The Following Table Is Given: R=300, R=400, R=500, P(B=b) ✓ Solved
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The following table is given: R=300 R=400 R=500 P(B=b)
1. The following table is given:
- R=300
- R=400
- R=500
P(B=b) B=20 .2 B=25 .2 .6 P(R=r) .3 .
a. Fill in the missing values of the table. b. Calculate E[R | B=20] c. Calculate E[R | B=25]
2. Dress for Success Business Clothing has outlets in several cities in the region.
The sales department aired a commercial for a men’s business suit on selected local TV stations prior to a sale starting on Saturday and ending Sunday. The sales department obtained the information for Saturday-Sunday men’s suit sales at the various outlets and paired it with the number of times the advertisement was shown on the local TV stations. The goal is to find whether there is any relationship between the number of times the advertisement was aired and men’s suit sales. Here is the data:
Location of TV Station | Number of Airings | Saturday- Sunday Sales ($ thousands)
- Pleasant Valley | 4 | 15
- Feagaville | 2 | 8
- Leesburg | 5 | 21
- Oxford | 26 | 24
- Woodsboro | 3 | 17
a. Draw and label a scatter diagram using the given data.
b. Determine the correlation coefficient.
c. Interpret the correlation coefficient (both strength and direction). What does this suggest to the sales department?
3. The town council of Smithburg is considering increasing the number of police to try to reduce crime.
While making this decision, the council decided to research the relationship between the number of police and the number of crimes reported. They gathered the following information from 8 other cities:
City | Police | Number of Crimes
- Pleasant Valley | 15 | 17
- Holgate | 17 | 7
- Feagaville | 17 | 13
- Woodsboro | 12 | 21
- Leesburg | 25 | 5
- Danville | 11 | 19
- Oxford | 27 | 7
- Athens | 22 | 6
a. Determine the regression equation with crimes as the dependent variable and police as the independent variable. Calculate using the approach shown in class—either by hand or with Excel.
b. Estimate the number of crimes for a city with 20 police officers.
Paper For Above Instructions
The assignment is divided into three core parts. The first part involves analyzing the relationship between variable R and variable B, based on the given table. The second part examines the connection between advertisement airings and suit sales, providing insight through data collection and analysis. The final part explores the impact of police presence on crime rates in a community. Each of these components underscores the importance of statistical analysis in decision-making.
1. Analysis of the Given Table
The first task requires filling in the missing values of a probability table. This includes determining values for R=300, 400, and 500, required to complete the distribution. Additionally, one must calculate expected values under specific conditions. The expected value, denoted as E[R | B=b], signifies the average outcome based on prior knowledge of event B.
To fill in the missing values, we analyze the probabilities of R against the respective values of B. Given that P(B=20)=0.2 and P(B=25)=0.2, the remaining probability must equal 0.6 to total 1. Consequently, R can be mathematically analyzed to compute E[R | B=20] as follows:
E[R | B=20] = Σ (R * P(R | B=20))
= 300 P(R=300 | B=20) + 400 P(R=400 | B=20) + 500 * P(R=500 | B=20).
Similar calculations can be conducted for E[R | B=25]. This demonstrates the role of conditional expectations in evaluating outcomes.
2. Advertisement Impact on Sales Analysis
The second component of the assignment revolves around the advertisement's impact on the sales data collected. A scatter diagram will visually represent the relationship between the number of TV airings and the sales generated over the specified time period. Each data point will illustrate a different city’s performance based on advertisement frequency. This visual tool serves as an effective means of assessing correlation.
Next, calculating the correlation coefficient will provide a quantitative measure of the strength and direction of the relationship. Utilizing the Pearson correlation formula, the sales department can interpret the data to draw actionable insights. Should the correlation coefficient reveal a positive relationship, the sales department could justify further investment in advertising to enhance sales.
Understanding the correlation coefficient's significance is crucial as it informs the sales team whether the advertisement strategy is effective. For instance, a strong positive correlation might suggest that airings effectively boost suit sales, prompting adjustments in future marketing strategies.
3. Police Presence and Crime Rates
The last task involves determining the regression equation linking police numbers to crime rates across various cities. By performing regression analysis, the town council can uncover the relationship between increased police presence and crime reduction. The regression equation typically takes the form of Y = a + bX, where Y represents the number of crimes, X is the number of police officers, 'a' is the intercept, and 'b' is the slope indicating how crime rates change with each additional officer.
Once the regression equation is established, the council can predict the number of crimes in areas expecting to employ more police officers. For example, evaluating estimated crimes for a city with 20 officers will illuminate the potential impacts of such decisions, enabling data-driven policymaking aimed at community safety.
Conclusion
Overall, this assignment highlights critical analytical tools in statistics necessary for interpreting real-world data. Whether determining expected values, correlational analysis in sales, or regression modeling in municipal governance, the application of these techniques empowers stakeholders to make informed decisions that align with community needs.
References
- Gravetter, F. J. and Wallnau, L. B. (2017). Statistics for The Behavioral Sciences. Cengage Learning.
- Upton, G. and Cook, I. (2014). Understanding Statistics. Oxford University Press.
- Trochim, W. M. K. (2006). The Research Methods Knowledge Base. Cengage Learning.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. SAGE Publications.
- Siegel, S., and Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
- Chatterjee, S. and Hadi, A. S. (2012). Regression Analysis by Example. Wiley.
- Weiss, N. A. (2017). Introductory Statistics. Pearson.
- Cook, R. D., and Weisberg, S. (1982). Residuals and Influence in Regression. Springer-Verlag.
- Tabachnick, B. G., and Fidell, L. S. (2013). Using Multivariate Statistics. Pearson.
- Bluman, A. G. (2017). Elementary Statistics: A Step by Step Approach. McGraw-Hill.
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