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Lightning strikes The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation \(2.4 .\) (These values are typical of much of the United States.) 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 \(\overline{x},\) the sample mean number of strikes per square kilometer? (b) Explain why you cannot safely calculate the probability that \(\overline{x}<5\) 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.

Step by step solution

01

Mean of the Sample Mean

The mean of the sample mean, \( \overline{x} \), is equal to the population mean. Since the mean number of strikes per square kilometer is 6, the mean of \( \overline{x} \) is also 6.

02

Standard Deviation of the Sample Mean

The standard deviation of the sample mean is calculated using the formula for the standard error of the mean: \( \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} \). Here, \( \sigma = 2.4 \) and \( n = 10 \). Substitute these values to get: \[ \sigma_{\overline{x}} = \frac{2.4}{\sqrt{10}} \approx 0.76. \]

03

Explanation of Probability Calculation for n=10

When \( n = 10 \), the sample size is not large enough to assume that \( \overline{x} \) is normally distributed, especially since the original distribution is unknown. The Central Limit Theorem requires a sample size of at least 30 to safely apply a normal approximation.

04

Central Limit Theorem for n=50

For \( n = 50 \), the sample size is large enough for the Central Limit Theorem to apply. This theorem states that the sampling distribution of \( \overline{x} \) will be approximately normal, even if the population distribution is not.

05

Calculate Probability for XÌ„ < 5 with n=50

Now that the Central Limit Theorem applies, we can calculate the probability that \( \overline{x} < 5 \) by finding the Z-score and using the standard normal distribution table. First, calculate the standard error for \( n = 50 \): \[ \sigma_{\overline{x}} = \frac{2.4}{\sqrt{50}} \approx 0.34. \] Then calculate the Z-score for \( \overline{x} = 5 \): \[ Z = \frac{5 - 6}{0.34} \approx -2.94. \] Use standard normal distribution tables to find the probability corresponding to \( Z = -2.94 \): the probability is approximately 0.0016.

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Key Concepts

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

Sampling Distribution

The concept of sampling distribution is central to understanding statistics. When we take a sample from a population, the sample mean differs slightly each time due to random chance. If we repeatedly take samples of size \( n \) and calculate their means, these means form what we call the sampling distribution of the sample mean.
The sampling distribution helps us make inferences about the population mean. In our lightning strike example, if you were to repeatedly sample different areas and calculate the mean number of strikes each time, each of these sample means would form the sampling distribution. It's important to note that the mean of this distribution is the same as the mean of the population, which in this example is 6. This predictability is vital because it allows statisticians to make informed predictions.

• Mean of sample distribution equals the population mean.
• Variation in the sample means decreases with a larger sample size.

Standard Error

The standard error measures variability in the sampling distribution. It can be thought of as the "average" amount by which sample means differ from the actual population mean. For a sample mean, the standard error is calculated as \( \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} \).
In our example, with a population standard deviation \( \sigma = 2.4 \) and a sample size of 10, the standard error was found to be approximately 0.76. This number gives an indication of how much the sample means can be expected to differ from the population mean of 6.

• Larger sample sizes lead to smaller standard errors.
• Smaller standard errors imply that sample means are closer to the population mean.

Z-score Calculation

The Z-score is a statistical measure that tells us how many standard deviations an element is from the mean. In this context, we use it to determine the probability of a sample mean being a certain value.To calculate the Z-score, we use the formula \( Z = \frac{\overline{x} - \mu}{\sigma_{\overline{x}}} \). In the lightning strike problem, when calculating the probability of \( \overline{x} < 5 \) using a sample size of 50, the Z-score was calculated as approximately -2.94.
This means that 5 is 2.94 standard deviations below the mean of the sampling distribution. Using Z-scores and standard normal distribution tables allows us to find how likely it is that a sample mean will fall below a given value.

• Z-scores standardize values across different data sets.
• A negative Z-score indicates a value below the mean.

Normal Approximation

Normal approximation uses the properties of the normal distribution to approximate the sampling distribution of the sample mean when certain conditions are met. Thanks to the Central Limit Theorem, even if the population distribution isn't normal, the distribution of the sample mean will be approximately normal if the sample size is large enough.
In this exercise, the original distribution is unknown. But, with a sample size of 50, the Central Limit Theorem allows us to assume that the distribution of the sample mean is approximately normal. This is a fundamental concept because it enables the use of normal probability tables to make statistically accurate predictions.

• Good approximation with sample sizes \( n \geq 30 \).
• Allows for use of normal distribution properties in analysis.