91Ó°ÊÓ

Larger Sample, More Accurate Estimate. Suppose that, in fact, the total cholesterol level of all men aged 20-34 follows the Normal distribution with mean \(\mu=182\) milligrams per deciliter (mg/dL) and standard deviation \(\sigma=37\) \(\mathrm{mg} / \mathrm{dL}\). (a) Choose an SRS of 100 men from this population. What is the sampling distribution of \(\mathrm{x}^{-} \vec{x}\) ? What is the probability that \(\mathrm{x}^{-} \vec{x}\) takes a value between 180 and \(184 \mathrm{mg} / \mathrm{dL}\) ? This is the probability that \(\mathrm{x}^{-} \bar{x}\) estimates \(\mu\) within \(\pm 2\) \(\mathrm{mg} / \mathrm{dL}\). (b) Choose an SRS of 1000 men from this population. Now what is the probability that \(\mathrm{x}^{-}=\)falls within \(\pm 2 \mathrm{mg} / \mathrm{dL}\) of \(\mu\) ? The larger sample is much more likely to give an accurate estimate of \(\mu\).

Step by step solution

01

Identify given parameters

The mean of cholesterol level is given as \( \mu = 182 \, \text{mg/dL} \), and the standard deviation is \( \sigma = 37 \, \text{mg/dL} \).

02

Determine the sampling distribution for SRS of 100 men

For a sample size \( n = 100 \), the mean of the sampling distribution remains \( \mu = 182 \), and the standard deviation of the sampling distribution, or the standard error, is \( \sigma/\sqrt{n} = 37/\sqrt{100} = 3.7 \, \text{mg/dL} \).

03

Calculate the Z-scores for SRS of 100 men

We want the probability that \( \bar{x} \) is between 180 and 184. Calculate \( Z \) for 180: \( Z = (180 - 182) / 3.7 \approx -0.54 \), and \( Z \) for 184: \( Z = (184 - 182) / 3.7 \approx 0.54 \).

04

Find the probability for SRS of 100 men

Look up the Z-values in the standard normal distribution table or use a calculator: \( P(-0.54 < Z < 0.54) \approx 0.4112 \).

05

Determine the sampling distribution for SRS of 1000 men

Now, for a sample size \( n = 1000 \), the mean remains \( 182 \), and the standard error is \( 37/\sqrt{1000} \approx 1.17 \, \text{mg/dL} \).

06

Calculate the Z-scores for SRS of 1000 men

For the range \( 180 \leq \bar{x} \leq 184 \), calculate Z for 180: \( Z = (180 - 182) / 1.17 \approx -1.71 \), and for 184: \( Z = (184 - 182) / 1.17 \approx 1.71 \).

07

Find the probability for SRS of 1000 men

Use the Z-table or calculator: \( P(-1.71 < Z < 1.71) \approx 0.9114 \).

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Key Concepts

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

Sampling Distribution

In statistics, the concept of a sampling distribution is central to understanding how sample data relates to the population from which it is drawn. When you draw a simple random sample (SRS), like choosing 100 or 1000 men from a population with known cholesterol levels, each possible sample has a sample mean—this is a realization of the sampling distribution. The sampling distribution of the sample mean \( \bar{x} \) provides a way to approximate the behavior of the sample mean across different samples.
For a larger sample size, such as 1000 men, the sampling distribution of \( \bar{x} \) becomes more narrowly centered around the population mean, \( \mu = 182 \, \mathrm{mg/dL}\). This means the estimates are more reliable, which we can also refer to as having lower variability, making the larger sample preferable for estimating population parameters accurately.

• The mean of the sampling distribution, denoted as \( E(\bar{x}) \), is equal to the population mean \( \mu \).
• The spread of this distribution is measured by the standard error, which decreases as the sample size increases.

Standard Deviation

A fundamental concept in statistics, the standard deviation \( \sigma \) quantifies the amount of variation or dispersion in a set of values. In the context of the original problem, it's given that the cholesterol level has a standard deviation of \( 37 \, \mathrm{mg/dL} \).
The standard deviation tells us how spread out our data points are from the mean. A larger standard deviation indicates that the data points are spread out over a wider range of values, whereas a smaller standard deviation suggests they are closer to the mean.

• It is crucial in defining the shape of the normal distribution.
• In a normal distribution, about 68% of values fall within one standard deviation (above or below) the mean.

Standard Error

The standard error is a key statistic when considering a sampling distribution. It indicates how much the sample mean \( \bar{x} \) deviates from the population mean \( \mu \). When you have a sample size \( n\) the standard error is calculated with the formula \( \sigma/\sqrt{n} \), where \( \sigma \) is the population standard deviation.
As seen in the exercise, for 100 men, the standard error was \( 3.7 \, \mathrm{mg/dL} \), whereas for 1000 men, it dropped to \( 1.17 \, \mathrm{mg/dL} \). This decrease shows that we expect less variation in sample means as the sample size increases.

• The standard error is smaller when the sample size is larger, indicating more reliable mean estimates.
• This concept is critical for constructing confidence intervals and hypothesis tests.

Normal Distribution

Normal distribution, often called the Gaussian distribution, is one of the most important concepts in statistics. It is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
In our exercise, it is stated that the total cholesterol level of men follows a normal distribution with a mean \( \mu = 182 \) and a standard deviation \( \sigma = 37 \). This distribution is used to calculate probabilities concerning sample means. The normal distribution is key because

• It provides a basis for making inferences about population parameters.
• The properties of a normal distribution allow for the use of Z-scores when determining probabilities.

Since 1000 men's sample estimates follow a normal distribution closely, we can use the Z-table for probabilities like \( P(-1.71 < Z < 1.71) \) effectively. The larger sample size makes the distribution of the sample mean conform more to the normal curve, enhancing accuracy.