Question

A certain type of automobile battery is known to last an average of 1110 days with a standard deviation of 80 days. If 400 of these batteries are selected, find the following probabilities for the average length of life of the selected batteries: a. The average is between 1100 and 1110 . b. The average is greater than 1120 . c. The average is less than 900 .

Short answer

a. The average is between 1100 and 1110. b. The average is greater than 1120. c. The average is less than 900. Answer: a. The probability that the average is between 1100 and 1110 is 0.4938. b. The probability that the average is greater than 1120 is 0.0062. c. The probability that the average is less than 900 is approximately 0.

Step-by-step solution

Calculate the Sample Mean and Standard Deviation

For a given sample size (n), we can calculate the sample mean (µ_x) and the sample standard deviation (σ_x) using the following formulas: µ_x = µ = 1110 days σ_x = σ / √n = 80 days / √400 = 80 days / 20 = 4 days

Calculate Z-scores for Each Case

Now, we will calculate the z-scores using the formula: z = (x - µ_x) / σ_x a. The average is between 1100 and 1110: z1 = (1100 - 1110) / 4 = -10 / 4 = -2.5 z2 = (1110 - 1110) / 4 = 0 / 4 = 0 b. The average is greater than 1120: z3 = (1120 - 1110) / 4 = 10 / 4 = 2.5 c. The average is less than 900: z4 = (900 - 1110) / 4 = -210 / 4 = -52.5

Find Probabilities Using the Standard Normal Table

a. The probability that the average is between 1100 and 1110: P(-2.5 < z < 0) = P(z < 0) - P(z < -2.5) = 0.5 - 0.0062 = 0.4938 b. The probability that the average is greater than 1120: P(z > 2.5) = 1 - P(z < 2.5) = 1 - 0.9938 = 0.0062 c. The probability that the average is less than 900: The z-score is far below the standard normal distribution range; thus, the probability is very close to 0. #Final Answers# a. The probability that the average is between 1100 and 1110 is 0.4938. b. The probability that the average is greater than 1120 is 0.0062. c. The probability that the average is less than 900 is approximately 0.

Key concepts

z-scores

Z-scores are an essential part of understanding statistical probability. They indicate how many standard deviations an element is from the mean. In practical terms, a z-score helps you determine the position of a value within a distribution. You calculate a z-score using the formula:

• \( z = \frac{x - \mu_x}{\sigma_x} \)

Where \( x \) is the value you're evaluating, \( \mu_x \) is the mean, and \( \sigma_x \) is the standard deviation.
In the given problem, we computed z-scores for three scenarios:

• The average is between 1100 and 1110 days, where we found z-scores of -2.5 and 0.
• For averages greater than 1120 days, the z-score is 2.5.
• For the improbable scenario of averages less than 900 days, we calculated a z-score around -52.5.

Z-scores help us visualize where your data stand in comparison to the average, providing insight into the likelihood of certain outcomes.

standard deviation

Standard deviation is a measure used to quantify the amount of variation or dispersion in a set of data values. The smaller the standard deviation, the closer the data points are to the mean. Conversely, a larger standard deviation suggests data are spread out over a wider range.

• In this context, the standard deviation is 80 days.
• This provides a range to gauge how each battery deviates from the average lifespan of 1110 days.

For a sample, we adjust the standard deviation by dividing it by the square root of the sample size using the formula:

• \( \sigma_x = \frac{\sigma}{\sqrt{n}} \)

Here, we have:

• \( \sigma_x = \frac{80}{\sqrt{400}} = 4 \) days

Understanding standard deviation equips you with the ability to appreciate the consistency or variability of the battery lifespans compared to the average.

sample mean

The sample mean is an average of a set of data points, allowing us to summarize a larger data set with a single value, providing insight into the dataset's center. In probability, the sample mean is critical in hypothesis testing as it’s used to estimate the population mean.

• For this exercise, the given sample mean (\( \mu_x\)) is the same as the population mean, which is 1110 days.

When dealing with large samples, understanding the sample mean becomes more reliable due to the law of large numbers. This statistical principle states that as more samples are taken, the sample mean will converge closer to the population mean.
The sample mean serves as a useful point of reference in both academic exercises and real-world statistical analysis, helping determine probabilities for different outcomes, such as the lifespan of automobile batteries in our example.