Question

A random sample of \(n=100\) measurements has been selected from a population with unknown mean \(\mu\) and known standard deviation \(\sigma=10 .\) Calculate the width of the confidence interval for \(\mu\) for the confidence levels given. What effect do the changing confidence levels have on the width of the interval? \(95 \%(\alpha=.05)\)

Short answer

The width of the 95% confidence interval for the population mean μ is 3.92. Changing the confidence level affects the width of the interval in such a way that increasing the confidence level results in a larger interval width, whereas decreasing the confidence level results in a smaller interval width. This is because, as the confidence level increases, there is a greater probability of capturing the true population mean μ within the interval.

Step-by-step solution

Calculate the standard error

Given the standard deviation \(\sigma=10\) and sample size \(n=100\), we can calculate the standard error (SE) using the formula: \( SE = \frac{\sigma}{\sqrt{n}} \) \( SE = \frac{10}{\sqrt{100}} \) \( SE = \frac{10}{10} \) \( SE = 1 \)

Find the critical value for 95% confidence level

For a \(95\%\) confidence level, the \(\alpha = 0.05\). Thus, we want to find \(z_\frac{\alpha}{2}\) corresponding to \(\frac{\alpha}{2} = 0.025\). From a standard normal distribution table or calculator, we find the critical value \(z_\frac{\alpha}{2} = 1.96\).

Calculate the width of the 95% confidence interval

Now we can calculate the width of the confidence interval using the formula: \( Width = 2 * z_\frac{\alpha}{2} * SE \) \( Width = 2 * 1.96 * 1 \) \( Width = 3.92 \) The width of the \(95\%\) confidence interval for \(\mu\) is \(3.92\).

Analyzing the effect of changing confidence levels on the width of the interval

As the confidence level increases, \(\alpha\) decreases and \(z_\frac{\alpha}{2}\) increases. This means that the confidence interval width will also increase, as it directly depends on \(z_\frac{\alpha}{2}\). In other words, a higher confidence level requires a larger interval width to ensure a greater probability of capturing the true population mean \(\mu\). Conversely, a lower confidence level will have a smaller interval width.

Key concepts

Standard Error

Understanding the standard error (SE) is crucial when conducting statistical analysis. It measures the variability in your sample mean, indicating how much the sample mean deviates from the true population mean. In the exercise, we calculated the SE by dividing the population standard deviation, \(\sigma=10\), by the square root of the sample size, \(\sqrt{n}\), where the sample size was \(n=100\).To calculate the SE, we used the formula:\[ SE = \frac{\sigma}{\sqrt{n}} \]With \(\sigma=10\) and \(n=100\), the calculation simplified to \(SE = 1\). This low SE implies our sample mean is likely to be a good estimate of the population mean.

Critical Value

The critical value is a factor used to widen your confidence interval to a point where, based on your desired confidence level, you expect it to contain the true population mean. A 95% confidence level was used in our exercise, corresponding to an \(\alpha=0.05\) (5% significance level). We find the critical value, denoted by \(z_\frac{\alpha}{2}\), by looking at a z-table or using statistical software, which corresponds to the point on the standard normal distribution that has \(\frac{\alpha}{2}\) area in the tails.In our exercise, we found the critical value for a 95% confidence level to be 1.96. This value scales the SE to set the width of our confidence interval.

Sample Size

The sample size, denoted by \(n\), plays a key role in determining the precision of our confidence interval. A larger sample size provides a smaller standard error, leading to a narrower confidence interval. It signifies more precision in our estimation of the population mean.In the given problem, we worked with a sample size of \(n=100\). If we were to increase the sample size, we would see a decrease in the standard error and thus a smaller interval width, all else being constant. Smaller sample sizes, on the other hand, lead to greater variability and thereby wider intervals.

Population Mean

The population mean, represented by \(\mu\), is the average value of a variable in the population. It is what we aim to estimate using our sample. Since \(\mu\) is unknown in the exercise, we use the sample mean as an estimator. Confidence intervals provide a range within which we can confidently say the population mean lies.Our confidence interval is constructed around the sample mean and extends \(z_\frac{\alpha}{2} * SE\) in both directions. The actual value of \(\mu\) remains unknown but with the confidence interval, we estimate it with a stated level of confidence.

Standard Deviation

Standard deviation, symbolized as \(\sigma\), is a measure of the amount of variation or dispersion of 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. In our case, the standard deviation of the population is known and is set at 10. This value is used to calculate the standard error, which is integral to finding the width of the confidence interval.The smaller the standard deviation of a given data set, the closer the sample means are to the population mean, making our estimation more precise.