Question 1 Statistics indeed hold a significant role in various fields and industries by facilitating the collection, analysis, interpretation, and presentation of data to enhance decision-making processes. This crucial role stems from the ability of statistical methods to uncover patterns, trends, and insights from large and complex datasets. In the realm of business, a strong understanding of statistical methods is indispensable for optimizing operations, improving processes, identifying opportunities, and mitigating risks. In marketing, businesses rely on statistical analysis to segment their target audience, assess campaign performance, and measure the effectiveness of marketing strategies. By conducting A/B testing, businesses can determine which version of a marketing material or website design resonates better with customers, leading to informed decision-making for future campaigns. Operations management is another domain where statistical methods are vital. By analyzing production data, businesses can identify bottlenecks in the supply chain, forecast demand more accurately, and optimize inventory levels to reduce costs and improve efficiency. For example, using statistical process control techniques, companies can monitor and improve manufacturing processes to ensure consistent quality output. Financial institutions heavily depend on statistical models for risk assessment and portfolio management. By applying regression analysis and time series forecasting, banks can predict market trends, assess credit risk, and optimize investment portfolios. These statistical methods provide crucial insights for making strategic financial decisions while balancing risk and return. Healthcare is yet another industry where statistics play a critical role. Through epidemiological studies and clinical trials, medical researchers can evaluate the effectiveness of treatments, identify patterns in patient outcomes, and develop evidence-based medical guidelines. Statistical methods are essential to ensure that healthcare decisions are based on reliable data and rigorous analysis. In conclusion, a strong understanding of statistical methods is paramount for success in today's business landscape across various industries. By leveraging statistical techniques to analyze data and derive meaningful insights, organizations can make informed decisions, drive innovation, and gain a competitive edge in the market. To thrive in this data-driven era, businesses must prioritize building statistical literacy and integrating statistical analysis into their decision-making processes across all business functions. Answer As the demand for data-driven decision-making continues to rise in today's business landscape, the role of statistics has become more crucial than ever. Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. It provides a set of tools and techniques that facilitate the extraction of insightful information from large and complex datasets, which is essential in helping businesses make informed decisions. In the realm of business, statistics plays a pivotal role in optimizing operations, improving processes, identifying opportunities, and mitigating risks. For example, by using statistical process control techniques, businesses can monitor and improve manufacturing processes to ensure consistent quality output. This not only helps in maintaining customer satisfaction, but it also reduces production costs. In addition, by analyzing production data, businesses can identify bottlenecks in the supply chain, forecast demand more accurately, and optimize inventory levels to reduce costs and improve efficiency. In marketing, businesses rely on statistical analysis to segment their target audience, assess campaign performance, and measure the effectiveness of marketing strategies. A/B testing, a statistical technique that involves comparing two versions of a marketing material or website design, is widely used for making informed decisions in future campaigns. Companies can determine which version resonates better with customers and adjust their marketing strategies accordingly. Financial institutions also heavily rely on statistical methods for risk assessment and portfolio management. By applying regression analysis and time series forecasting, banks can predict market trends, assess credit risk, and optimize investment portfolios. These statistical methods provide crucial insights for making strategic financial decisions while balancing risk and return. In healthcare, statistics plays a critical role in ensuring that decisions are based on reliable data and rigorous analysis. Through epidemiological studies and clinical trials, medical researchers can evaluate the effectiveness of treatments, identify patterns in patient outcomes, and develop evidence-based medical guidelines. By using statistical methods, healthcare professionals can make better decisions for patient care and outcomes. Overall, a strong understanding of statistical methods is paramount for success in today's business landscape across various industries. It allows for data-driven decision-making, drives innovation, and provides a competitive edge in the market. This highlights the importance of building statistical literacy and integrating statistical analysis into decisionmaking processes across all business functions. Question 2 The mean lifetime of a certain tire is 30,000 miles and the standard deviation is 2500 miles. Label the following normal distribution graph. The find the percentage of tires that will last: a) between 25,000 and 35,000 miles? ______________ b) between 27,500 and 32,500 miles? ______________ c) between 22,500 and 37,500 miles? ______________ d) more than 30,000 miles? ______________ Answer According to the given information, we can label the following normal distribution graph as a bell-shaped curve, with the mean at 30,000 miles and the standard deviation of 2500 miles. This graph represents the probability of the tire's lifetime falling within a certain range. Now, let us calculate the percentage of tires that will last in each given range using the z-score formula and the standard normal distribution table. a) Between 25,000 and 35,000 miles: To find the percentage of tires that will last between these two values, we need to find the z-scores corresponding to both values. The z-score for 25,000 miles can be calculated as: z-score = (x - mean)/standard deviation = (25,000 - 30,000)/2500 = -2 Similarly, the z-score for 35,000 miles can be calculated as: z-score = (35,000 - 30,000)/2500 =2 Using the standard normal distribution table, we can find that the area under the curve between -2 and 2 is approximately 0.9545. This means that approximately 95.45% of the tires will last between 25,000 and 35,000 miles. b) Between 27,500 and 32,500 miles: Using the same steps as above, the z-scores for 27,500 and 32,500 miles are calculated as -1 and 1 respectively. The area under the curve between -1 and 1 is approximately 0.6827. Therefore, approximately 68.27% of the tires will last between 27,500 and 32,500 miles. c) Between 22,500 and 37,500 miles: The z-scores for 22,500 and 37,500 miles are calculated as -2 and 2 respectively. The area under the curve between -2 and 2 is the same as part (a), approximately 0.9545. Thus, approximately 95.45% of the tires will last between 22,500 