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Chapter 6: Problem 8

Assume that the sample is taken from a large population and the correction factor can be ignored. Life of Smoke Detectors The average lifetime of smoke detectors that a company manufactures is 5 years, or 60 months, and the standard deviation is 8 months. Find the probability that a random sample of 30 smoke detectors will have a mean lifetime between 58 and 63 months.

Short Answer

Expert verified
The probability is approximately 0.8945.

Step by step solution

01

Identify the Given Values

Given: the population mean \( \mu = 60 \) months, the population standard deviation \( \sigma = 8 \) months, the sample size \( n = 30 \), and the sample mean range of interest is between 58 and 63 months.
02

Calculate the Standard Error of the Mean

The standard error of the mean (SE) is calculated using the formula: \( SE = \frac{\sigma}{\sqrt{n}} \). Substitute the given values: \( SE = \frac{8}{\sqrt{30}} \approx 1.46 \) months.
03

Find the Z-Scores for the Sample Mean Range

Calculate the Z-scores for 58 months and 63 months using the formula: \( Z = \frac{\bar{x} - \mu}{SE} \).For \( \bar{x} = 58 \): \( Z = \frac{58 - 60}{1.46} \approx -1.37 \).For \( \bar{x} = 63 \): \( Z = \frac{63 - 60}{1.46} \approx 2.05 \).
04

Calculate the Probabilities Using the Z-Scores

Look up the probabilities corresponding to the Z-scores in a standard normal distribution table or use a calculator:\( P(Z < -1.37) \approx 0.0853 \).\( P(Z < 2.05) \approx 0.9798 \).
05

Calculate the Probability of the Sample Mean Range

Find the probability that the sample mean is between 58 and 63 months by subtracting the probabilities: \( P(-1.37 < Z < 2.05) = P(Z < 2.05) - P(Z < -1.37) \). This equals approximately \( 0.9798 - 0.0853 = 0.8945 \).

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Key Concepts

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

Standard Error
When you hear about the Standard Error, think about it as a measure of how "spread out" the sample means will be from the true population mean. It's crucial to remember that Standard Error (SE) is different from Standard Deviation (SD). While SD measures the spread of all data points in your sample, SE indicates how much the sample mean will vary from the actual population mean, depending on the sample size.
  • Standard Error provides insight into the accuracy of a sample mean when estimating the population mean. Smaller SE means more precision.
  • Calculate SE using the formula: \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) represents the population standard deviation, and \( n \) is the sample size.
  • In our exercise, this calculation helps determine how close the sample means of 30 smoke detectors are likely to be to the actual mean of 60 months.
Always aim to understand what SE "tells" you about your data rather than just doing the calculation.
Z-Score
The Z-Score is a way to identify how far (in terms of standard deviations) a data point is from the mean. It allows us to understand where a particular value sits within a normal distribution and to calculate probabilities.
  • Calculate a Z-Score using: \( Z = \frac{\bar{x} - \mu}{SE} \). Here, \( \bar{x} \) is the sample mean, \( \mu \) the population mean, and SE the standard error of the mean.
  • In our scenario, determining Z-Scores for 58 and 63 months provides a standardized way to compare these values in the context of the distribution of smoke detector lifespans.
  • The Z-Score helps us use standard normal distribution tables or statistical software to find probabilities.
By using Z-Scores, we can translate a sample mean into a context that allows for comparing and contrasting with other data points.
Normal Distribution
Normal Distribution, often referred to as a "bell curve", describes how data is spread out in a way that most values cluster around the mean, with fewer found as you move further away.
  • It's symmetrical, meaning the left and right sides are mirror images.
  • Most data in a normal distribution falls within three standard deviations of the mean, with about 68% within one, 95% within two, and 99.7% within three standard deviations.
  • In the given exercise, our understanding of normal distribution allows us to use Z-Scores to find probabilities for sample means.
Understanding normal distribution is essential for interpreting a variety of statistical analyses, including probabilities and hypothesis testing.
Sample Mean
The Sample Mean is the average of the observations in a sample and serves as an estimate of the population mean. It gives a snapshot of the "central tendency" of the data you are analyzing.
  • Calculating the Sample Mean is straightforward: add up all sample values and divide by the number of values.
  • In our smoke detector example, the sample mean helps in determining the probability that the average lifespan of sampled detectors meets or exceeds certain thresholds (58 and 63 months).
  • When combined with standard error, the sample mean becomes a powerful tool for making inferences or predictions about the greater population.
Knowing your Sample Mean is integral for anyone looking to simplify real-world dynamics into digestible numeric terms.

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