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Chapter 22: Problem 16

Use the following data. It has been previously established that for a certain type of AA battery (when newly produced), the voltages are distributed normally with \(\mu=1.50 \mathrm{V}\) and \(\sigma=0.05 \mathrm{V}\). What percent of the batteries have voltages above \(1.64 \mathrm{V} ?\)

Short Answer

Expert verified
0.26% of the batteries have voltages above 1.64 V.

Step by step solution

01

Identify the Problem

We need to find the percentage of AA batteries that have a voltage above 1.64 V, given that the voltage distribution is normal with \(\mu = 1.50 \mathrm{V}\) and \(\sigma = 0.05 \mathrm{V}\).
02

Find the Z-score

The Z-score measures how many standard deviations an element is from the mean. It is calculated using the formula \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the value we're interested in (1.64 V), \(\mu\) is the mean (1.50 V), and \(\sigma\) is the standard deviation (0.05 V). So, \(Z = \frac{1.64 - 1.50}{0.05} = 2.8\).
03

Use the Standard Normal Distribution

To find the percentage of batteries with a voltage above 1.64 V, we need to find the area to the right of the Z-score (2.8) on the standard normal distribution curve.
04

Find the Area Beyond the Z-score

Using a standard normal distribution table or a calculator, the area to the left of Z = 2.8 is approximately 0.9974. The area to the right, representing voltages above 1.64 V, is \(1 - 0.9974 = 0.0026\).
05

Convert the Area to Percentage

To express the probability as a percentage, multiply by 100. Therefore, the percentage of batteries with voltages above 1.64 V is \(0.0026 \times 100 = 0.26\%\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution
In the realm of statistics, the normal distribution is a common probability distribution pattern that pops up frequently in real-world data. It is also called a Gaussian distribution, and it resembles a symmetric bell-shaped curve. In a normal distribution:
  • The mean, median, and mode are all located at the center of the curve.
  • The spread of data points around this central peak is dictated by the standard deviation.
The significance of a normal distribution lies in its application. It is used as an essential tool for assessing probabilities and patterns in datasets, just like in the exercise involving AA battery voltages. Understanding the normal distribution helps in making predictions about data and inferring statistical conclusions from small samples.
Z-Score
The Z-score is a key concept in statistics. It helps in understanding how far an individual data point is from the mean of a set of data, measured in terms of standard deviations. The Z-score formula is:\[Z = \frac{X - \mu}{\sigma} \]Where:
  • \(X\) is the data point of interest.
  • \(\mu\) is the mean of the dataset.
  • \(\sigma\) is the standard deviation.
A Z-score helps us determine if a data point is within the expected range or is considered an outlier. In our voltage example, calculating the Z-score for 1.64 V gave us a Z of 2.8. This means that the voltage of 1.64 V is 2.8 standard deviations above the mean. This understanding is crucial in judging the relative position of data within a distribution, aiding in probability calculations.
Standard Deviation
Standard deviation is a measure of how spread out numbers are around the mean in a dataset. It tells you whether the data is closely packed or more dispersed. A small standard deviation means the values are clustered closely around the mean, while a large standard deviation indicates wider spread. Calculating standard deviation involves:
  • Finding the difference between each data point and the mean.
  • Squaring these differences.
  • Calculating the average of these squared differences.
  • Taking the square root of this average.
In our battery voltage problem, the standard deviation of 0.05 V indicates that most battery voltages will lie fairly close to the 1.50 V mean. Understanding standard deviation is essential in predicting how much variation exists within a data set.
Probability Calculation
Probability calculation allows us to quantify how likely it is for an event to occur. With a normal distribution, once we find the Z-score of an event, we can use statistical tables or software to discover the corresponding probability. This process involves:
  • Calculating the Z-score for the event of interest.
  • Using the Z-score to find the probability from a standard normal distribution table.
  • Determining the area under the curve to the right or left of the Z-score, depending on the scenario.
For the exercise with the AA battery, we wanted the probability above 1.64 V. With a Z-score of 2.8, using the table showed us the area to the right is 0.0026. This probability translates to a 0.26% chance of a battery having a voltage higher than 1.64 V. Probability calculations like these are valuable for decision-making and predicting future events.

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Most popular questions from this chapter

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