Question

If it is assumed that the heights of men are normally distributed with a standard deviation of 2.5 inches, how large a sample should be taken to be fairly sure (probability .95) that the sample mean does not differ from the true mean (population mean) by more than .50 in absolute value?

Short answer

Answer: The required sample size is 62 men.

Step-by-step solution

Identify the Margin of Error and Confidence Level

The margin of error is 0.50 inches and the confidence level is 95%. We'll need to find the critical value (z-score) corresponding to this confidence level.

Find the Critical Value (z-score)

For a 95% confidence level, the corresponding z-score can be found from a standard normal distribution table, or by using statistical software. A 95% confidence level means that the area between the two tails is 95%, so each tail contains 2.5% of the area. Thus, we want to find z-score corresponding to 97.5% percentile. The z-score for a 97.5% percentile is approximately 1.96.

Use the Formula for Margin of Error

The formula for margin of error (E) in a sample mean is given by: E = z * (σ/√n) where z is the critical value, σ is the population standard deviation, and n is the sample size. We have the margin of error (0.50 inches), z-score (1.96) and standard deviation (2.5 inches). We'll solve for n: 0.50 = 1.96 * (2.5/√n)

Solve for Sample Size (n)

We can now solve for n: (0.50 / (1.96 * 2.5))^2 = n n ≈ 61.29 Since we cannot have a fraction of a person in our sample, we will round up to the nearest whole number, which is 62.

Interpret the Result

We need a sample size of 62 men to be 95% confident that the sample mean is within 0.50 inches of the true population mean.

Key concepts

Normal Distribution

A Normal Distribution is a key concept in statistics that appears often when discussing datasets, as it's a common model used to describe how values are scattered around a mean. It is recognizable by its bell-shaped curve, which is symmetric around its central point, known as the mean. This distribution occurs naturally in many datasets, such as heights, weights, and test scores.

Each normal distribution can be defined with two parameters: the mean, which dictates where the center of the bell is, and the standard deviation, which controls how spread out the data is. The larger the standard deviation, the flatter and more spread out the bell curve is. Conversely, a smaller standard deviation results in a steeper peak.

For example, in our exercise concerning men's heights, it is mentioned that these heights are normally distributed with a standard deviation of 2.5 inches. This tells us that most men's heights are within 2.5 inches of the average.

Confidence Level

The Confidence Level in statistical terms represents how confident we can be that a parameter lies within a calculated interval. In simpler terms, it's our degree of certainty that the mean of the sample is close to the mean of the population.

Common confidence levels include 90%, 95%, and 99%. In the context of our exercise, we have a confidence level of 95%. This means that if we were to take many samples and compute the confidence interval for each one, 95% of those intervals would contain the true population mean.

To put it practically, the higher the confidence level, the more confident we are in our results. However, this often requires a larger sample size because a higher confidence level also means a wider interval.

Margin of Error

The Margin of Error is an important statistical term that quantifies the range within which we expect our sample statistic to fall relative to the true population parameter. It acts like a safety cushion that accounts for sampling variability.

In our exercise, the margin of error is set at 0.50 inches. This means that we want our sample mean to be within plus or minus 0.50 inches of the actual population mean. The margin of error is critical in determining the required sample size, as a smaller margin calls for a larger sample size to ensure greater precision in estimates.

The margin of error is directly tied to the confidence level and standard deviation. It can be found using the formula: \[ E = z \times \left(\frac{\sigma}{\sqrt{n}}\right) \]where \(E\) is the margin of error, \(z\) is the critical value, \(\sigma\) is the standard deviation, and \(n\) is the sample size we wish to determine.

Critical Value

A Critical Value is derived from the standard normal distribution and is used to set the boundaries for confidence intervals. It's represented as a z-score, which tells us how far from the mean of the distribution our decision threshold is.

To find it, you first decide your desired confidence level. Here, we're working with a 95% confidence level. Given the bell-shaped curve, 95% confidence represents the area under the curve from one tail to the other, leaving only 2.5% at each extreme tail. The corresponding z-score typically used for a 95% confidence interval is 1.96, marking the point beyond which only 2.5% of values lie.

This critical value is then used in the margin of error formula to calculate how big of a sample you need for your estimates to be within a certain range of the true population mean. Understanding critical values helps in creating more accurate and reliable statistical analyses.