Question:

A non-profit research organization conducted a study in which they collected data regarding people's incomes and level of happiness. The organization is interested in examining the relationship between people's happiness and their income, given the common assumption that more money results in more happiness. Although overall the researchers were able to determine a signiicant relationship between people's incomes and happiness score, the researchers speculate that once people have enough money to pay their expenses and build their savings, that more income will have little or no effect on their overall happiness. They collected data for those people's whose income was $70,000 per year or more; these data are shown below with Income shown in units of $10,000 and a happiness score between 0 and 10.

a. Draw a scatterplot of the data and explain what the graph indicates about the appropriateness of simple linear regression for these data.

b. Determine the estimated simple linear regression model for these data.

c. Interpret the coefficient from part b above.

d. What Is the value of coefficient of determination and what does it indicate about the simple linear regression model from part b?

e. Explain what you would conclude about the researchers' speculation that once people have enough money to pay their expenses and build their savings, that more income will have little or no effect on their overall happiness. Be sure to justify your conclusions with statistical evidence.

data : the income and score for 40 persons

Income 7.00439884 7.02551769 7.0320571 7.06279211 7.07616637 7.08158906 7.10161959 7.11158364 7.11722011 7.11947859 7.11950538 7.13675997 7.15367409 7.16187287 7.17639965 7.18090719 7.18611226 7.19440696 7.19503729 7.19640925 7.2070597 7.22519186 7.22826521 7.241313 7.24675483 7.26037414 7.29772228 7.30090272 7.31050264 7.31091603 7.34724552 7.36421338 7.4238325 7.43924761 7.44811656 7.45150065 7.46350981 7.4666532 7.47844662 7.48152138

Score 5.36244826 4.81153338 5.04017458 6.86338795 5.01057743 4.12164775 5.34835294 6.08647828 5.56254547 5.9518141 4.98198112 5.50662154 4.72098734 6.1604341 5.02995166 5.60796663 5.15159829 5.83619666 5.23170297 5.59639827 4.20947057 4.98525493 5.03400379 4.68281699 5.8901487 5.38139847 6.3842745 6.00498747 5.92334607 5.7474441 4.39662232 6.61827984 4.90968847 6.3596 5.96342209 4.10994465 4.50344538 5.96054731 4.87772554 6.19629

Which of the following statements regarding the Central Limit Theorem is FALSE? (a) The Central Limit Theorem assumes that the sample drawn from the population consists of i.i.d random variables. (b) The Central Limit Theorem states that, for large sample sizes, the sample mean has an approximately normal distribution. (c) The Central Limit Theorem states that, for large sample sizes, the random variable YLE has a H standard normal distribution where T, ,u,, and or are the sample mean, population mean, and population standard deviation respectively. ((1) The Central Limit Theorem assumes that the distribution of the population from which the sample is drawn must be normal. Problem 1: A Central Limit Theorem Simulation Here we will perform a Central Limit Theorem simulation similar to the ones done in class. That is: . Pick a distribution (that was not presented in class) . Justify that the distribution will abide by the central limit theorem . Find a parameter set and value for / where we can see that the central limit theorem clearly applies . Find a parameter set and value for / where we can see that the central limit theorem does not apply . Code must be submitted (preferably in R, but MATLAB, Python, ForTran, and C++ will also be accepted).riate Normal Distribution: Properties Marginal Distributions of Bivariate Normal Random Variables If X and Y have a bivariate normal distribution with joint probability density fxy(x, y, ox. Oy, Hy, Hy, p), the marginal probability distributions of X and Y are normal with means Hy and py and standard deviations ox and oy, respectively. (5.22) Conditional Distribution of Bivariate Normal Random Variables If X and Y have a bivariate normal distribution with joint probability density fxy(x, y, ox. Or, Hx, Hy, p), the conditional probability distribution of Y given X = x is normal with mean Hylx = Hy + P- (x- Hx) and variance ONLY = o? (1 - p? )\f