Question 1 You may need to use the appropriate technology to answer this question. How long it takes paint to dry can have an impact on the production capacity of a business. An auto body & paint business invested in a paint-drying robot to speed up its process. An interesting question is, "Do all paint-drying robots have the same drying time?" To test this, suppose we sample five drying times for each of different brands of paint-drying robots. The time in minutes until the paint was dry enough for a second coat to be applied was recorded. Suppose the following data were obtained. Robot 1 Robot 2 Robot 3 Robot 4 129 143 134 149 137 133 142 143 134 141 138 136 123 146 136 139 142 132 130 153 At the 𝛼 = 0.05 level of significance, test to see whether the mean drying time is the same for each type of robot. State the null and alternative hypotheses. H0: At least two of the population means are equal. Ha: At least two of the population means are different. H0: 𝛼1 ≠ 𝛼2 ≠ 𝛼3 ≠ 𝛼4 Ha: 𝛼1 = 𝛼2 = 𝛼3 = 𝛼4 H0: 𝛼1 = 𝛼2 = 𝛼3 = 𝛼4 Ha: Not all the population means are equal. H0: Not all the population means are equal. Ha: 𝛼1 = 𝛼2 = 𝛼3 = 𝛼4 H0: 𝛼1 = 𝛼2 = 𝛼3 = 𝛼4 Ha: 𝛼1 ≠ 𝛼2 ≠ 𝛼3 ≠ 𝛼4 Correct: Your answer is correct. Find the value of the test statistic. (Round your answer to two decimal places.) 2.53 Incorrect: Your answer is incorrect. Find the p-value. (Round your answer to four decimal places.) p-value = .08 Answer This question involves testing the hypothesis that the mean drying time for each type of paint-drying robot is the same. To do this, we will use a one-way ANOVA (analysis of variance) test. This test compares the variability between groups with the variability within groups to determine if there is a significant difference in the means. Null and Alternative Hypotheses: The null hypothesis, denoted as H0, states that there is no significant difference between the mean drying times of the four different types of paint-drying robots. The alternative hypothesis, denoted as Ha, suggests that there is at least one pair of population means that are different from each other. H0: 𝛼1 = 𝛼2 = 𝛼3 = 𝛼4 Ha: At least two of the population means are different. Calculations: To conduct the one-way ANOVA test, we first need to calculate the mean drying time for each group (Robot 1, Robot 2, Robot 3, Robot 4). The mean for each group is: Robot 1: 134.6 Robot 2: 139.8 Robot 3: 137 Robot 4: 137.4 Next, we need to calculate the grand mean, denoted as G. This is calculated by taking the sum of all observations and dividing it by the total number of observations, which in this case is 20. The grand mean is 137.4. Now, we can calculate the sum of squares between groups (SST), sum of squares within groups (SSE), and the total sum of squares (SSTO). These calculations are used to determine the F-statistic, which is then used to determine the p-value. The sum of squares between groups is calculated by finding the difference between the mean of each group and the grand mean and then squaring the difference. This is then multiplied by the number of observations in each group. SST = (5(134.6-137.4)^2) + (5(139.8-137.4)^2) + (5(137-137.4)^2) + (5(137.4-137.4)^2) = 280.32 The sum of squares within groups is calculated by finding the difference between each observation and the mean of its respective group, then squaring it. These values are then added together. SSE = (5(129-134.6)^2) + (5(143-134.6)^2) + (5(134-137)^2) + (5(149-137)^2) + (5(137-137)^2) + (5(133-139.8)^2) + (5(142-139.8)^2) + (5(143-139.8)^2) + (5(134139.8)^2) + (5(141-139.8)^2) + (5(138-137)^2) + (5(136-137)^2) + (5(123-137.4)^2) + (5(146-137.4)^2) + (5(136-137.4)^2) + (5(139-137.4)^2) + (5(142-137.4)^2 Question 2 Hight of a flower in cm is discrete or continuous Answer The height of a flower in centimeters can be considered a continuous variable. This means that the measurement can take on any value within a specific range, rather than being limited to certain distinct values. When we measure the height of a flower, we can use a ruler or measuring tape that is marked off in centimeters. This allows us to measure the height to the nearest millimeter, giving us a more precise value. This falls under the category of a continuous measurement, as we can measure any height between two whole numbers. Furthermore, the height of a flower can range from very small, such as a few centimeters, to very large, up to several meters. This wide range of possible values also supports the idea that height of a flower is a continuous variable. Question 3 questions on the Wald statistic. (a) In a linear regression you wish to test yi = x0 i + i; 2Rk; H0 : R =cagainst H1 :H0 is not