Lightning strikes The number of lightning strikes on a...

Lightning strikes The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation $2.4 .$ The National Lightning Detection Network (NLDN) uses automatic sensors to watch for lightning in a random sample of 10 one-square-kilometer plots of land. (a) What are the mean and standard deviation of the sampling distribution of $\bar{x}$, the sample mean number of strikes per square kilometer? (b) Explain why you cannot safely calculate the probability that $\bar{x}<5$ based on a sample of size 10 . (c) Suppose the NLDN takes a random sample of $n=50$ square kilometers instead. Explain how the central limit theorem allows us to find the probability that the mean number of lightning strikes per square kilometer is less than $5 .$ Then calculate this probability. Show your work.

Lightning strikes The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation $2.4 .$ The National Lightning Detection Network (NLDN) uses automatic sensors to watch for lightning in a random sample of 10 one-square-kilometer plots of land.
(a) What are the mean and standard deviation of the sampling distribution of $\bar{x}$, the sample mean number of strikes per square kilometer?
(b) Explain why you cannot safely calculate the probability that $\bar{x}<5$ based on a sample of size 10 .
(c) Suppose the NLDN takes a random sample of $n=50$ square kilometers instead. Explain how the central limit theorem allows us to find the probability that the mean number of lightning strikes per square kilometer is less than $5 .$ Then calculate this probability. Show your work.

Key Concepts

Sampling Distribution

This concept refers to the probability distribution of a statistic (commonly the sample mean) obtained through repeated sampling from a population. It has important properties, such as its mean being equal to the population mean and its standard deviation (known as the standard error) being equal to the population standard deviation divided by the square root of the sample size.

Standard Error

The standard error is the measure of how much the sample mean is expected to vary from the true population mean. It is calculated as the population standard deviation divided by the square root of the sample size. A smaller standard error indicates more precise estimates of the population mean.

Central Limit Theorem

The central limit theorem states that regardless of the distribution of the underlying population, the distribution of sample means will tend toward a normal distribution as the sample size increases. This theorem is critical because it justifies the use of normal probability models for inference about the sample mean even when the population distribution is not normal, provided the sample size is large enough.

Sample Size and its Impact

The size of the sample plays a crucial role in determining the accuracy and reliability of estimates. With small samples, the sampling distribution may noticeably deviate from normality, making it unsafe to apply normal approximations to probability calculations. As the sample size increases, these approximations become more valid thanks to the central limit theorem.

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