Question

A random sample of \(n\) measurements is 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 these values of \(n\) : a. \(n=100\) b. \(n=200\) c. \(n=400\)

Short answer

Answer: The width of the 95% confidence interval for the population mean decreases as the sample size increases.

Step-by-step solution

Identify the relevant values

From the problem, we have: - \(\alpha = 0.05\) (100% - 95% = 5%) - \(Z_{\alpha/2} = 1.96\) (z-score for a 95% confidence interval) - \(\sigma = 10\) (known population standard deviation) We are asked to find the width of the 95% confidence interval for the given values of \(n\).

Calculate the width for each value of n

We will use the formula for the width of the confidence interval: $$Width = 2 \cdot Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$ a) For \(n = 100\): $$Width = 2 \cdot 1.96 \cdot \frac{10}{\sqrt{100}} = 3.92$$ The width of the confidence interval for a sample size of 100 is 3.92. b) For \(n = 200\): $$Width = 2 \cdot 1.96 \cdot \frac{10}{\sqrt{200}} = 2.77$$ The width of the confidence interval for a sample size of 200 is 2.77. c) For \(n = 400\): $$Width = 2 \cdot 1.96 \cdot \frac{10}{\sqrt{400}} = 1.96$$ The width of the confidence interval for a sample size of 400 is 1.96. Thus, the width of the 95% confidence interval for the population mean decreases as the sample size increases.

Key concepts

Population Mean

The population mean, often represented by the Greek letter \( \mu \), is a key concept in statistics that refers to the average value of all the data points in an entire population. In practice, we rarely have access to data from every single individual in a population and must instead estimate the population mean using a sample. The sample mean serves as an estimate of the population mean, but because it's based on only a portion of the population, there's some uncertainty associated with it. This uncertainty is precisely what confidence intervals aim to quantify. By constructing a confidence interval around the sample mean, we provide a range of values that are likely to include the true population mean. The width of this interval is an indicator of precision; narrower intervals suggest more precise estimates of the population mean. However, the true value of the population mean will remain unknown, and we rely on confidence intervals to provide a degree of certainty regarding where this unknown value lies based on the sample data.

Standard Deviation

The standard deviation, denoted as \( \sigma \) for a population and \( s \) for a sample, is a measure of the amount of variation or dispersion in a set of values. It tells us, on average, how much the individual data points differ from the mean. When the standard deviation is high, the values spread out more widely from the mean; a low standard deviation indicates that the values are more tightly clustered around the mean.In the context of confidence intervals, the standard deviation comes into play to help assess the variability within the population. Since we're using it to estimate the range in which the population mean might fall, it is directly proportional to the width of the confidence interval. This means that as the standard deviation increases, so does the width of the confidence interval, implying more uncertainty about the exact value of the population mean. In our exercise, with a known standard deviation of \( \sigma = 10 \) we use this value to help determine the spread of our confidence interval for the population mean.

Sample Size

Sample size, often denoted as \( n \), is a determining factor in statistical analyses. It represents the number of observations or data points collected from the population of interest. The sample size plays a crucial role in the precision of statistical estimates, including the confidence interval for the population mean.As we can see in our exercise, as the sample size increases, the width of the confidence interval decreases. This is because the margin of error associated with the estimate reduces when we have a larger sample. Mathematically, we see this relationship in the formula for the confidence interval width, where the sample standard deviation is divided by the square root of the sample size. By taking larger samples, we can make a more accurate estimation of the population mean, thus increasing our confidence in the precision of our measurement.To summarize, larger samples provide more information, and therefore, a smaller margin of error, leading to narrower confidence intervals. Consequently, researchers often aim for the largest sample size possible within the constraints of their resources to achieve more precise estimates.