality in this case by viewing the number of cases in our variable set (twelfth graders who have always been rated at grade level) and mean social studies score. We can assume tenability of the normality assumption in this instance because the number of observations (n=149) meets the requirements of being sufficiently large (n>30) by viewing the level of skewness within our data. Table 1. Descriptive Statistics for Social Studies Achievement in Twelfth Grade. The skewness reported here is between-0.5 and 0.5, so the distribution is approximately symmetric. b) The null hypothesis is H 0=  =55 while the alternative is H 1=  >55. c) The sample mean given by our t-test is 57.32 which is larger than our anticipated difference that the population of college bound twelfth graders have who have always been at grade level have a mean social studies achievement score larger than 55. Table 2. Two Sample T-test of Social Studies Achievement Score, by Gender d) The p-value associated with our one-sample t-test is 0.0006, which we get from the one sample t- test where t (148) =3.31 with a p value of 0.006. e) Our results based on the one-sample t test are significant because the p-value of 0.006 is less than the alpha of .05. f) According to our two –sample t test, the mean social studies achievement score for this population [ M =57.32, SD =8.56) was found to be statistically significantly larger than a score of 55 [ t (148) =3.31, p =.0006} g) The mean reported achievement score for this population is .28 standard deviations larger than 55. We arrive at this observation by utilizing the formula for Cohen’s D [(57.32-55)/8.56]. This is a small to moderate effect considering the rules for Cohen’s d. 1

Rachel Ludeke January 28, 2017 Stats II Homework I h) The standard deviation value of 8.56 represents an estimate of the population standard deviation, which measures the spread within the population of social studies achievement scores, while the standard error of the mean value .70 is an estimate of the sample mean. The standard error of .70 measures how spread the data are for social studies achievement means based on a sample size of 149 randomly selected individuals from this population. i) I believe that the p-value would have been larger if we were given a standard deviation of 20 instead of 8.56, which would mean that the t-statistic would be larger. In turn, the t-value would have decreased and the area to the right of the t-value would have been larger. 11.16 a) The new variable Bmidiff showed that the difference in BMI for the first person in the data set is -.64. This result shows that this person’s BMI decreased by .64 units from the first examination in 1956 to the third examination in 1968. b) Figure 1. Boxplot of BMI Differences, By Gender The boxplot shows that the distributions between men and women are almost symmetric with outliers occurring in each observation, but when looking at the distribution for women, there are more outliers that seem to suggest that women increased in their BMI over time while men seem to have decreased their BMI over time. c) The independent samples t-test is robust if the sample size is greater than 30. In our case, the sample size for males and females is greater than 30, so we do not have to be concerned about any violations to the normality assumption. d) Using the results from the Levene’s test, there is tenability in the equality of our data and the condition for homogeneity of variances is met ( F (1,302) = 1.96, p=. 16). 2

Rachel Ludeke January 28, 2017 Stats II Homework I Table 3. Levene’s Test of Homogeneity for Sex, by Gender e) The p-value given from the two-sample t-test indicates that on average the body mass index increased for women between 1956 and 1968, but decreased for men during this same time period Table . Two Sample T-test of BMI Differences, by Gender Our test shows that the average change in body mass index for women (M=.50, SD=2.34) is significantly different than the body mass index reported for men (M=-.13, SD=1.91), with the p- value equaling .01, which is less than our stated alpha of .05, which allows us to reject the null hypothesis that there is no difference between men and women in terms of BMI in the years between 1956 and 1968. f) Using Cohen’s D, we can see that the mean change in the body mass index reported for women is .29 standard deviations higher than what is reported for men, but this is a small or moderate effect (-.632/2.164). This result is given by dividing the mean of the differences between the samples by the combined standard deviation of difference reported from our two sample t test. 11.26 a) A paired-samples t-test is more appropriate in this case to measure the level of science achievement change from eighth to twelfth grade because we are using the same sample to test the results over time. Thus, the one-sample t-test or independent t-test because the variables we are interested in studying are related to each other. b) The sample size for science achievement in eighth grade and for science achievement for 12 th grade is more than n=30. The skewness reported for each variable is less than 1, which allows us to assume normality. 3

