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Sampling students, continued. To estimate the mean score \(\mu\) of those who took the Medical College Admission Test on your campus, you will obtain the scores of an SRS of students. From published information, you know that the scores are approximately Normal with standard deviation about 10.4. You want your sample mean \(x^{-} \nu\) to estimate \(\mu\) with an error of no more than one point in either direction. (a) What standard deviation must \(x^{-} \bar{x}\) have so that \(99.7 \%\) of all samples give an \(x^{-} \bar{x}\) within one point of \(\mu\) ? (Use the 68-95-99.7 rule.) (b) How large an SRS do you need in order to reduce the standard deviation of \(x^{-} \bar{x}\) to the value you found in part (a)?

Step by step solution

01

Understanding the Problem

We need to find the required standard deviation of the sample mean \( \bar{x} \) (part a), and then determine the sample size \( n \) to achieve this standard deviation (part b). The scores follow a normal distribution with a known standard deviation of the population, \( \sigma = 10.4 \). Finally, we need the error for \( \bar{x} \) to be no more than one point from \( \mu \).

02

Determine the Desired Standard Deviation

To ensure that 99.7% of samples have their mean within one point of \( \mu \), we use the 68-95-99.7 rule. This implies that \( 3 \times \) the standard deviation of \( \bar{x} \), \( \sigma_{\bar{x}} \), should equal 1 point. Thus, we solve: \[ 3\sigma_{\bar{x}} = 1 \] for \( \sigma_{\bar{x}} \).

03

Solve for the Desired Standard Deviation

Solving \( 3\sigma_{\bar{x}} = 1 \) gives us: \[ \sigma_{\bar{x}} = \frac{1}{3} \approx 0.333 \] Thus, the required standard deviation of the sample mean \( \bar{x} \) is approximately 0.333.

04

Relate Sample Size to Standard Deviation

The standard deviation of the sample mean \( \bar{x} \) is calculated as \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \). To find \( n \), we need to rearrange this equation to solve for \( n \): \[ n = \left( \frac{\sigma}{\sigma_{\bar{x}}} \right)^2 \] where \( \sigma = 10.4 \) from the problem setup and \( \sigma_{\bar{x}} = 0.333 \) from Step 3.

05

Calculate the Required Sample Size

Substitute the known values into the formula to get: \[ n = \left( \frac{10.4}{0.333} \right)^2 \] This evaluates to \( n \approx 973.64 \). Since \( n \) represents the number of students, we take the smallest whole number that is greater than 973.64, which gives us \( n \approx 974 \).

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Key Concepts

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

Sample Size Determination

Determining the appropriate sample size is crucial in statistical studies. It helps ensure the results are accurate and representative. When you're trying to determine the sample size, you are looking for the number of observations you need to achieve reliable results from your sample.
In the case of the Medical College Admission Test scores, we want the sample mean to have a small enough standard deviation so it reflects the true population mean as closely as possible. Here, we ascertain
the sample size using the formula:

• The standard deviation of the sample mean \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \).
• Rearranging, the formula for the required sample size becomes \( n = \left( \frac{\sigma}{\sigma_{\bar{x}}} \right)^2 \).
• With \( \sigma = 10.4 \) and \( \sigma_{\bar{x}} = 0.333 \), the calculated sample size is approximately 974.

This ensures the sample mean is within one point of the actual mean in 99.7% of the samples.

Standard Deviation of Sample Mean

The standard deviation of the sample mean quantifies how much the sample mean is expected to vary from the true population mean. The formula to calculate this is \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \).
Here, \( \sigma \) is the population standard deviation and \( n \) is the sample size.
The smaller the standard deviation of the sample mean, the closer the sample statistics will be to the population parameter.

• Let's say the population standard deviation is 10.4, as noted in the exercise. To achieve a sample mean with a standard deviation of 0.333, indicating small variability, a calculation must be performed.
• This involves rearranging the formula to solve for \( n \), enabling us to understand how many samples are needed to achieve the desired precision.

These calculations ensure research findings are valid and the conclusions drawn are based on reliable data.

Normal Distribution

A normal distribution is a bell-shaped curve that is symmetrical around the mean. In the context of statistics, the normal distribution is crucial because it predicts how data values are likely to be dispersed.
Often in studies, variables like test scores follow a normal distribution.
Key characteristics of a normal distribution include:

• Mean, median, and mode are all equal and located at the center of the distribution.
• Approximately 68% of data falls within one standard deviation, about 95% within two, and about 99.7% falls within three standard deviations.
• It forms the basis for many statistical tests and methods, including determining sample sizes and intervals.

In our problem, test scores are normally distributed with a known standard deviation, highlighting the importance of understanding this distribution when calculating sample sizes and estimating population parameters.

68-95-99.7 Rule

The 68-95-99.7 rule, also known as the empirical rule, is a statistical guideline for normal distributions. It suggests:

• About 68% of data points lie within one standard deviation from the mean.
• About 95% lie within two standard deviations.
• About 99.7% fall within three standard deviations.

This rule is central to determining sample size accuracy and predicting data variability.
Using this rule in our example, we want the triplet of standard deviations to cover an error of no more than one point from the mean, defining \( 3\sigma_{\bar{x}} = 1 \).
From this, we calculate \( \sigma_{\bar{x}} = \frac{1}{3} \), ensuring almost all sample means will lie within this range. This highlights the power of the 68-95-99.7 rule in assessing data consistency and reliability.