Larger sample Suppose that the blood cholesterol level of...

Larger sample Suppose that the blood cholesterol level of all men aged 20 to 34 follows the Normal distribution with mean $\mu=188$ milligrams per deciliter $(\mathrm{mg} / \mathrm{dl})$ and standard deviation $\sigma=41 \mathrm{mg} / \mathrm{dl}$ (a) Choose an SRS of 100 men from this population What is the sampling distribution of $\overline{x} ?$ (b) Find the probability that $\overline{x}$ estimates $\mu$ within $\pm 3 \mathrm{mgldl}$ . This is the probability that $\overline{x}$ takes a value between 185 and 191 $\mathrm{mg} / \mathrm{dl} .$ ) Show your work. (c) Choose an SRS of 1000 men from this population. Now what is the probability that $\overline{x}$ falls within $\pm 3 \mathrm{mg} / \mathrm{dl}$ of $\mu ?$ Show your work. In what sense is the larger sample "better"?

Larger sample Suppose that the blood cholesterol level of all men aged 20 to 34 follows the Normal distribution with mean $\mu=188$ milligrams per deciliter $(\mathrm{mg} / \mathrm{dl})$ and standard deviation $\sigma=41 \mathrm{mg} / \mathrm{dl}$
(a) Choose an SRS of 100 men from this population What is the sampling distribution of $\overline{x} ?$
(b) Find the probability that $\overline{x}$ estimates $\mu$ within $\pm 3 \mathrm{mgldl}$ . This is the probability that $\overline{x}$ takes a value between 185 and 191 $\mathrm{mg} / \mathrm{dl} .$ ) Show your work.
(c) Choose an SRS of 1000 men from this population. Now what is the probability that $\overline{x}$ falls within $\pm 3 \mathrm{mg} / \mathrm{dl}$ of $\mu ?$ Show your work. In what sense is the
larger sample "better"?

Key Concepts

Effect of Sample Size on Estimation Accuracy

Increasing the sample size decreases the standard error, which in turn makes the sampling distribution of the sample mean more concentrated around the population mean. This improvement in precision means that larger samples yield a higher probability that the sample mean will be close to the true population mean, thereby making estimates more reliable and accurate.

Z-Scores and Probability Calculations

Z-scores are used to standardize values from a normal distribution by subtracting the mean and dividing by the standard deviation (or standard error in the context of the sample mean). This transformation makes it possible to use standard normal distribution tables or software to calculate probabilities and determine how likely the sample mean is to fall within a specified range.

Sampling Distribution of the Sample Mean

This concept describes the probability distribution of sample means obtained from multiple samples of the same size drawn from the same population. When the underlying population is normal, the distribution of the sample mean is also normal with the same mean as the population and a standard deviation equal to the population standard deviation divided by the square root of the sample size. This concept is foundational in inferential statistics for making probability statements about the sample mean.

Central Limit Theorem (CLT)

The Central Limit Theorem states that regardless of the original population distribution, the distribution of the sample mean will tend to be approximately normal as the sample size becomes large. This theorem justifies the use of the normal model for the sample mean in many situations and underpins many statistical methods and probability calculations, especially when dealing with large samples.

Standard Error

The standard error quantifies the variability of the sample mean and is calculated as the ratio of the population standard deviation to the square root of the sample size. It reflects how much the sample mean is expected to fluctuate from sample to sample and plays a critical role in determining the precision of estimates and in constructing confidence intervals.

Transcript

00:01 This problem deals with the blood cholesterol level of men between the ages of 20 and 24, and we know that this population follows a normal distribution.

00:09 The population mean is given as 188 milligrams per deciliter, and we know that there's a standard deviation of 41 milligrams per deciliter.

00:24 Part a says that there's a simple random sample of 100 men, so that means our sample size is equal to 100.

00:32 And we need to find the sampling distribution of the sample mean.

00:37 So the first thing we need to find is the mean of the sampling distribution of the sample mean, which i'll denote like this.

00:45 And then we need to find the standard deviation of the sampling distribution of the sample mean, which i'll denote like this.

00:55 The mean of the sampling distribution of the sample mean is the same value as the mean of the population.

01:01 So these two are equal to each other.

01:07 The standard deviation of the sampling distribution of the sample mean follows this formula.

01:13 It's the population standard deviation over the square root of the sample size.

01:21 In this case, i'm going to have the standard deviation of the population is 41 over the square root of the sample size, which is 100.

01:46 So that's 41 over 10, which is equal to 4 .1.

01:54 So in part b, we're trying to determine the probability that the mean of the sampling distribution of the sample mean estimates the population mean within plus or minus 3 milligrams per deciliter.

02:08 So in this case, some things that i know are that the population mean is equal to 188.

02:15 I know that the standard deviation of the sampling distribution of the sample mean is equal to 4 .1.

02:22 And if i have 188 and i want to go plus or minus three from either side, i'm going to get 191 and 185.

02:36 So the value is i'm trying to find the probability that the mean of the sampling distribution of the sample mean falls between 185 and 191.

02:51 So if i'm looking at a curve, i have that.

02:57 That the mean is 188 and i want to know that the prop, what is the probability that the mean of the sampling, distribution of the sample mean falls between 191 and 185.

03:17 So basically i'm looking for the area between these two curves.

03:24 So to find the probability represented between these two curves, i have to find something called a z score.

03:31 Now the c score gives how to the c score, how many standard deviations above or below the population mean that a data point is.

03:39 So the formula for z score is the value minus the mean over the standard deviation.

03:47 And i want to find the z score for my data point, 191.

03:52 So i have 191 minus the mean 188 divided by my standard deviation of 4 .1, which is going to give me 3 over 4 .1, which is equal to 0 .731.

04:10 Then i want to find the z score of my other data point 185, which is 185 minus 188 over 4 .1, which is going to be negative 3 over 4 .1, which will be negative 0 .731.

04:28 So i'm going to look up in something called a z table.

04:33 And the z table gives me the percent of values to the left of the z score.

04:40 So it gives me the probability of values, the percent chance of values that are going to be left of the z score.

04:48 So when i look up the probability of the z score of 191, i'm going to look up this number in the chart, and then i'll get the probability.

05:01 So let's look up .731 in our chart.

05:06 So up here i have 0 .7 and then up here is 3.

05:16 So this value right here, 0 .76730, is going to be the one value.

05:22 And then when i need the negative of that, i get 0 .2370.

05:27 So the z score probability for 191 is going to be 0 .76731.

05:39 And what that says is that the probability from here all the way over here, because it's all to the left of that point, is .76731...

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