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 $2.4 .$ (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.

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.

Key Concepts

Sample Size Considerations

Sample size plays a critical role in statistical inference; smaller samples may not adequately approximate the underlying distribution, which can lead to unreliable probability estimates. Larger samples tend to produce a more accurate and normally distributed sample mean due to the Central Limit Theorem, which makes statistical analysis and hypothesis testing more reliable.

Central Limit Theorem

The Central Limit Theorem (CLT) states that, regardless of the shape of the population distribution, the distribution of the sample means will tend to approximate a normal distribution as the sample size increases, provided that the original distribution has a finite variance. This principle is crucial for making probabilistic inferences about the sample mean when the sample size is sufficiently large.

Sample Mean

The sample mean is the average of a set of observations drawn from a population. It serves as an unbiased estimator of the population mean when the sample is random and representative. In statistical analysis, the behavior of the sample mean is central to estimating population parameters and making inferences about the larger group.

Standard Error

The standard error is the standard deviation of the sample mean's distribution, calculated as the population standard deviation divided by the square root of the sample size (?/?n). It quantifies the variability of the sample mean from sample to sample and provides insight into the precision of the estimation of the population mean.

Transcript

00:02 This problem deals with the typical number of lightning strikes on one square kilometer of land.

00:08 So if i do a distribution with that information, i have that the mean is equal to six.

00:19 And i'm given that there's a standard deviation of 2 .4.

00:23 So this distance, standard deviation of 2 .4.

00:34 And this is the population of lightning strikes on.

00:49 One square kilometer of land.

00:55 So we get a random sample from 10 square kilometers of land, so that means that our sample size is going to be 10.

01:04 And i want to do a sampling distribution.

01:08 So i know that the mean is going to be the same as the population mean.

01:14 So that's going to equal to six.

01:17 And my sample size of 10 lets me figure out the standard deviation of the sampling mean.

01:24 Standard deviation is equal to the standard deviation of the population divided by the square root of the sample size.

01:33 So the standard deviation of my population was 2 .4 over the square root of 10.

01:41 So when i put that in the calculator, i get 0 .7589.

01:48 So there were two things i needed to find in this problem.

01:51 I needed to find the mean of a sample.

01:57 And i needed to find a standard deviation when the sample size went to 10.

02:02 Now, part b says, why can't you calculate that the sampling, the means of the sample is less than five using a sample size of 10? and the problem is that my population information, i don't know if it's normal or not.

02:31 And so if i don't know if it's normal, then a sample size of 10 is not big enough to ensure that the sampling distribution is going to be normal.

03:41 So to make sure that the sampling distribution is normal, we need a sample size of at least 30.

03:49 So in part c, i say, okay, we're going to give you a sample size of 50, and we want you to compute...

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