The t distribution approaches the standardised normal distribution when the number of degrees of freedom increases.Question 14Select one:a.Falseb.True

Break Down the Problem

1. Identify the concept being examined: the behavior of the t distribution as degrees of freedom (df) change.
2. Determine the implication of increasing degrees of freedom on the distribution.

Relevant Concepts

• The t distribution is used in statistics, particularly in hypothesis testing and confidence intervals when sample sizes are small.
• As the sample size increases, the t distribution converges to the standard normal distribution (Z distribution).

Analysis and Detail

1. Understanding the t Distribution: The t distribution is similar to the standard normal distribution but has heavier tails. This means it's more prone to producing values that fall far from the mean, particularly with smaller sample sizes.
2. Effect of Degrees of Freedom: The degrees of freedom in a t distribution are typically equal to the sample size minus one (df = n - 1). As the sample size increases, this value increases, and the differences between the t distribution and the standard normal distribution decrease.
3. Conclusion: When the degrees of freedom are large, the t distribution becomes indistinguishable from the standard normal distribution due to the reduction in the impact of the extremes.

Verify and Summarize

Given the understanding that the t distribution approaches the standard normal distribution as the degrees of freedom increase, we can affirm the statement presented in the question.

Final Answer

b. True