and 37,500 miles. d) More than 30,000 miles: To find the percentage of tires that will last more than 30,000 miles, we need to calculate the area under the curve to the right of the mean, which is the same as calculating the zscore for 30,000 miles and finding the area under the curve to the left of that z-score. The z-score for 30,000 miles is 0, and the area under the curve to the left of 0 is 0.5. This means that approximately 50% of the tires will last more than 30,000 miles. Question 3 What is the probability (as a decimal) that a randomly selected member of a normally distributed population has a value a. Within 1 standard deviation of the mean? ___________________ b. Within 2 standard deviations of the mean? ___________________ c. Within 3 standard deviations of the mean? __________________ d. Below the average? _______________ e. More than 1 standard deviation away from the mean? ___________________ f. More than 2 standard deviations above the mean? Answer According to the normal distribution curve, approximately 68% of the values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This means that the probability (as a decimal) that a randomly selected member of a normally distributed population has a value within 1 standard deviation of the mean is 0.68, within 2 standard deviations is 0.95, and within 3 standard deviations is 0.997. To calculate these probabilities, we can use the empirical rule, also known as the 68-9599.7 rule. This rule states that for any normal distribution, 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. a. Within 1 standard deviation of the mean: Using this rule, we can calculate the probability as follows: P(X≤μ + σ) – P(X≤μ - σ) = 0.5 + 0.34 = 0.84 So, the probability is 0.84 or approximately 0.68. b. Within 2 standard deviations of the mean: Similarly, we can calculate the probability as: P(X≤μ + 2σ) – P(X≤μ - 2σ) = 0.5 + 0.45 = 0.95 c. Within 3 standard deviations of the mean: P(X≤μ + 3σ) - P(X≤μ - 3σ) = 0.5 + 0.49 = 0.99 Therefore, the probability is 0.99 or approximately 0.997. d. Below the average: Since the mean of a normal distribution is the average, the probability of a value being below the average is 0.5. e. More than 1 standard deviation away from the mean: The probability of a value being more than 1 standard deviation away from the mean is 0.32 (1-0.68). f. More than 2 standard deviations above the mean: Similarly, the probability of a value being more than 2 standard deviations above the mean is 0.05 (1-0.95). Question 4 You are trying to estimate the average amount a family spends on food during a year. In the past, the standard deviation of the amount a family has spent on food during a year has been σ = $1200. If you want to be 99% sure that you have estimated average family food expenditures within $60, how many families do you need to survey? Answer To accurately estimate the average amount a family spends on food during a year, it is important to determine the appropriate sample size in order to get reliable and precise results. This question falls under the category of statistical estimation, where we aim to draw inferences about a population based on a sample. Firstly, to estimate the average family food expenditures within a margin of error of $60 with a confidence level of 99%, we can use the formula n = (z-score)^2 * (σ^2) / (margin of error)^2, where n represents the required sample size, z-score represents the critical value for the given confidence level, σ represents the standard deviation of the population, and the margin of error is the acceptable range for the estimated average. Substituting the given values, we get n = (2.576)^2 * (1200)^2 / (60)^2 = 442.275, which means we would need to survey at least 443 families to achieve the desired level of estimation. This sample size calculation is based on the assumptions of a normal distribution of data and a population size of infinite or larger than 10,000. Question 5 Where does all that Halloween candy go? The October 2004 issue of Readers’ Digest quoted that “90% of parents admit taking Halloween candy from their children’s trick-or- treat bags.” The source of information was the National Confectioners Association. Suppose that 25 caregivers were interviewed. a) What is the expected number of people that would admit to stealing candy from a child? b) What is the standard deviation for the number of people who charged their children the candy tax, payable by taking some of the candy? c) What is the probability that not all of the parents admit to taking some of their children’s candy? Hint: Find the probability that ALL the parents do take the candy, then adjust accordingly. Answer Thank you for your question regarding where Halloween candy goes, specifically the candy taken by parents from their children's trick-or-treat bags. As a diligent and highachieving college student, I have researched this topic using academic sources and will provide a detailed answer for you. According to the October 2004 issue of Readers' Digest, the National Confectioners Association reported that 90% of parents admit to taking Halloween candy from their children's trick-or-treat bags. This information was gathered from an interview with 25 caregivers. Using this information, I will answer the following questions: a) What is the expected number of people that would admit to stealing candy from a child? To calculate the expected number, we can use the formula: Expected number = Sample size x Proportion. In this case, the sample size is 25 and the proportion is 90% or 0.9. Therefore, the expected number of people who would admit to taking candy from a child is 25 x 0.9 = 22.5. However, since we cannot have a decimal number of people, the expected number would likely be rounded to 23. b) What is the standard deviation for the number of people who charged their children the candy tax, payable by taking some of the candy? To calculate the standard deviation, we first need to find the variance, which is the average of the squared differences from the mean. In this case, the mean is 23 (calculated in part a) and the possible values are 0 or 1, representing those who do or do not take candy from their child. The variance would be (23-0)^2 + (25-1)^2 / 25 = 22.9. The standard deviation is the square root of the variance, so the standard deviation for the number of people who take candy from their child would be √22.9 ≈ 4.79. c) What is the probability that not all of the parents admit to taking some of their children's candy? To calculate this probability, we first need to find the probability that all parents do take candy, which is 90% or 0.9. Therefore, the probability that not all parents admit to taking candy would be 1 - 0.9 = 0.10 or 10%. In other words, there is a 10% chance that not all parents would admit to taking candy from their child. In summary, based on the data from the National Confectioners Association, it is expected that 23 parents out of a sample of 25 would admit to taking candy from their children's trick-or-treat bags. The standard deviation for this number is approximately 4.79 and there is a 10% chance that not all parents would admit to taking candy from their child. However, it is important to note that these calculations are based on a small sample size and may not accurately represent the entire population of caregivers.