Statistics Answer Note #7

Question 1 A market research firm supplies manufacturers with estimates of the retail sales of their products from samples of retail stores. Marketing managers are prone to look at the estimate and ignore sampling error. An SRS of 25 stores this year shows mean sales of 60 units of a small appliance, with a standard deviation of 14.4 units. During the same point in time last year, an SRS of 25 stores had mean sales of 75.696 units, with standard deviation 12.3 units. A decrease from 75.696 to 60 is a drop of about 26%. Construct a 95% confidence interval estimate of the difference u1-u2, where u1 is the mean of this year's sales and u2 is the mean of last year's sales. Answer As a college student with high marks, it is important to have a thorough understanding of statistics and how to interpret and analyze data. In this question, we are presented with two sets of data from a market research firm that provides manufacturers with estimates of retail sales for their products. It is evident that marketing managers tend to overlook any potential sampling errors in these estimates. To tackle this question, we first need to understand the concept of sampling error. It refers to the difference between the sample and the population mean. In this case, our sample is the 25 retail stores and the population is all retail stores. Sampling error is an important consideration when interpreting data and making inferences about a population. Ignoring sampling error can lead to inaccurate conclusions. Based on the data provided, this year's sample of 25 retail stores had a mean sales of 60 units of a small appliance with a standard deviation of 14.4 units. On the other hand, last year's sample of 25 retail stores had a mean sales of 75.696 units, with a standard deviation of 12.3 units. This indicates a decrease of about 26% in sales from last year. To calculate the 95% confidence interval for the difference in means (u1-u2), we first need to calculate the standard error. The standard error (SE) for the difference between two means can be calculated as: SE = √ ( (s1^2 / n1) + (s2^2 / n2) ) Where s1 and s2 are the standard deviations of the two samples and n1 and n2 are the sample sizes. Plugging in our values, we get: SE = √ ( (14.4^2 / 25) + (12.3^2 / 25) ) = √ (3.84 + 4.752) = 3.786 Next, we need to calculate the margin of error (ME) which is given by: ME = t * SE Where t is the critical value obtained from the t-table based on the degrees of freedom and the desired level of confidence. In this case, we will use a two-tailed test with a level of confidence of 95%. Therefore, our degrees of freedom will be 25-1 = 24. Consulting the t-table, we find that the critical t-value is 2.064. Thus, the margin of error (ME) is: ME = 2.064 * 3.786 = 7.816 Finally, we can construct our 95% confidence interval (CI) by adding and subtracting the margin of error from the difference in means (u1-u2). 95% CI = ( -26 + 7.816, -26 - 7.816 ) = (-19.184, -32.816) Therefore, we can be 95% confident that the true difference in means between this year's sales and last year's sales falls within the range of -19.184 to -32.816. This indicates that there is a statistically significant decrease in sales from last year to this year. Question 2 Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. Click to view page 1 of the table. LOADING... Click to view page 2 of the table. LOADING... zz0.86650.8665 A graph with a bell-shaped curve, divided into 2 regions by a line from top to bottom on the right side. The region left of the line is shaded and is labeled 0.8665. Question content area bottom Part 1 The indicated z score is enter your response here. (Round to two decimal places as needed.) Answer The indicated z score is 1.08. Z score, also known as the standard score, is a single number that tells us how many standard deviations a particular value is above or below the mean in a normal distribution. It is calculated by subtracting the mean from the given value and dividing the result by the standard deviation. In this case, the given value is 0.8665 and the mean is 0, so the z score can be calculated as (0.8665 - 0) / 1 = 0.8665. Calculating z scores can be a useful tool in statistics as it allows us to compare different sets of data with different means and standard deviations. It helps us standardize the values and make meaningful comparisons. In this scenario, the graph depicts a standard normal distribution with a mean of 0 and a standard deviation of 1. This is also known as the standard normal curve, which is a bellshaped curve symmetrically distributed around the mean. The area under the curve is equal to 1, and the z score corresponds to the shaded area on the curve. To find the indicated z score of 0.8665, we can use a z-score table or a calculator. The table gives us the cumulative probability, or the area under the curve, and we can find the z score that corresponds to that probability. Using the table, the closest value to 0.8665 is 0.8664, which corresponds to a z score of 1.08. This means that 0.8665 is 1.08 standard deviations above the mean. Question 3 quantatitative method management advanatages and disadvantages, importance, probabilitiies kinds with definition Answer Quantitative method management involves the use of numerical data and statistical analysis to make decisions and solve problems within an organization. This approach has both advantages and disadvantages, and understanding them is crucial for effective management. Advantages: 1. Objectivity and accuracy: Quantitative data is based on numbers and facts, eliminating the risk of personal bias. This makes the analysis more accurate and objective, leading to more informed decision-making. 2. Replicability: The quantitative approach allows for replication of the research process, which increases the reliability of the findings. This is beneficial for making predictions and informing future decision-making. 3. Comprehensive analysis: With the use of statistical tools, quantitative analysis enables in-depth analysis of large datasets. This helps in identifying patterns and trends, which can inform management strategies. 4. Comparability: Quantitative data can be easily compared across different variables, making it easier to identify correlations and relationships. This helps in understanding the impact of different factors on the outcome, and in turn, informs decision-making. Disadvantages: 1. Limited understanding: Although quantitative data provides a comprehensive analysis, it does not provide an in-depth understanding of the underlying factors. In some cases, it may be necessary to complement it with qualitative data to gain a better understanding. 2. Dependence on accurate data: The accuracy of quantitative analysis is highly dependent on the accuracy of the data collected. If the data is flawed or biased, it can lead to inaccurate conclusions and decisions. 3. Cost and time-consuming: Gathering and analyzing quantitative data can be a costly and time-consuming process. This may not be feasible for organizations with limited resources and tight deadlines. 4. Lack of context: Quantitative data does not take into account the context and nuances of a situation, which may be important for decision-making. This can sometimes result in oversimplification of complex issues. Importance of Quantitative Method Management: Quantitative method management is essential for effective decision-making in organizations. By utilizing numerical data and statistical analysis, it provides accurate and reliable information that informs crucial management strategies. It helps in identifying problems and making predictions, which can guide future actions for the organization. Probabilities and Types: Probabilities are used in quantitative analysis to predict the likelihood of an event occurring based on the available data. There are two types of probabilities: subjective and objective. 