true, where R is a r k constant matrix with r < k and rank(R) = r, and c 2 Rr is a constant vector. Show that the Wald statistic Wn = R^ c 0h R n s2(X0X) 1 o R0 i 1 R^ c can be rewritten as where Wn =(n k) ~2 ^2 ^2 ^2 = 1 n^ e0^ e; ~2 = 1 n~ e0~ e; and ^ e and ~e are the OLS and CLS (= LS estimation under H0 : R = c) residuals, respectively. [Hint: Apply the result of e0^ e ~ e0~. Answer The Wald statistic is a statistical test used to evaluate the significance of coefficients in a linear regression model. It is commonly used to test the null hypothesis of no relationship between the independent variable(s) and the dependent variable. In this scenario, we are interested in testing whether there is a significant linear relationship between the dependent variable yi and the independent variables x0i and i. To test this null hypothesis, we can rewrite the Wald statistic as follows: Wn = R^ c 0h R n s2(X0X) 1 o R0 i 1 R^ c Where: - R is a r x k constant matrix, where r < k - c is a constant vector of length r - n is the sample size - s2 is the mean squared error (MSE) of the regression model - X is the n x k matrix of independent variables - 1 is a k x r matrix of ones - ^ e is the OLS (ordinary least squares) estimation - ~ e is the CLS (constrained least squares) estimation under the null hypothesis To understand how this statistic can be rewritten, we need to understand the OLS and CLS estimations. OLS estimation is the most common method for estimating the parameters in a linear regression model. It minimizes the sum of squared residuals (SSR) between the observed and predicted values. On the other hand, CLS estimation is used when the null hypothesis imposes restrictions on the coefficients. It is similar to OLS estimation, but the sum of squared residuals is minimized under the constraint that the coefficients satisfy the null hypothesis. Given the above information, we can rewrite the Wald statistic as: Wn =(n-k) ^2 = ^ e0^ e; ~2 = ~ e0~ e (1/ n) Where: - n-k is the degrees of freedom of the regression model - ^2 = s2/(n-k) is the MSE of the regression model - ^ e0^ e and ~ e0~ e are the sum of squared residuals for the OLS and CLS estimations respectively - 1/n adjusts for the degrees of freedom in the CLS estimation This rewritten form of the Wald statistic is easier to interpret as it expresses the sum of squared residuals in terms of the OLS and CLS estimations. The difference between the two estimations is multiplied by 1/ n and then divided by the MSE, which is a measure of the overall variability in the model. This allows us to compare the two estimations on a similar scale and determine the statistical significance of the coefficients. Question 4 You are the customer service director of a mid-tier hotel chain, named “Stay Here for Best Memories” (SBM). The hotel chain has a mission statement to provide memorable experiences for its two (2) target market segments- business and leisure travellers. To obtain up-to-date feedback from hotel guests, a hotel experience survey is conducted electronically to obtain feedback from its guests about their stay at the hotel. The customer service director will update management on the key findings every quarterly. Timely and cost-effective survey findings are the key priorities of the customer service director. Survey Objectives 1. Compare the rating scores and qualitative feedback across different two (2) target market segments-business vis-à-vis leisure on their overall experience, room cleanliness & comfort and staff service & professionalism. 2. Identify areas of improvement and increase business opportunities for the hotel. Survey Questionnaire (Partial) Q1. Please rate the following aspects of your stay at our hotel on a scale from 1 to 5. (a) Overall experience (b) Room Cleanliness & Comfort (e.g., bed quality, temperature control) (c) Staff Service & Professionalism (e.g., responsiveness, helpfulness) Tasks Write a 900-word reflection on how the concepts and theorem(s) taught in the topic “Sampling Distribution of the Sample Means” and “Discrete Probability Distribution” can facilitate your decision-making process in conducting the hotel experience survey, such as choosing an appropriate sample size for the survey as well as other considerations, costs and benefits. Your reflection should provide discussion and explanation for the following areas: (a) Sampling Plan If the population standard deviation of the guest overall experience rating score is 1.5 points, would you advise the customer service director to randomly (i) select a hotel guest; (ii) select a sample of 20 hotel guests; (iii) select a sample of 30 hotel guests; or (iv) select a