Problem 30 - An Introduction to Mathematical Statistics and Its Applications

Question

What "confidence" is associated with each of the following random intervals? Assume that the \(Y_{i}\) 's are normally distributed. (a) \(\left[\bar{Y}-2.0930\left(\frac{S}{\sqrt{20}}\right), \bar{Y}+2.0930\left(\frac{S}{\sqrt{20}}\right)\right]\) (b) \(\left[\bar{Y}-1.345\left(\frac{S}{\sqrt{15}}\right), \bar{Y}+1.345\left(\frac{S}{\sqrt{15}}\right)\right]\) (c) \(\left[\bar{Y}-1.7056\left(\frac{S}{\sqrt{27}}\right), \bar{Y}+2.7787\left(\frac{S}{\sqrt{27}}\right)\right]\) (d) \(\left[-\infty, \bar{Y}+1.7247\left(\frac{S}{\sqrt{21}}\right)\right]\)

Step-by-Step Solution

Answer

The confidence levels are respectively about 95%, 80% and 95% for intervals (a), (b) and (d). For interval (c), there is not enough information to give a precise confidence level without additional context or information.

Step 1: Analyzing the Confidence Interval formula

The general formula for a confidence interval is \(\bar{Y} ± Z α/2 * \frac{S}{\sqrt{n}}\), where \(Z α/2\) is the Z-score (or T-score) for the desired level of confidence, \(S\) is the sample standard deviation, and \(n\) is the sample size.

Step 2: Evaluate the confidence level for interval (a)

In interval a, the multiplier of \(\frac{S}{\sqrt{20}}\) is 2.0930, which corresponds to a confidence level of approximately 95% in a two-tailed test.

Step 3: Evaluate the confidence level for interval (b)

In interval b, the multiplier of \(\frac{S}{\sqrt{15}}\) is 1.345, which corresponds to a confidence level of approximately 80% in a two-tailed test.

Step 4: Evaluate the confidence level for interval (c)

In interval c, we have asymmetric limits of the confidence interval, -1.7056 and +2.7787. For usual symmetric intervals, these limits are the same. This doesn't represent a standard confidence interval directly. It needs additional information about the distribution or an asymmetric confidence level. We need more information to provide a precise confidence level for this.

Step 5: Evaluate the confidence level for interval (d)

Interval d is a one-sided interval with infinite lower bound. The upper bound follows the pattern \(\bar{Y}+Z_{α}*(S/\sqrt{21})\), where Z_α = 1.7247 matches to confidence level of approximately 95% for a one-tailed test.

Key Concepts

Statistical Confidence

Statistical confidence is a measure of how much assurance we can have in the results of a statistical analysis. Often expressed as a percentage, it corresponds to the probability that a given confidence interval contains the true population parameter. For instance, a 95% confidence interval suggests that if the same population is sampled multiple times and intervals are calculated, about 95% of them would capture the true population mean.

The choice of confidence level is a balance between certainty and precision: higher confidence levels tend to yield broader intervals, but they decrease the risk of excluding the true parameter. It's crucial to note that the confidence level does not indicate the probability that a single interval contains the parameter.

Normal Distribution

The normal distribution, often called the 'bell curve', is a continuous probability distribution that is symmetric around its mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. In real-world scenarios, characteristics such as heights, test scores, and measurement errors often follow a roughly normal distribution.

When constructing confidence intervals for normally distributed data, we can use the normal distribution's properties to determine how far the sample mean is likely to be from the population mean. The central limit theorem provides the foundation for using normal distribution in confidence interval calculations, asserting that the distribution of sample means approximates a normal distribution as the sample size grows.

Sample Standard Deviation

The sample standard deviation (often represented as 'S') measures the dispersion or variation of sample data points from the sample mean. It is a central concept in calculating confidence intervals since it partly determines the width of the interval. The larger the standard deviation, the wider the confidence interval, reflecting greater uncertainty about the precise value of the population mean. Therefore, it's important to calculate it accurately.

When finding confidence intervals, it is this sample standard deviation that we use to estimate the variability within the population, as opposed to the population standard deviation, which would be used if the entire population's data were known.

Z-score

A Z-score is a statistical measurement describing a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. When constructing confidence intervals, the Z-score provides the critical value required to determine the margin of error. The chosen Z-score is associated with the desired confidence level. For example, a Z-score of approximately 1.96 corresponds to a 95% confidence level for a two-tailed test.

It's most applicable when the sample size is large or the population standard deviation is known because Z-scores are based on the assumption that the sampling distribution of the mean is normally distributed.

T-score

While Z-scores are used when the population standard deviation is known or the sample size is large, T-scores are a similar concept used when the sample size is small and the population standard deviation is unknown. T-scores are based on the t-distribution, which varies according to the degree of freedom (typically the sample size minus one).

The t-distribution is more spread out and has fatter tails than the normal distribution, which accounts for the increased variability expected in smaller samples. As the sample size gets larger, the t-distribution approaches the normal distribution. In hypothesis testing, the T-score helps determine if a result is significant relative to the variability in the sample.