Question

How does the t distribution compare with the normal distribution? How does this difference affect the size of confidence intervals constructed using \(\mathrm{z}\) relative to those constructed using t? Does sample size make a difference?

Short answer

The t distribution has thicker tails than the normal distribution, making t-based confidence intervals wider, especially with small samples. Larger samples reduce this difference.

Step-by-step solution

Understand the Distributions

The t distribution is similar to the normal distribution but has thicker tails. This means that there is more probability in the tails of a t distribution compared to a normal distribution. The t distribution is used when dealing with smaller sample sizes or when the population standard deviation is unknown.

Comparison of Confidence Intervals

Confidence intervals using the t distribution are generally wider than those using the normal distribution, especially for smaller sample sizes. This is because the t distribution accounts for the increased variability that comes with estimating the population standard deviation from a small sample.

Impact of Sample Size

As the sample size increases, the t distribution approaches the normal distribution due to the Central Limit Theorem. When the sample size is large (typically n > 30), the difference between the t and normal distributions becomes negligible, and the width of the confidence intervals converges.

Key concepts

Normal Distribution

The normal distribution is a fundamental concept in statistics and is often called the Gaussian distribution. This distribution forms the classic bell-shaped curve that many physical and social phenomena tend to follow. When graphed, it shows that most data points cluster around a central region, with fewer values appearing as you move away from the center. This property is known as the mean, where about 68% of the data falls within one standard deviation of the mean.

• Symmetrical shape: The curve is perfectly symmetric around the mean.
• Mean, median, and mode are equal: All these central measures coincide at the center.
• No skew: The tails of the distribution approach the horizontal axis but never touch it.

The normal distribution is central to various statistical methods, including hypothesis testing and constructing confidence intervals.

Confidence Intervals

Confidence intervals provide a range of values that are used to estimate a population parameter, such as a mean or proportion. When we calculate a confidence interval, we’re saying that we are a certain percent "confident" that the true population parameter falls within this range.

To construct confidence intervals, we often use the normal distribution or the t distribution. The choice of distribution largely depends on the size of the sample and whether we know the population standard deviation.

• Using Normal Distribution: Applied when the sample size is large, typically greater than 30, or the population standard deviation is known.
• Using t Distribution: Used when the sample size is smaller, or if the population standard deviation is unknown.

The wider the confidence interval, the more uncertainty we have about the parameter's true value. This is why confidence intervals from the t distribution tend to be wider than those from the normal distribution.

Sample Size

Sample size is a crucial concept in statistics that affects the reliability and validity of your results. A larger sample size tends to give more accurate estimates of the population parameter. This is because as sample size increases:

• Margin of error decreases, leading to narrower confidence intervals.
• The sample mean becomes a better estimate of the population mean.
• Variability in the sample estimates is reduced.

When the sample size is small, we often use the t distribution to account for increased variability and less known information about the population. As the sample size grows, the t distribution increasingly resembles the normal distribution, making the z and t distributions nearly equivalent for large samples.

Central Limit Theorem

The Central Limit Theorem is a foundational principle in statistics, which states that the distribution of the sample mean will tend to approximate a normal distribution, regardless of the population's distribution, as the sample size becomes large. This theorem is critical because:

• It allows for normal distribution inference with large sample sizes even when the original data is not normally distributed.
• Provides justification for using normal distribution-based techniques like confidence intervals and hypothesis tests when the sample size is large.
• Explains why the t distribution approaches the normal distribution as sample size increases.

Because of the Central Limit Theorem, statisticians can make inference about population parameters using sample data, even if the population usually does not follow a normal distribution, making it a powerful tool in the analysis of statistical data.