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The simple regression model (SRM) is model for association in the population between an explanatory variable X and response Y. The SRM states that these averages align on a line with intercept ? 0 and slope ? 1: µy|x = E(Y|X = x) = ? 0 + ? 1x Deviation from the Mean The deviation of observed responses around the conditional means µy|x are called errors (? ). The error’s equation: ? = y – µy|x Errors can be positive or negative, depending on whether data lie above (positive) or below the conditional means (negative). Because the errors are not observed, the SRM makes three assumptions about them: * Independent.

The error for one observation is independent of the error for any other observation. * Equal variance. All errors have the same variance, Var(? ) = ?? 2. * Normal. The errors are normally distributed. If these assumptions hold, then the collection of all possible errors forms a normal population with mean 0 and variance ?? 2, abbreviated ? ?? N (0, ?? 2). Simple Regression Model (SRM) observed values of the response Y are linearly related to values of the explanatory variable X by the equation: y = ? 0 + ? 1x + ? , ? ?? N (0, ?? 2)

The observations: 1. re independent of one another, 2. have equal variance ?? 2 around the regression line, and 3. are normally distributed around the regression line. 21. 2 Conditions for the SRM ( Simple Regression Model ) Instead of checking for random residual variation, we have three specific conditions. Checklist for the simple regression model * Is the association between y and x linear? * Have we ruled out obvious lurking variables? Errors appears to be a sample from a normal population. | * Are the errors evidently independent? * Are the variances of the residuals similar? Are the residuals nearly normal?

The estimated standard error of b1 substitutes the sample standard deviation of the residuals Se for the standard deviation of the errors ?? , Se (b1) = sen-1 x 1Sx ? Sen x 1Sx The residual standard deviation sits in the numerators of the expression for se (b1). Since the regression line estimates the conditional mean of Y given X, it is the residuals that measure the variation around the mean in regression analysis. The sample size n is again in the dominator. The larger the sample grows, the more information we have and the more precise the estimate of the slope becomes.

Confidence Intervals T = b1- ? 1se(b1) If the errors ? is normally distributed or satisfies the CLT condition, then sampling distribution b1 is approximately normal. Since we substitute Se for ?? to calculate the standard error, we use a t-distribution for inference. The 95% confidence interval for the slope ? 1 in the simple regression model is the interval [b1 – t0. 025,n-2 x se(b1), b1 + t0. 025,n-2 x se(b1)] The 95% confidence interval for the intercept ? 0 is [b0 – t0. 025,n-2 x se(b0), b0 + t0. 025,n-2 x se(b0)] Hypothesis Test

Equivalent Inferences for the SRM We reject the claim that a parameter in the SRM (? 0 or ? 1) equals zero with 95% confidence (or a 5% chance of a Type I error) if a. Zero lies outside the 95% confidence interval for the parameter; b. The absolute value of the associated t-statistic is larger than t0. 025,n-2 ? 2; or c. The p-value reported with the t-statistic is less than 0. 05. Regression is often used to predict the respons for new, unobserved cases. In this case, the explanatory variables (xnew) is known but the response (ynew) is unknown.

To solve this case, the SRM provides a framework that predict ynew and anticipates the accurancy of this prediction. The SRM models implies that ynew is determined by the equation : Ynew = ? 0 + ? 1 xnew + ? new (? new is a random error term that represent the influences of other factors on the new observation). Prediction interval is an interval designed to hold a fraction (usually 95%) of the values of the response for a given value x of the explanatory variable in a regression.

The 95% prediction interval for the response ynew in the SRM is the interval [ynew – t0. 025,n-2 se (ynew), ynew – t0. 25+2 se (ynew)] se ynew=se 1+1n+xnew- x2n-1sx2 the standard error of a prediction se (ynew) is tedious to calculate because it adjust for the position of xnew relative to the observed data. The fathers xnew is from x, the longer the prediction interval becomes. Reliability of Prediction Intervals A prediction interval measures the accuracy of predictions of new observations. Provided the SRM (Simple Regression Model) holds, the approximate 95% prediction interval for an observation at x is ynew ± 2se. Prediction intervals are reliable within the range of observed data, the region in which the approximate interval [y – 2se, y + 2se ] holds.

Prediction intervals are also sensitive to the assumptions of constant variance and normality. If the variance of the errors around the regression line increases with the size of the prediction, then the prediction intervals will be to narrow for large items. So, before using prediction intervals, verify that residuals of the regression are nearly normal. The method: * Identify x and y * Link b0 and b1 to problem * Describe data. It can be linear, no obvious lurking variable, evidently independent, similar variances, or nearly normal * Check linear condition and lurking variable condition

Brand’s advertisement strategy through Social Media

Brand’s advertisement strategy through Social Media.

First Draft For your first draft, you need to hit the specified minimums for the first draft (if you go beyond this, that is totally fine, but these are here to give us a substantive amount of material to supply you with the necessary feedback). Your final paper serves as a culmination of what we’ve been covering throughout this course. It is aimed at pushing you to position yourself as a researcher, conceptualizing, approaching, and proposing a potential study that you could feasibly complete. Minimums for First Draft: Minimum of 6 full pages of text. A fully fleshed out Introduction The foundational aspects of the literature review Here, work out the possible subsections (pulling heavily from the direction you took in your bibliography project) Work these out from the literature you’ve already found. Go from just the annotations you’ve supplied into assembling them together for a specific purpose and point. The full final paper needs to be formatted in the following manner/contain the following requisite support: Size 12 Times New Roman font Minimum 12 pages of content (i.e. 12 pages minimum plus cover page and references) 1” margins all around Double-spaced A uniformly applied citation method (both in-text and in the references list) You can use ASA, APA, MLA, etc. A minimum of 15 resources 13 of these resources must be from academic peer-reviewed resources 2 of these resources must come from a non-academic (though trustworthy) resource. These are especially useful for your introduction To that end, the Research Proposal contains the following, required items: Cover Page (this is not included in the 12 page minimum) Introduction Literature Review Synthesis Conclusion References (this is not included in the 12 page minimum) Each of these components break down as follows: Cover Page On the cover page you need to have the following items: title, your name, course number, and date submitted Introduction The introduction will serve to introduce the topic and idea to your audience (i.e. me). In this case, you need to construct an introduction that motivates an interest in the topic as well as introduces the basic necessities of understanding to your reader. To that end, your introduction should focus primarily on your issue of interest (the pay gap, for example). Make sure there is a clear thesis of what your overall paper is as well as the research question you’re attempting to answer. It should give some insight into the components that underlie that issue (define it, give a bit of history on it, and also present basic statistics for it, which you can obtain through places like newspaper articles or research organizations like Pew). Remember: the introduction motivates the reader to want to know more, so it needs to stay focused on the issue that you want to address. If you have an annual review article, it can give you some great basic details for your introduction. As such, you should enter in with a specific claim or point. Don’t go so broad as “Since the beginning of time” because… no. Rather, you want it to be something that is clear and direct. “Determining why people buy what they buy has been a core concern of economic theorizing.” Note the differences. From there, you would want some general, broader points such as “Americans spend approximately ### billion dollars a year on grocery shopping” or something else that is tied into your specific claims and topic. Obviously, make sure you cite these claims. Work down in this fashion allows your introduction to go from your broader point/interest down into the specific manifestations of what you’ll be arguing, which thesis will explicitly articulate.

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