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Rachel Ludeke January 28, 2017 Stats II Homework I c) The null hypothesis is H 0 =  2 -  1 and the alternative hypothesis is H 1 =  2 ≠ 1 . d) The p-value that results from using the paired sample t-test is 0.74. Table 4. Paired T-Test of Science Achievement, by Grade e) The paired sample t-test shows that the change in science achievement in the change in science achievement in the 8 th grade as M =55.67, ( SD =9.39) to M =55.77 ( SD =8.57) in the twelfth grade for this population. This result is not statistically significant, since our given p-value of 0.74 is larger than the desired alpha of .05. f) The 95 % confidence interval for the difference in means in science achievement between the eighth and twelfth grade is (-.76, .54) or (-.54, and. 76) g) Zero is contained within the confidence interval, so this means that is possible for zero to be the difference in reported mean for science achievement between the eighth and twelfth grade. This means that our testing was unable to prove a statistically significant difference between means in science achievement between 8 th and 12 th grade for this population of students. 13.4 a) Using the boxplot, it seems that the normality assumption is tenable for those who have a bachelor’s degree or less. There are outliers present for those who have more than a bachelor’s degree, but the sample size for each of the variables are large enough that the outliers won’t impact the results of statistical testing. Figure 2. Boxplot of Perceived Academic Achievement Homogeneity of variance is present based on our boxplot and the interquartile range for each of the expected education types seems similar. The ANOVA will most likely indicate that if people 4

Rachel Ludeke January 28, 2017 Stats II Homework I obtain less than a college degree that they will statistically take less units of math than those in the other groups. b) The sample sizes for our variables that we want to examine are all above n=30. There are 48 students who believe they will obtain less than a college degree, 159 who believe they will obtain a bachelor’s degree, 190 who will obtain a master’s degree and 103 who will obtain a professional/terminal degree. Since the sample sizes reported here are larger than 30, the ANOVA we will conduct is to be robust against violations of normality. We can conduct Levene’s test to ensure that the homogeneity of variance is not violated. The results of this test are below: Table 5. Levene’s Test of Homogeneity for Educational Attainment Levene’s test shows that we are not in danger of violating the homogeneity of variance, as the results yield F (3, 496) = .37, p=.77. c) Conducting a one-way ANOVA yields the following results: Table 6. One-way ANOVA of Perceived Academic Achievement The average number of years of math taken according to the ANOVA of college-bound students have been at grade level shows variance in terms of their expected educational level, F(3, 496) =15.70, with a p-value of .0005. d) Using the formula for effect size (Measure of Squares Between Groups/Sum of Total Squares), we get an effect size of 8.67, which means that 8.67 percent of the variance in the amount of math 5

Rachel Ludeke January 28, 2017 Stats II Homework I taken by the students can be explained by the highest level of education that each student believes they will earn. 8.67 represents a moderate effect in our data. e) When using the bootstrap option, we get a different p-value (p=0.004) than we did when using the ANOVA test, but both approaches prove to be statistically significant. Table 7. Bootstrap Test of the Years of Math of Students taken in Twelfth Grade. f) It is still necessary to conduct post hoc tests since the results of the ANOVA were found to be significant and we are comparing more than two means. We can use the Tukey test to report the level of significance: Table 7. Tukey Test of Significance for Perceived Academic Achievement The Tukey tests shows that students who believe that they will earn less than a college degree are taking significantly fewer math courses than the other degree types we compared. Students who believe that they will earn a bachelor’s degree are taking statistically significantly less classes than those who believe that they will earn a PhD or other terminal professional degree. The test does not find any other significant differences between these groups. 6

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