In the EAI sampling problem (see Figure 7.8 ), we showed...

In the EAI sampling problem (see Figure $7.8 ),$ we showed that for $n=30$ , there was. 5064 probability of obtaining a sample mean within $\pm \$ 500$ of the population mean. a. What is the probability that $\overline{x}$ is within $\$ 500$ of the population mean if a sample of size 60 is used? b. Answer part (a) for a sample of size 120 .

In the EAI sampling problem (see Figure $7.8 ),$ we showed that for $n=30$ , there was. 5064 probability of obtaining a sample mean within $\pm \$ 500$ of the population mean.
a. What is the probability that $\overline{x}$ is within $\$ 500$ of the population mean if a sample of size 60 is used?
b. Answer part (a) for a sample of size 120 .

Key Concepts

Z-Scores and Probability Calculations

Z-scores standardize the difference between the sample mean and the population mean in terms of the standard error, allowing for the use of the standard normal distribution to calculate probabilities. By converting a specific margin (such as ±500) into a corresponding z-score, one can determine the probability that the sample mean lies within that margin of the population mean.

Standard Error

The standard error is a measure of the variability or spread of the sampling distribution of the sample mean. It is calculated as the population standard deviation divided by the square root of the sample size (?/?n). A larger sample size results in a smaller standard error, indicating that the sample mean is likely to be closer to the population mean.

Sampling Distribution of the Mean

This concept refers to the probability distribution of all possible sample means drawn from a population. It is central to inferential statistics, as it underpins the idea that a sample mean can estimate a population mean, and its shape and spread are determined by the underlying population distribution and the sample size.

Central Limit Theorem

The Central Limit Theorem is a fundamental statistical principle stating that, given a sufficiently large sample size, the distribution of the sample mean approximates a normal distribution regardless of the population’s original distribution. This theorem justifies using the normal distribution to calculate probabilities associated with sample means.

Transcript

0:00 All right.

00:01 Earlier in the section, there was a problem about eai sampling given these two population statistics, a standard deviation of 4 ,000 and mean of 51 ,800.

00:12 And we want to know the probability of obtaining obtaining, oops, did not proofread that one, a point estimate of the mean with an error of plus or minus 500 if the population size is either 60 or 120.

00:29 So let's start with 60.

00:32 First off, let's find an estimate for the standard error.

00:39 Since we aren't really given a population, we can assume it is infinite.

00:52 Yeah.

00:53 So knowing that, we are going to calculate our...

00:58 That's not the formula.

01:02 Point estimate at 60, so that's going to be 4 ,000 of the square root of 60.

01:09 We get that this is approximately equal to 516 .3978.

01:18 All right.

01:19 Now let's z score plus or minus 500s against our point estimate of our mean.

01:27 Actually, no, this doesn't matter because the top is just going to be positive 500 and negative 500.

01:33 And then we're going to divide that by our point estimate.