# Read attached file.Part 1Come up with an example of a hypothesized correlation between the quantity of a product consumed and a specific background variable of consumers. Clarify the operational defin

UNIT 4: QUANTITATIVE METHODS QUESTION #4.1: How does quantitative research “prove” hypotheses? SHORT ANSWER: by calculating the probability of the results occurring by pure chance Social scientists don't like to speak of their research as “proving” th eir hypotheses. The preferred term is “confirming” the hypotheses. The way that this is done is rather backward: we calculate (or estimate) the probability of the observed results occurring by random variation (i.e., pure chance, luck) and then if that pro bability is sufficiently low enough, we say that some alternate hypothesis is a better explanation for the results. http://www.youtube.com/watch?v=G60Hp_iFW5I This approach to statistical sign ificance is called null hypothesis testing. A null hypothesis is a statement that that we did not really prove anything because the observed results could be explained by random variation. In order to prove something, I have to come up with an initial hypo thesis that I will test. I do this by stating my hypothesis as an alternative to the null. The only way I can confirm my hypothesis is to reject the null. The only way I can reject the null is to show that it is a very improbable explanation of the results . Many times, the null hypothesis is obviously the best explanation for the observed results. For example, suppose I say that the majority of the shoppers at a certain grocery store on Tuesday afternoons are women. (In other words, my alternate hypothesis is that most of the observed shoppers will be female.) So, I stand outside the store and observe the first three customers to exit the store. Number one is a woman in her thirties, pushing a basket with two little children inside; number two appears to be an older woman, alone, looks like she just finished some activity at the senior center across the street; number three is a younger women in her late teens or twenties, looks like she just got off of her office job. So, I'm three for three; does that mean that I confirmed my alternate hypothesis that most shoppers are women? If you are thinking like a scientist, your reply would be to stick with the null hypothesis as plausible explanation.

Women are half of the population in the city where I did my observ ations, so the odds of observing three woman would be ½ times ½ times ½ which would be 0.125. Most scientists would not reject the null at that probability of random variation explaining the results. Statistical significance is expressed in terms of a p value, which stands for probability (of the null hypothesis). P values range from 0.00 (indicating that something is impossible) to 1.00 (indicating that something is certain). The more improbable that the null hypothesis is, the more likely our alternate hypothesis is. Scientists usually accept the following cut off points for rejecting the null hypothesis: p below .05, reject the null with fair confidence; p below .01, reject the null with good confidence; p less than .001 reject the null with excellent c onfidence. http://www.youtube.com/watch?v=G60Hp_iFW5I STATISTICAL SIGNIFICANCE (probability of the null hypothesis) p = 1.00 - - - - - - - - - - - - - - - - - - - - - - - - - (certainty ) p > .10 not significant ACCEPT THE NULL p = .10 - - - - - - - - - - - - - - - - - - - - - - - p < .10 marginal ACCEPT THE NULL p = .05 - - - - - - - - - - - - - - - - - - - - - - - p < .05 fair REJECT NULL p = .01 - - - - - - - - - - - - - - - - - - - - - - - p < .01 good REJECT NULL p = .001 - - - - - - - - - - - - - - - - - - - - - - - p < .001 excellent REJECT NULL p = 0.00 - - - - - - - - - - - - - - - - - - - - - - - (impossibility) In the case of the above example, the probability of getting three women out of three observations was p = .0125. We look at the above chart and find where that would place is: we are still in the area that is not significant. We must accept the null hypothesis and admit that our (alternate) initial hypothesis was not confirmed. This process of calculating (or estimating) the probability of t he null hypothesis is known as inferential statistics . They give us the p value that tells us if we have confirmed our hypothesis or whether we have to accept the null (and admit that we have proved nothing). - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - GRAMMAR LESSON: Do not use the word significant unless you mean p < .05. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - If we reject the null hypothesis when there really isn’t an underlying difference or co rrelation, we have committed a Type I error. We can reduce the number of Type I errors by gathering our data carefully, selecting the right statistical tests or even by imposing a more stringent standard for the rejection of the null (e.g., the .01 level). However, we would not want to increase our vulnerability to Type II error (which means failure to reject the null even though real differences do exist. QUESTION #4.2: How important is sample size? SHORT ANSWER: sample size is v ery important in quantitative research In doing qualitative research, it is less important how many are in our sample, and more important how much we get from them. In quantitative research, sample size is very important because the larger the sample siz e, the easier it is for an observed trend to be statistically significant. In the above example, I observed a definite trend: all shoppers leaving the store were women, but my sample size was small (n = 3). Suppose I had observed four customers leaving the store: all women. Now, the probability of that occurring would by half of 0.125 (p = .0625) but the above chart shows that although we are marginal (getting close to statistical significance) we would still have to accept the null. However, if my observat ion was five our of five, that is even less likely to occur by pure chance (p = .03125) we could then reject the null with fair confidence. Let us be clear, you cannot just stand out in front of the store and wait for four women in a row to exit, ignorin g all the men who came before and after. That would be as bad as flipping a coin, ignoring the tails, and claiming that you got a lot of heads. You should include all observations in your sample. However, with a sample size of a thousand, if you observed 5 5% females exiting the store (compared to only 45% males) those results might be statistically significant.

Notice that most polling companies and marketing research companies usually have a sample size of at least a thousand. However, for student research , a sample size of fifty might be more realistic, but you might need to observe almost 40 women in order to reject the null hypothesis. QUESTION #4.3: What other kinds of statistics are used in research on consumer behavior? SHORT ANSWER: descriptive s tatistics such as mean, median, mode, percent, standard deviation Most (alternate) hypotheses are phrased in terms of measures of central tendency about the dependent variable: mean, median, mode, percent. Some hypotheses are phrased in terms of the dispe rsion (e.g., standard deviation, variance, range). Which of these measures is appropriate depends upon the quantitative scale in which the dependent variable is measured. Statistics textbooks explain the distinctions between ratio and interval scales (and whether they are continuous or discrete). Most of that doesn't really matter in selecting the best measure to describe the central tendency of the dependent variable. What matters most is whether the distribution of the variable is close to symmetrical (i. e., normal, standard, Gaussian) such that most of the scores are close to the middle of the range. In that case, we can use the arithmetic mean as the average. http://www.youtube.com/watch?v=16hUQ rX8akI If there are a few extremely high (or low) outliers, we say that the distribution is skewed , and it would be wiser to use the median , which would be the score attained by a person in the middle of the distribution . http://www.youtube.com/watch?v=FwsImyWiqjY Income distribution is a good example of a skewed variable. Suppose a men's group at a church has ten members. Nine of them are small business owners or professionals and make close to \$100,000 annually. The tenth member is the C.E.O. of a large corporation, and his income last year was over ten million dollars. This would produce an extreme right skew in our distribution.