1. Subjective probabilities are based on an individual's personal beliefs and perceptions. These may be influenced by personal biases and can vary from person to person. 2. Objective probabilities are based on empirical data and mathematical calculations. They are more reliable as they are based on facts and numbers. In conclusion, quantitative method management has its advantages and disadvantages, but it plays a crucial role in effective decision-making in organizations. By understanding the probabilities and types associated with this approach, and utilizing reliable sources, managers can make informed and evidence-based decisions for their organizations. Question 4 Descriptive methods involve the use of numerical data to summarize and describe characteristics of a population or sample, while inferential methods involve the use of statistical analysis to make inferences and predictions about a larger population based on a smaller sample. give me problem solving examples Answer Descriptive and inferential methods are essential tools in understanding and analyzing data in almost all fields of study, including business. These methods help researchers to identify patterns and relationships among variables, as well as make predictions and inferences about a larger population based on a smaller sample. To fully understand their applications and benefits in problem-solving, let us take a closer look at each method and provide some examples. Descriptive methods involve the use of numerical data to summarize and describe characteristics of a population or sample. This method is primarily used to describe the basic features of the data in a study, such as central tendency and variability of a particular variable. One example of its application is in market research. Suppose a company wants to understand the characteristics of its target market, such as age, income, and spending habits. In that case, descriptive methods can be used to summarize and present the data in the form of graphs, charts, or tables, providing a clear understanding of the target market's characteristics. On the other hand, inferential methods involve the use of statistical analysis to make inferences and predictions about a larger population based on a smaller sample. Unlike descriptive methods, inferential methods allow researchers to draw conclusions and make predictions about a larger population based on a smaller sample by utilizing probability and hypothesis testing. One example of its application is in predicting consumer behavior. Suppose a company wants to find out if a new product will have successful sales in an entirely new market. In that case, inferential methods can be used to test a hypothesis and make predictions about the product's potential success based on a small sample of consumers. In conclusion, descriptive and inferential methods are essential in problem-solving as they provide researchers with a deeper understanding of the data and allow them to make predictions and inferences about a larger population based on a smaller sample. By utilizing these methods, businesses can make well-informed decisions and improve their overall performance. Question 5 regression analysis, correlation analysis, time series analysis, and forecasting give examples of each including problem sovings Answer Regression analysis, correlation analysis, time series analysis, and forecasting are four commonly used statistical techniques in the field of business. Each of these techniques has its own unique purpose and application, and when used correctly, they can provide valuable insights and solutions to various business problems. Regression analysis is a statistical method used to examine the relationship between a dependent variable and one or more independent variables. It is often used to identify the nature and strength of the relationship between variables. For example, a business might use regression analysis to determine the effect of advertising expenditure on sales revenue. A good example of regression analysis in action can be seen in a study by Nguyen and Zheng (2018), who examined the relationship between marketing efforts and the performance of listed firms in their study "Does a Top-tier Ranking Promote Firm Performance? Evidence from Publicly Listed Firms in Vietnam". They used regression analysis to analyze the impact of different marketing activities on firm performance and found a significant positive relationship between marketing efforts and firm performance. Correlation analysis is another statistical technique that is used to measure and analyze the strength and direction of the relationship between two or more variables. It is often used to determine how changes in one variable affect another. For example, a business might use correlation analysis to determine the relationship between customer satisfaction and repeat purchases. A practical use case of correlation analysis can be seen in a study by Gao et al. (2019) titled "Relationship between Customer Experience, Customer Satisfaction and Repurchase Intention: The Role of Service Quality in the Chinese Hotel Industry". The researchers used correlation analysis to explore the relationship between customer experience, satisfaction, and repurchase intention. They found a strong positive correlation between these variables, highlighting the importance of providing a positive customer experience for customer retention. Time series analysis is a statistical technique used to analyze data over a period of time in order to identify patterns and trends. It is commonly used to make forecasts or predictions based on past data. For example, a business might use time series analysis to predict future sales based on previous sales data. One example of time series analysis can be seen in a study by Singh and Sharma (2018) titled "Forecasting Monthly Electrical Energy Demand in India using Time Series Analysis". The researchers utilized a time series model to predict the monthly electrical energy demand in India for the next five years. Their study concluded that the time series model was accurate and effective in forecasting future demand. Forecasting is the process of using historical data and statistical techniques to predict future trends and patterns. It is often used in business decision making to help with planning and resource allocation. For instance, a business might use forecasting to predict future demand for a product in order to optimize production and inventory levels. An example that demonstrates the use of forecasting can be found in a study by Owusu et al. (2020) titled "Forecasting Stock Market Prices Using Hybrid ARIMA-GARCH Model". The researchers used a hybrid ARIMA-GARCH model to forecast stock market prices for five leading stock exchanges. The results showed that this model outperformed other forecasting techniques, providing valuable insights for investors and businesses. In conclusion, regression analysis, correlation analysis, time series analysis, and forecasting are powerful statistical techniques that can provide valuable insights and solutions for various business problems. By utilizing these techniques correctly and interpreting the results accurately, businesses can make better informed decisions and improve their overall performance.