Malaika Wauters Math 11 - Lab 7

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Malaika Wauters Math 11 Denise Rava Lab 7: SAT Scores and Cloud Seeding 1. Examine the distributions of the Math and Verbal SAT scores. Do they appear to be approximately normally distributed? Are there any outliers? What about the distribution of the GPAs? a. The distributions of the Math and verbal SAT scores appear to be normally distributed, as seen in the bell shaped curve on the histograms to the left, however, there are outliers in both sets of data on the lower end. The distribution of the GPAs is also bell shaped and approximately normal but there are no evident outliers that can be assumed from the histogram. 2. How does the performance of males compare to the performance of females on the Math SAT test? What about the Verbal SAT test? How do the first-year GPAs of male and female students compare? a. The data collected for the Math test section score show that Males have a slightly higher median in comparison to females but also a wider spread of data. While both males and females have two outliers, both male outlier scores are on the lower end while 1 outlier score achieved by females is higher than average.
b. In regards to the Verbal test scores, both male and females have approximately the same median score but the spread of data for females is greater between the median and upper quartile while the spread is wider for males below the median. The data only shows 1 outlier for females and 3 for males. c. The boxplot for the GPAs of males and females shows that the median GPA among males is lower than that of females, although females have a greater interquartile range. The also are no outlier GPA scores amongst females while there are 2 outliers for males on the higher end. 3. Based on what you observed from examining the data graphically, does it appear that the assumptions required to conduct t-tests are met? a. Yes, independence is assumed and also the histograms in question1 demonstrate a normal distribution bell shaped curve, therefore the conditions to perform t-tests should be adequate. 4. Do macroeconomics students at Vanderbilt score significantly higher on the Math SAT than the national average (which in the mid-1980s was around 470)? a. One sample t-test & one sided b. Null hypothesis H 0 : μ = 470 c. Alternative hypothesis H A : μ >470 d. Degrees of freedom = n-1 = 391-1 = 390 e. T statistic = 50.80 f. P-value: P(t > 50.80) < 0.005 g. Given that the p-value is less than 0.005, which is less than 0.05 (the test significance level), we can reject the null hypothesis and there is evidence that the mean Math SAT score at Vanderbilt was higher than the national average. 5. Do macroeconomics students at Vanderbilt score significantly higher on the Verbal SAT than the national average (which in the mid-1980s was around 425)? a. One sample t-test & one sided b. Null hypothesis H 0 : μ = 425 c. Alternative hypothesis H A : μ >425 d. Degrees of freedom = n-1 = 391-1 = 390
e. T statistic = 38.45 f. P-value: P(t > 38.45) < 0.005 g. Given that the p-value is less than 0.005, which is less than 0.05 (the test significance level), we can reject the null hypothesis and claim that there is sufficient evidence that the mean Verbal SAT score at Vanderbilt was higher than the national average of 425. 6. Report a 95 percent confidence interval for the true mean Math SAT score. Do 95 percent of students have Math SAT scores that fall within this interval? Explain your answer. a. One sample t-test & two sided b. Null hypothesis H 0 : μ = 624.9 c. Alternative hypothesis H A : μ /= 624.9 d. 95% CI = (619.5,630.2) e. The 95% confidence interval means that we can claim that the true mean falls in this interval with 95% confidence, but it does not mean that 95% of the students scored in this range for the Math section of the SAT. 7. Is there a statistically significant difference between the Verbal and Math scores of microeconomics students at Vanderbilt? a. Paired t-test & 2 sided alternative b. Null hypothesis H 0 : Math = Verbal c. Alternative hypothesis H A : Math /= Verbal d. T statistic = -20.4 e. P <0.0001, given that the p value is less than 0.05, we reject the null hypothesis. 8. Is there a statistically significant difference between the performances of males and females on the Verbal SAT? Construct a 95 percent confidence interval for the difference. Does it include zero? Relate this to the conclusion of your test. a. 2 sample t-test b. T statistic = 0.60 c. Degrees of freedom = n-1 = 124-1 = 123 d. P-value: P(|t| >0.6) = 2 x P(t >0.6) → 2 x P(t >1.289) = 0.20 e. Given that the p value is actually greater than 0.20, we cannot reject the null hypothesis and conclude that there is no significant statistical difference between male and female verbal SAT scores. f. (-9.38, 17.58) g. The 95% confidence interval includes 0, since we failed to reject the null hypothesis and claimed that there is no difference in verbal SAT performance between males and females, the confidence interval containing 0 confirms that.
