MATH 3070: Test I (Part 1)

Answer

Question 1. (10 points) Assume that Z is a standard normal variable. Answer the following questions. 1) Find the probability that Z is between 0.47 and 1.45. 0.9265 – 0.6808 = 0.2457 2) What value separates the rest of 90% from the largest 10%? The exact area of 0.90 is between 1.28 and 1.29. Thus, it can be approximated by 1.28, 1.29, or 1.285 Question 2. (5 points) The annual precipitation amounts are normally distributed with a mean of 107 inches and a standard deviation of 10 inches. 3) What is the probability that the annual precipitation will be no greater than 120inches? The Z-score is 1.3, and the corresponding probability is 0.9032. Question 3. (10 points) The annual yield of various investment options has a normal distribution with mean 5% and standard deviation 5%, and are assumed to be independent. 4) If one chooses a single investment option, what is the probability that the annual yield is more than 7%? The Z-score is (7 – 5)/5 = 0.4. Thus, the corresponding probability is 1 – 0.6554 = 0.3446 5) A fund combines 16 different investment options and guarantees the average annual yield of them at the end of term. What is the probability that the yield of the combined investment fund is more than 7%. According to the central limit theorem, 5 / [square root of 16] = 5/4 = 1.25 is the standard deviation for the average annual yield. The zscore is z = (7 – 5)/1.25 = 1.6, and therefore, the probability is 1 – 0.9452 = 0.0548. Question 4. (10 points) Answer the following questions. 6) The temperatures (in degrees Fahrenheit) in 7 different cities on New Year's Day are listed below. Find the median temperature. The median is the 4th smallest value 51. 78

51

33 65 26

29

68

7) According to the empirical rules, what percentage of data can be found within four standard deviations 4s? That is, the percentage in the interval (y−2s, y+2s) around the mean value y. How about 5s? That is the percentage in the interval (y−2.5s, y+2.5s). The corresponding intervals of Z-score are given by (−2, 2) and (−2.5, 2.5), and their probabilities are 0.9772−0.0228 = 0.9544 and 0.9938−0.0062 = 0.9876. Thus, they are 95%, and 99%, respectively.

MATH 3070: Test I: Data sheet Billionaires Data. Fortune magazine publishes a list of the world's billionaires each year. The 1992 list includes 225 individuals. Their wealth are reported and summarized in the following table (in billions of dollars). Variable Mean S.D wealth 2.725778 3.368334

Min L.Quartile 1 1.3

Median U.Quartile 1.8 3

Max 37

The frequency histogram for their wealth (in billions of dollars) is given in the following figure.

The box plot for their wealth (in billions of dollars) is given as follows.

Hubble Data. In 1929 Edwin Hubble investigated the relationship between distance and velocity of extra-galactic nebulae (celestial objects). He published the data about how galaxies are moving away from us no matter which direction we look, and hypothesized the so-called “Hubble's law” as follows: Recession velocity = (Hubble's constant) * Distance

Given here are two plots which were produced from 23 data points Hubble published in 1929. The correlation coefficient was 0.8 between two variables. The intercept −40 and the slope 454 were estimated for the regression line.

Cancer Treatment with Ascorbate. Patients with advanced cancers of the stomach, bronchus, colon, ovary or breast were treated with ascorbate. Survival times were recorded and collected for the study. The purpose of the study was to determine if the survival times differ with respect to the organ affected by the cancer.

Given here is the ANOVA table obtained from the above cancer treatment data. Source Group Error Total

df 4 59 63

SS 11535761 26448144 37983905

MS 2883940 448273.6

MATH 3070: Test I (Part 2) Question 5. (20 points) Answer the following questions regarding the world's billionaires in 1994 (see Billionaires Data). 8) Describe the shape of wealth distribution. Are there any potential outliers? If so, explain how you found them. It is skewed to the right. The box plot indicates about 12 potential outliers.

9) Explain when the empirical rule is applicable. Does the empirical rule apply for the wealth distribution of the billionaires? Justify your answer. The empirical rule is applicable when a distribution is symmetric and bell-shaped. Thus, the empirical rule does not apply for the wealth distribution.

10) Identify all the measures of center in Billionaires Data Sheet. Which measure of center would you recommend to use for the billionaires' wealth? Justify your answer. Mean 2.7 and median 1.8. We recommend the median because the wealth distribution is skewed to the right.

11) Identify all the measures of variation in Billionaires Data Sheet. Calculate or find the value for each of the measures of variation you have identified. The standard deviation (SD) is 3.37, and the interquartile range (IQR) is 3 – 1.3 = 1.7. [You could also add the range 37−1 = 36, and the coefficient of variation (3.37/2.73)(100)=123%.]

Question 6. (30 points) Answer the following questions regarding Hubble Data. 12) Which variable, Recession velocity or Distance, should be the explanatory variable? Distance should be the explanatory variable according to the Hubble’s law equation. 13) Which of the plots, the left or the right one in Hubble Data Sheet, is the residual plot? Does it indicate that the residuals are homogeneous? Or, heterogeneous? The right one is the residual plot. It seems homogeneous. (But you could say heterogeneous, or indicate some pattern in the residual plot, also known as heteroscedasticity).

14) Is the 16th data point an extreme value? Should it be removed? Why or why not? The 16th data point is identified as an extreme value in the residual plot. It should be removed if one has indicated heterogeneity of residuals in (13). 15) After removing the 16th data point, the correlation coefficient was 0.82. The intercept −58 and the slope 449 were estimated. Does the result indicate a significantly stronger association between Recession velocity and Distance? Justify your answer. The correlation coefficient 0.82 does not indicate a significantly stronger relationship in comparison with the previous value of 0.80. 16) Is the 16th data point influential? Why or why not? No. The association did not improved significantly by removing the 16th data point.

17) The Hubble's constant is now thought to be about 75. Does the data published in 1929 support this Hubble's constant up to date? Justify your answer. No, since the slope coefficient was 454 (or 449 after removing the 16th data point) from the data published in 1929, and it is significantly higher than the current estimate of the slope which is about 75. Question 7. (15 points) Answer the following questions regarding Cancer Treatment with Ascorbate. 18) Which of the five groups has the lowest survival times? Justify your answer. The box plots indicate that either bronchus or stomach cancer has the lowest median. If the summary statistics are available, you will find the stomach cancer has the lowest median (124 days) compared to the median survival time (155 days) of bronchus cancer. 19) The objective of the study is to evaluate whether the survival times differ with respect to the organ affected by the cancer. Write your observation based on the box plots in Cancer Treatment with Ascorbate. In the boxplots we can observe that five groups appear to have some difference in survival times.

20) Write your conclusion of study based on the ANOVA table. Does it change your observation stated above, or reaffirm it? From the output the ratio of mean squares is 6.4, which is significantly larger than one. Thus, there is a significant difference in survival times among the five groups. Thus, we reaffirm the observation made in (19).