Central Limit Theorem And Steel Rods Production

Central Limit Theorem Meana Company Produces Steel Rods The Lengths

The company produces steel rods with lengths normally distributed, having a mean of 185.4 cm and a standard deviation of 1 cm. For shipment, 49 rods are bundled together. We are asked to determine the distribution of the sample mean, calculate probabilities concerning individual rods and bundles, and assess the normality assumption's necessity. Additionally, similar analyses are applied to data on miles traveled before tire replacement, laptop replacement times, graduate salaries, unemployment durations, Facebook friends, and pollutants in waterways, all assuming normal distributions with specified means and standard deviations.

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The Central Limit Theorem (CLT) is fundamental in understanding the behavior of sample means, particularly when dealing with normally distributed data. In examining the steel rods, tire miles, and other scenarios, the CLT provides the basis for approximating the sampling distribution of the mean as normal, regardless of the population distribution, provided the sample size is sufficiently large.

Distribution of the Sample Mean for Steel Rod Lengths

The lengths of individual steel rods are normally distributed with a mean (μ) of 185.4 cm and a standard deviation (σ) of 1 cm. The sampling distribution of the sample mean (\(\bar{X}\)) for a sample size of n=49 is also normally distributed, with mean \(\mu_{\bar{X}}\) equal to the population mean and standard deviation \(\sigma_{\bar{X}}\) equal to the population standard deviation divided by the square root of the sample size.

Thus, \(\bar{X} \sim N(185.4, \frac{1}{\sqrt{49}}) = N(185.4, 1/7) \approx N(185.4, 0.1429)\).

Probability for a Single Steel Rod

For a single randomly selected steel rod, the probability that its length is between 185.3 cm and 185.4 cm requires calculating the z-scores for these values and then using the standard normal distribution.

The z-score for 185.3 cm is \(z = \frac{185.3 - 185.4}{1} = -0.1\), and for 185.4 cm, \(z=0\).

The probability that the length is between 185.3 cm and 185.4 cm is \(P(-0.1 \leq Z \leq 0) = \Phi(0) - \Phi(-0.1) \approx 0.5 - 0.4602 = 0.0398\).

Probability for the Mean of 49 Rods

For the bundle of 49 rods, the sample mean (\(\bar{X}\)) has distribution \(N(185.4, 0.1429)\). To find the probability that the average length is between 185.3 cm and 185.4 cm, convert these bounds to z-scores:

For 185.3 cm: \(z = \frac{185.3 - 185.4}{0.1429} \approx -0.7\), and for 185.4 cm: \(z=0\).

Probability: \(P(-0.7 \leq Z \leq 0) = \Phi(0) - \Phi(-0.7) \approx 0.5 - 0.2420 = 0.2580\).

Necessity of Normality Assumption

Because the population distribution is normal, the sampling distribution of the mean is also exactly normal regardless of sample size. Thus, the assumption of normality is not necessary here; it is inherently satisfied.

Additional Scenarios

Similarly, for tires, laptop replacement times, salaries, unemployment durations, Facebook friends, and pollutants—each with their specific means and standard deviations—the CLT allows approximation of the sampling distribution of the sample mean as normal when the sample size is sufficiently large (typically n ≥ 30). Calculations involve determining the mean and standard deviation of the sample mean, transforming bounds into z-scores, and then finding the corresponding probabilities using standard normal distribution tables or software.

Implications of Normality Assumption

In cases where the underlying distribution is not known to be normal, large sample sizes help ensure the validity of the normal approximation due to the CLT. For small samples or skewed distributions, the assumption may be less valid, and alternative methods or distribution-specific analyses are necessary.

Conclusion

Overall, the CLT provides a robust foundation for probabilistic inference in various real-world contexts involving means, as long as assumptions about sample size and distribution are adequately considered. Recognizing when normal approximation is valid simplifies calculations and supports effective decision-making in quality control, business analytics, and scientific research.

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