This paper works through a series of applied statistics problems covering correlation analysis, regression modeling, ANOVA interpretation, and research ethics. Topics include the limitations of formal education as a variable in historical context, the relationship between gas prices and passenger mile ratios, correlation matrices for sales force data, linear demand functions, F-statistic significance testing, the coefficient of determination (r²), sports team performance correlations, and the ethical obligations of researchers using Likert-scale survey data. Each problem illustrates core concepts in inferential statistics and proper analytical methodology.
In the case of Abraham Lincoln, the researcher undoubtedly expected to find a correlation between education level and lifetime achievement. However, the researcher made two mistakes. The first was expecting such a correlation in the first place. In Lincoln's time, the correlation between education and achievement was weaker than it is today, and a man could become President without a strong formal education. Education was often informal, and much of the relevant learning for all citizens came outside of the formal education system. Thus, the variable of formal education was not a particularly useful variable during Lincoln's era. It is not unreasonable for a researcher to test the variable of education, but if that variable proves not to be useful, the researcher should not be surprised. The researcher should view the hypothesis as equally likely to be disproven as proven.
The second mistake the researcher made was to assume that there would be no outliers in the survey results. With any measure of correlation, it is reasonable to expect exceptions or outliers. For other individuals, the correlation between education and achievement may have been strong, but it is equally reasonable that the correlation is not universal. Thus, the researcher should not have been surprised at the presence of an outlier in the survey results.
The researcher had clearly expected a linear relationship. However, as the flaws in the thinking reveal, Abraham Lincoln did not exhibit that relationship. Furthermore, the researcher did not test for the assumptions of a linear relationship. An assumption was made, but it does not appear that the different variables in the study were tested against each other. Therefore, the researcher did not adequately test for the conclusion before reaching it.
What the figures show is that the price of gas is highly correlated with the passenger mile ratio. There were three main relationships tested statistically. In addition to the relationship between price and passenger mile ratio, each of those variables was tested against "year," representing normal growth. From these figures, we can see that normal growth — the change in passenger mile ratio year over year — is less strongly correlated than the change in relation to gas price changes.
The model shows that gas prices increase as time increases, such that on average there is an increase in the gas price with each passing year. The model also shows an increase in the passenger mile ratio with each passing year. This indicates to management that there may be a correlation between gas prices and passenger mile ratio, and more clearly that the passenger mile ratio improves over time, on average. The data further shows an increase in passenger mile ratio with an increase in gas prices.
The strength of these correlations tells a more specific story. Although business improves each passing year, some of that improvement is attributable to natural growth and some is directly related to the increasing price of gas. The correlation between gas prices and passenger mile ratio is stronger and exhibits less volatility than the other correlations studied. This indicates that a certain portion of the increase in the passenger mile ratio is attributable directly to increases in the price of gas. This data therefore helps management understand the impact that gas prices have on their business growth potential by allowing them to separate the strength of that relationship from the strength of the relationship between time and the passenger mile ratio.
This correlation matrix reveals interesting results. The first set of variables — age and years of service — has the highest correlation and the highest probability. This is not unexpected, since age is a limiter to years of service. A thirty-year-old sales representative will not have twenty-five years of service. The correlation is only moderately strong, but that again is not unexpected. Some sales representatives will have joined the firm mid-career, while others will have started with the company at an early age, such that a 40-year-old and a 30-year-old representative could both have eight years of experience.
The second set of variables — years of service and current sales — reveals a moderately strong correlation. The correlation is not as strong as between the previous set of variables, but there is a clear correlation between these two variables. Again, this is not unexpected, since a more experienced sales representative will not only know the business better, but will also have a deeper and more manageable client list. The correlation is only moderately strong because experience is only one variable in determining current sales; the other key variable is talent.
The third set of variables contains the most interesting results. Age does not correlate well with current sales. Given that a moderate correlation was found in the other two sets of variables, the weakness of this correlation is somewhat surprising. If this matrix is used to undertake a path analysis, we can see that while there is some correlation between age and experience, the only one of those two that correlates to sales is experience. We would expect, given this matrix, that the raw data will show some younger but experienced representatives outperforming older, less experienced ones. For management, this information is valuable because it indicates they should hire younger sales representatives so that the company can benefit from additional years of experienced service from them.
The first relationship can be broken down into its constituent parts. The 3.5 represents the baseline — the average likelihood of a new car purchase. To this is added a variable that impacts that likelihood: family income. The relationship illustrates that as family income increases, the likelihood of a new car purchase also increases. The relationship is not perfectly linear; however, the correlation coefficient of 0.7 illustrates the strength of the linear relationship. Thus, there is a moderate-to-strong linear correlation between family income and the likelihood of a new car purchase.
The second relationship begins with the same baseline demand. To this, the variable of age is added. However, for rock concerts, the variable's correlation differs from that of new cars in two key ways. First, the strength of the correlation between rock concert ticket purchases and age is weaker, at 0.4. This indicates that there is some correlation, but it is not particularly strong. The other difference is that the correlation is negative, which would be represented graphically with a downward-sloping line rather than an upward-sloping one. Thus, the second relationship illustrates that there is a weak-to-moderate increase in the likelihood of a rock concert ticket purchase as a person gets younger — or, equivalently, that the likelihood of a rock concert ticket purchase declines as a person gets older.
Combined, these two relationships show that the independent variable of family income has more of an impact on the car purchase decision than age has on the decision to purchase rock concert tickets. The two relationships together illustrate differences in both the intensity and direction of the demand slope for the products and variables in this comparison.
An ANOVA summary table outlines the results of a regression analysis. The usefulness of this is that these results can then be weighed for statistical significance according to different parameters. In this case, the parameter is 5%. The relationship illustrated in this ANOVA summary table is statistically significant at 5%.
The F-value as found on the table is 3.12. The degrees of freedom listed on the ANOVA table are 1 for the numerator and 8 for the denominator. According to the F-distribution table, the upper critical value for the 5% level where ν₁ = 1 and ν₂ = 8 is 5.318. Therefore:
p (F = 3.12 < .05)
The F value falls within the acceptable range for statistical significance at 5% variance. Therefore, we accept the null hypothesis, and the relationship is statistically significant at 5%.
The r² is a number between 0 and 1 that relates to the degree of predictability between two variables. The higher the number, the greater the degree of predictability. In this case, the regression provides an r² of 0.7824. This implies a relatively strong linear relationship between the variables tested. The correlation will produce a strong, upward-sloping line, and therefore the regression will be a good predictive model.
The prediction will not be 100% accurate, but there is reason to believe that the predictions generated from the model will be statistically significant. The economist will have a fairly reliable estimate of the average total budget of retired couples in Phoenix based on this model, given the relatively high level of r² obtained from the gathered data.
a) There is very little correlation between the percentage of games won and the other variables. The correlation between percentage of games won and season ticket sales is 0.015, which indicates almost no correlation at all. The correlation between percentage of games won and alumni support is negative and weak, suggesting no meaningful correlation — or, if any, that alumni support increases with declines in games won. There is, however, a high degree of correlation between alumni support and the number of season tickets sold. This correlation is 0.95, suggesting almost perfect co-movement between the two variables. It can reasonably be expected that as one increases, the other will move in the same direction and with very similar intensity.
"F-value interpretation at 5% significance level"
"r² value and predictive model strength"
"Win rate, ticket sales, and alumni correlations"
"Ethical obligations in bivariate correlation studies"
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