sample of 40 guests for each customer segment in the survey above? Evaluate the pros and cons of each of the 4 options above with appropriate calculations. Determine the sampling plan to meet the survey objectives above. (b) Survey Instrument Amend the survey questionnaire above and add further questions and/or sections to meet the survey objectives fully and elaborate on how you would analyse the questions in your survey questionnaire and present the key findings for the quarterly report to management. (c) Hotel Guest Complaint Situation Based on past data, the number of complaints received from hotel guests follow a Poisson Distribution with an average of 2 complaints per day. A day with more than 4 complaints is considered a high-complaint day. If the probability that there are more than 1 high-complaint days in a week exceed 0.05, it would warrant management’s attention. With appropriate calculations, determine if the complaint situation in the hotel warrant management's attention Answer Introduction: As the Customer Service Director of "Stay Here for Best Memories" (SBM), it is my responsibility to ensure the satisfaction of our hotel guests and uphold the mission of providing memorable experiences for both our business and leisure clientele. In order to obtain up-to-date feedback from our guests, we conduct an electronic hotel experience survey and analyze the results every quarter. The success of this survey is crucial in achieving our key priorities of timely and cost-effective feedback, as it allows us to identify areas of improvement and increase business opportunities for the hotel. In this reflection, I will discuss how the concepts and theorems taught in the topics of "Sampling Distribution of the Sample Means" and "Discrete Probability Distribution" can assist me in the decision-making process for conducting the hotel experience survey. Sampling Plan: In order to obtain accurate feedback, it is essential to have an appropriate sampling plan. This involves selecting a representative sample from our population of guests. As the population standard deviation of the overall experience rating score is 1.5 points, it is important to choose a sample size that will provide reliable results. This is where the concept of sampling distribution of the sample means comes into play. Selecting a random sample of one guest may not accurately represent the population, as it could be an outlier. On the other hand, selecting a sample of 20 or 30 guests will provide a better representation of the population and allow for more accurate results. However, if we select a sample of 40 guests, it may become redundant and not significantly improve the accuracy of our results. Therefore, I would suggest selecting a sample of 30 guests for each customer segment in the survey, as it strikes a balance between reliability and costeffectiveness. Pros and Cons of each Sampling Size: (i) Selecting a hotel guest: Pros: - Provides a diverse representation of the population - Low cost and minimal effort Cons: - May not accurately represent the population (ii) Selecting a sample of 20 hotel guests: Pros: - More reliable results than selecting a single guest - Still relatively low cost and effort Cons: - May still not accurately represent the population - Small sample size may lead to biased results (iii) Selecting a sample of 30 hotel guests: Pros: - Provides a higher level of reliability - More accurate representation of the population Cons: - May be more costly and time-consuming than selecting a smaller sample size (iv) Selecting a sample of 40 guests: Pros: - Maximum reliability - Highest accuracy in representing the population Cons: - Redundant and may not significantly improve accuracy - May be more costly and time-consuming than selecting a smaller sample size Based on the pros and cons of each option, selecting a sample size of 30 guests for each customer segment would be the most suitable sampling plan to meet the survey objectives. This will provide reliable and accurate results while also being cost-effective. Survey Instrument: In order to fully meet the survey objectives, it is important to ensure that the survey questionnaire is comprehensive and addresses all relevant aspects of the guests' stay. The existing survey questionnaire only covers three aspects, namely the overall experience, room cleanliness and comfort, and staff service and professionalism. Therefore, I would suggest adding additional questions and sections to the survey questionnaire. For the purpose of comparing rating scores and qualitative feedback among the two target market segments, it would be beneficial to include questions that specifically