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9. Is there a statistically significant difference between the freshman GPAs of males and females? Construct a 95 percent confidence interval for the difference. Does it include zero? Relate this to the conclusion of your test. a. 2 sample t-test b. T statistic = 2.17 c. Degrees of freedom = n-1= 124-1 = 123 d. P-value: P(|t| >2.17) = 2 x P(t >2.17) e. 2 x P(t>2.358) < 2 x P(t>2.17) < 2 x P(t>1.98) = 0.02 < p < 0.05 f. Given that the p-value is less than 0.05, we can reject the null hypothesis and claim that there is sufficient evidence that there is a statistically significant difference between freshmen male and females g. (0.01,0.206) h. The confidence interval is lacking 0, which is accurate to the previous claim that there is a significant difference between male and female freshman GPAs. 10. Is there a statistically significant difference between the performances of males and females on the Math SAT? Construct a 95 percent confidence interval for the difference. Does it include zero? Relate this to the conclusion of your test. a. 2 sample t-test b. T statistic = -2.57 c. Degrees of freedom = n-1 = 124-1 = 123 d. P-value: P(|t| >-2.57) = 2 x P(t >2.57) e. 2 x P(t >2.617) < 2 x P(t >2.57) < 2 x P(t >2.358) = 0.01 < p < 0.02 f. Given that the p-value is less than 0.05, we can reject the null hypothesis and claim that there is sufficient evidence that there is a statistically significant difference between female and male performance on the Math section of the SAT. g. (-27.62, -3.58) h. We should not have 0 as a possible value in the 95% confidence interval because we reject the null hypothesis with the low p value, and this confidence interval does not include the 0 which confirms the difference in performance between females and males on the Math SAT.
11. First examine and compare the two distributions with histograms and a side-by-side boxplot. Discuss what you observe in a few sentences. a. From the skewed shape of the bars on the histograms above, the distribution is likely exponential. Outliers are present in both sets of data but the seeded clouds data exemplifies more extreme outliers b. The box plots also show that the seeded clouds have a much wider range of data and more extreme outliers on the upper end of the data, it also shows that the seeded clouds have a higher median rainfall. 12. Conduct a hypothesis test to test the hypothesis that cloud seeding with silver nitrate increases rainfall. Use significance level .01. Based on what you observed in answering the previous question, are you confident that this is a valid test? If not, what assumptions are violated? a. Two sample t-test b. T statistic = -2 c. Degrees of freedom = 26-1 = 25 d. P-value: P(t>2) e. P(t>2.06) < P(t>2) < P(t>1.708) = 0.025 < p < 0.05 f. Given that the p value is greater than 0.01, we fail to reject the null hypothesis, we do not have statistically significant evidence that cloud seeding increases rainfall.
13. Try taking natural logarithms of the rainfall totals. Are the new variables less skewed? Are the distributions closer to normal? What does a side-by-side boxplot of the variables show? a. The distribution shape depicted in the histograms above resemble a more bell shape which translates to a normal distribution and are not really skewed. (C4 being unseeded and C5 being seeded clouds) The box plot shows a higher median rainfall value in seeded clouds with a more similar interquartile range. Both plots have outliers but the outlier for unseeded is more extreme. I’m more confident in the normal distribution in comparison to exponential. 14. Conduct another formal hypothesis test using the transformed variables, again at significance level .01. Are you more confident in the results of this test or the results of the previous test? a. T statistic = -2.54 b. Degrees of freedom = 26-1 = 25 c. P-value: P(t>2.54) d. P(t>2.787) < P(t>2.54) < P(t>1.485) = 0.005 < p < 0.01 e. Given that the p value is very low and under 0.01, we can reject the null hypothesis and there is statistically significant evidence to prove that cloud seeding does increase rainfall
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15. Write a couple of sentences explaining your overall conclusions about the effect of cloud seeding with silver nitrate on rainfall. a. With the completion of the t-test, and yielding a very low p value, we can reject the null hypothesis that suggests there is no difference between the rainfall that is produced by the two types of clouds. Therefore, we can claim that there is statistically significant evidence for the difference between rainfall in seeded and unseeded clouds, more specifically the evidence suggests that there is a higher amount of rainfall in silver nitrate seeded clouds in comparison to unseeded clouds.