Question

A random sample of \(n\) measurements has been selected from a population with unknown mean \(\mu\) and known standard deviation \(\sigma=10 .\) Calculate the width of a \(95 \%\) confidence interval for \mu for the sample sizes given. What effect do the changing sample sizes have on the width of the interval? \(n=100\)

Short answer

Answer: As the sample size increases, the width of the 95% confidence interval becomes narrower, resulting in a more precise estimate of the population mean. Conversely, a smaller sample size leads to a wider confidence interval and less precise estimate.

Step-by-step solution

Determine the z-score for 95% confidence level

To calculate the z-score for a 95% confidence interval, we can use a z-table or calculator to determine the z-score associated with an area of 0.975 (since we want to cover 95% of the area under the curve and there are two tails). The z-score is 1.96.

Calculate confidence intervals for sample size n=100

We're given the sample size \(n = 100\) and the standard deviation \(\sigma = 10\). To calculate the width of the 95% confidence interval, we use the formula: Width = \(2 * z * \frac{\sigma}{\sqrt{n}}\) Substituting the values, we get: Width = \(2 * 1.96 * \frac{10}{\sqrt{100}}\) = \(2 * 1.96 * 1\) Width = 3.92

Analyze the effect of changing sample sizes on the width of the confidence interval

In step 2, we calculated the width of the interval when \(n = 100\). Note that the term \(\frac{\sigma}{\sqrt{n}}\) in the width calculation has a direct, inverse relationship with the square root of the sample size. This means that when the sample size increases, the width of the confidence interval decreases, and vice versa. In summary, as the sample size increases, the width of the confidence interval becomes narrower, resulting in a more precise estimate of the population mean (μ).

Key concepts

Population Mean

The population mean, often symbolized as \(\mu\), represents the average of all measurements in a complete dataset for a particular group or entity. Imagine you were to measure the heights of all adult women in a small town; the population mean would be the average height of all these women.

In many real-world scenarios, it's impractical or impossible to measure every single individual in a population, so we rely on estimation methods. One of these methods is the confidence interval, which provides a range within which we expect the population mean to fall, based on a random sample from the population.

Standard Deviation

Standard deviation, denoted as \(\sigma\), measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

For instance, if you take a group of students' test scores, a low standard deviation would mean most students scored around the class average, whereas a high standard deviation would imply wide disparity in scores. In the context of confidence intervals, standard deviation plays a key role in determining the interval's width; a greater standard deviation will result in a wider confidence interval, assuming all other factors remain constant.

Sample Size

Sample size, represented as \(n\), is the number of observations or measurements taken from a population to form a sample. This sample acts as a representation of the population in statistical analysis.

As the sample size increases, the estimate of the population mean becomes more accurate. This inverse relationship between sample size and the confidence interval's width is pivotal: larger sample sizes yield narrower intervals, suggesting a more precise population mean estimate. This is because a larger sample size more accurately approximates the true population, reducing statistical uncertainty.

Z-score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. In the context of confidence intervals, the z-score corresponds to the confidence level and determines how many standard deviations an element is from the mean.

For a 95% confidence interval, the z-score ensures that we are capturing 95% of the data within our interval estimates. As your solution described, the z-score of 1.96 represents the point at which 95% of the dataset falls within two tails of the distribution, creating our confidence interval around the sample mean.

Confidence Level

The confidence level is the probability that the confidence interval actually contains the population mean. Common confidence levels include 90%, 95%, and 99%. A 95% confidence level suggests that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of those intervals to contain the population mean \(\mu\).

The choice of confidence level affects the width of the confidence interval: higher confidence levels result in wider intervals (providing more 'safety margin') and lower confidence levels lead to narrower intervals. Therefore, there is a balance between the degree of certainty we want in our estimates and the precision of the interval width.