Calculate The Probability That A Surecell Battery Will Last Longer Than 230 Hours. Calculate The Probability Of A Surecell Battery Lifetime Within A Specified Range.

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In this comprehensive analysis, we delve into the lifetimes of Surecell batteries, which are normally distributed with a mean of 200 hours and a standard deviation of 25 hours. Our primary goal is to calculate, with precision to three decimal places, the probability of specific battery lifetime scenarios. This exploration will cover two key areas: first, the probability that a battery chosen at random will last longer than 230 hours, and second, the probability of a battery having a lifetime within a specified range. Understanding these probabilities is crucial for both manufacturers and consumers, as it provides valuable insights into battery performance and reliability. By employing the principles of normal distribution and z-score calculations, we can accurately determine the likelihood of these events, offering a statistical perspective on battery longevity. This analysis not only highlights the practical application of statistical methods but also underscores the importance of quality control and performance evaluation in battery manufacturing. The implications of these calculations extend beyond mere theoretical exercises, influencing purchasing decisions, warranty policies, and product development strategies. Thus, a thorough understanding of battery lifetime probabilities is essential for making informed decisions and ensuring customer satisfaction.

(a) Probability of a Battery Lasting Longer Than 230 Hours

To determine the probability that a Surecell battery lasts longer than 230 hours, we need to utilize the properties of the normal distribution. The given parameters are a mean (μ) of 200 hours and a standard deviation (σ) of 25 hours. The first step is to calculate the z-score, which measures how many standard deviations away from the mean the value of 230 hours is. The z-score is calculated using the formula: z = (X - μ) / σ, where X is the value of interest (230 hours in this case). Plugging in the values, we get z = (230 - 200) / 25 = 30 / 25 = 1.2. This z-score indicates that 230 hours is 1.2 standard deviations above the mean. Next, we need to find the probability associated with this z-score. This is typically done using a standard normal distribution table or a calculator with statistical functions. The table provides the cumulative probability, which is the probability that a value falls below a given z-score. However, we are interested in the probability of a battery lasting longer than 230 hours, so we need to find the area to the right of z = 1.2 on the standard normal distribution curve. This is calculated by subtracting the cumulative probability from 1. Using a standard normal distribution table, we find that the cumulative probability for z = 1.2 is approximately 0.8849. Therefore, the probability of a battery lasting longer than 230 hours is 1 - 0.8849 = 0.1151. Rounding this to three decimal places, we get 0.115. This result signifies that there is an approximately 11.5% chance that a randomly chosen Surecell battery will last longer than 230 hours, providing a clear indication of the battery's performance beyond its average lifespan.

(b) Probability of a Battery Lifetime Between Specified Hours

To calculate the probability of a Surecell battery having a lifetime within a specified range, we need to define the range first. Since the range was not provided in the original question, let's assume we want to find the probability that a battery's lifetime falls between 180 hours and 220 hours. This range is centered around the mean of 200 hours, making it a practical scenario to analyze. Similar to the previous calculation, we start by finding the z-scores for both limits of the range. For 180 hours, the z-score is z1 = (180 - 200) / 25 = -20 / 25 = -0.8. This indicates that 180 hours is 0.8 standard deviations below the mean. For 220 hours, the z-score is z2 = (220 - 200) / 25 = 20 / 25 = 0.8. This indicates that 220 hours is 0.8 standard deviations above the mean. Next, we need to find the cumulative probabilities associated with these z-scores. Using a standard normal distribution table, the cumulative probability for z1 = -0.8 is approximately 0.2119, and the cumulative probability for z2 = 0.8 is approximately 0.7881. The probability of a battery's lifetime falling between 180 and 220 hours is the difference between these two cumulative probabilities: P(180 < X < 220) = P(Z < 0.8) - P(Z < -0.8) = 0.7881 - 0.2119 = 0.5762. Rounding this to three decimal places, we get 0.576. This result indicates that there is a 57.6% chance that a randomly chosen Surecell battery will have a lifetime between 180 and 220 hours. This probability provides a valuable insight into the typical performance range of the batteries, highlighting the consistency and reliability of their lifespan around the mean value. Understanding this probability is essential for both manufacturers and consumers in assessing the expected performance and longevity of Surecell batteries.

In conclusion, our analysis of Surecell battery lifetimes has provided valuable insights into the probabilities associated with their performance. By utilizing the principles of normal distribution and z-score calculations, we have successfully determined the likelihood of specific battery lifetime scenarios. Firstly, we calculated the probability that a battery chosen at random will last longer than 230 hours, finding it to be approximately 0.115. This result indicates that there is an 11.5% chance of a battery exceeding this lifespan, highlighting the potential for some batteries to outperform the average expectation. Secondly, we explored the probability of a battery having a lifetime within a specified range, assuming the range to be between 180 and 220 hours. The calculated probability of 0.576 suggests that there is a 57.6% chance of a battery's lifespan falling within this range, demonstrating the typical performance consistency around the mean. These probabilities are crucial for various stakeholders. For manufacturers, this information aids in quality control, product development, and warranty policy design. By understanding the distribution of battery lifetimes, they can identify potential issues, improve product reliability, and set appropriate warranty periods. For consumers, these probabilities offer a realistic expectation of battery performance, helping them make informed purchasing decisions. Knowing the likelihood of a battery lasting beyond a certain point or within a specific range allows consumers to assess the value and suitability of the product for their needs. Furthermore, these calculations underscore the importance of statistical analysis in understanding real-world phenomena. The normal distribution, with its well-defined properties, provides a powerful tool for predicting and interpreting battery performance. By applying statistical methods, we can gain a deeper understanding of product reliability and make data-driven decisions that benefit both manufacturers and consumers. In summary, the analysis of Surecell battery lifetimes demonstrates the practical application of statistical concepts and the significance of probability calculations in assessing product performance and making informed decisions.