cater to the needs and preferences of each segment. For business travellers, we can include questions about the availability and functionality of business facilities such as conference rooms and Wi-Fi. On the other hand, for leisure travellers, we can add questions about recreational activities and amenities. Furthermore, in order to obtain more detailed feedback, it would be beneficial to include open-ended questions where guests can elaborate on their experiences and provide specific suggestions for improvement. Additionally, a section for guests to rate their satisfaction with the hotel's food and beverage options could also be added. Analyzing and Presenting Findings: To analyze the questions in the survey questionnaire, I would use statistical methods such as mean, median and mode to identify patterns and trends in the data. These methods will assist in identifying areas of improvement and providing insights into guest satisfaction among the different market segments. In order to present the findings to management, the results could be compiled into a report with visual aids such as graphs and charts to provide a clear and concise overview. The key findings could also be presented in a presentation format, highlighting the areas of improvement and potential business opportunities for the hotel. Hotel Guest Complaint Situation: Based on past data, the number of complaints received from hotel guests follows a Poisson Distribution with an average of 2 complaints per day. In order to determine whether this complaint situation warrants management's attention, we need to calculate the probability of having more than 1 high-complaint day in a week. This can be calculated using the Poisson Distribution formula as follows: P(X>1) = 1 - P(X=0) - P(X= Question 5 Description and major strengths of R Software, Hardware and Software requirement for R Software Answer Description and major strengths of R Software: R is an open-source programming language commonly used for statistical computing and graphics. It was developed by Ross Ihaka and Robert Gentleman in the early 1990s and has since become one of the most popular languages for data analysis and machine learning. R offers a wide range of statistical and graphical techniques, making it a versatile tool for data analysis. One of the major strengths of R is its extensive library of packages. These packages provide pre-written code for various statistical techniques, making it easier for users to perform complex data analysis without having to write code from scratch. The Comprehensive R Archive Network (CRAN) is the official repository for R packages, containing over 15,000 packages as of 2020. This large library of packages allows users to perform a wide range of statistical analysis, from basic descriptive statistics to advanced machine learning algorithms. Another strength of R is its graphics capabilities. The language offers a variety of graphical techniques to visualize data, such as scatter plots, bar charts, and histograms. These graphical capabilities are particularly useful for exploratory data analysis, where analysts can visually identify patterns and relationships in the data. Additionally, R allows for customization of graphs, giving users more control over how their data is presented. R is also known for its strong community support. As an open-source language, R has a large and active community of users who constantly contribute to the development and improvement of the language. The community also provides extensive documentation, tutorials, and support forums for users to share their knowledge and troubleshoot any issues they may encounter while using R. Hardware and Software requirements for R Software: R can run on a variety of operating systems, including Windows, Mac, and Linux. The minimum hardware requirements for R are fairly modest, with no specific requirements besides having a computer with enough memory and storage space to install the software and data sets. However, for optimal performance, it is recommended to have at least 2GB of RAM and a 2 GHz processor. This will ensure that R can handle large datasets and complex calculations smoothly. Additionally, having a larger monitor can be beneficial for visualizing and analyzing data effectively. To install and run R, users will need to have the latest version of R software (currently 4.0.2) and RStudio, an integrated development environment (IDE) specifically designed for R. RStudio provides a user-friendly interface and several features to enhance the user's experience with R, such as code completion, debugging tools, and visualizations of objects and data structures.