Problem 72 - Applied Statistics and Probability for Engineers

Question

The time until recharge for a battery in a laptop computer under common conditions is normally distributed with a mean of 260 minutes and a standard deviation of 50 minutes. (a) What is the probability that a battery lasts more than four hours? (b) What are the quartiles (the \(25 \%\) and \(75 \%\) values) of battery life? (c) What value of life in minutes is exceeded with \(95 \%\) probability?

Step-by-Step Solution

Answer

(a) P = 0.6554. (b) Q1 = 226.25 min, Q3 = 293.75 min. (c) 177.75 min.

Step 1: Converting Time for Part (a)

To find the probability that a battery lasts more than four hours, convert hours to minutes: 4 hours equals 240 minutes.

Step 2: Standardizing the Value for Part (a)

Use the z-score formula to standardize 240 minutes: \( z = \frac{X - \mu}{\sigma} \) where \( X = 240 \), \( \mu = 260 \), and \( \sigma = 50 \). Calculate \( z = \frac{240 - 260}{50} = -0.4 \).

Step 3: Finding Probability for Part (a)

Find the probability that the z-score is greater than \(-0.4\) using the standard normal distribution table or a calculator: \( P(Z > -0.4) = 0.6554 \). The probability that the battery lasts more than four hours is approximately \( 0.6554 \).

Step 4: Calculating Quartile Q1 for Part (b)

To find the first quartile (\(Q_1\)), determine the z-score for the 25th percentile: \( z = -0.675 \). Use the z-score to find \(Q_1\) minutes: \( Q_1 = \mu + z \times \sigma = 260 - 0.675 \times 50 = 226.25 \) minutes.

Step 5: Calculating Quartile Q3 for Part (b)

To find the third quartile (\(Q_3\)), determine the z-score for the 75th percentile: \( z = 0.675 \). Use the z-score to find \(Q_3\) minutes: \( Q_3 = \mu + z \times \sigma = 260 + 0.675 \times 50 = 293.75 \) minutes.

Step 6: Finding Value Exceeded with 95% Probability for Part (c)

To find the value exceeded with 95% probability, use the z-score for the 5th percentile: \( z = -1.645 \). Calculate the value: \( X = \mu + z \times \sigma = 260 - 1.645 \times 50 = 177.75 \) minutes.

Key Concepts

Probability Calculation

Understanding probability in the context of a normal distribution can seem daunting, but it becomes straightforward with a step-by-step approach. When we say we want the probability of a battery lasting more than 240 minutes, we aim to find the area under the normal distribution curve to the right of this point. This is because probability corresponds to an area under the curve in this context. In practice, to determine this probability, we use the z-score, which helps us standardize our value and find its associated probability in the standard normal distribution table. In the exercise, after calculating the z-score for 240 minutes, we found that the probability for a battery to last more than four hours was approximately 0.6554, which implies there's about a 65.54% chance. This means, if you had 100 batteries, about 66 of them might last more than four hours.

Z-Score

The z-score is a critical concept in statistics, used to standardize data points to allow comparison to a standard normal distribution. It measures how many standard deviations a data point (in this case, the battery life) is from the mean.The formula is given by:

• \( z = \frac{X - \mu}{\sigma} \)

where \( X \) is the value to be standardized, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. By converting the original battery life in minutes to a z-score, we can use a standard normal distribution table to find its probability. For example, in the exercise, 240 minutes is equivalent to a z-score of -0.4, indicating it's less than the average battery life. Thus, the technique simplifies complex probability problems by reducing them to looking up or calculating from standard tables.

Quartiles

Quartiles divide a distribution into four equal parts, helping summarize and describe the distribution of data. There are three quartiles in any dataset:

• The first quartile \(Q_1\) is the 25th percentile, marking where 25% of the data lies below it.
• The second quartile \(Q_2\), or the median, is the 50th percentile.
• The third quartile \(Q_3\) is the 75th percentile, where 75% of the data is below it.

In the exercise, \(Q_1\) was calculated at approximately 226.25 minutes, and \(Q_3\) was 293.75 minutes. These quartiles give us an idea of the spread and central tendency of the battery life data. Typically, they are useful for identifying outliers or understand variability within the dataset.

Standard Deviation

Standard deviation is a measure that describes how spread out the values in a dataset are in relation to the mean. In a normal distribution, it tells us how much the individual data points typically deviate from the average battery life. The concept is essential because it helps convey whether data points tend to be close to or far from the mean. In this context, a larger standard deviation indicates more spread in the times until battery recharge, whereas a smaller standard deviation means the times are closer to each other. In the given exercise, the standard deviation is 50 minutes, suggesting that most battery life observations lie within 50 minutes above or below the average of 260 minutes. This context assists in comprehending the reliability and expectation of battery life you can compare using this statistical measure.