The distribution that is symmetric and bell-shaped but with thicker tails and that approaches the Z-distribution as degrees of freedom increase is the

The distribution that is symmetric and bell-shaped but with thicker tails and that approaches the Z-distribution as degrees of freedom increase is the

• A. Z-distribution
• B. Chi-square
• C. t-distribution
• D. Uniform

Answer: (C)

This distribution is used when you’re estimating a population mean but the population standard deviation is unknown. Because you’re substituting the sample standard deviation for the true one, there’s extra uncertainty, and the resulting distribution has heavier (fatter) tails than the normal distribution. It remains symmetric and bell-shaped, like the normal, but with these fatter tails to account for that extra variability. As the degrees of freedom increase (which happens with larger sample sizes), the estimate of the standard deviation becomes more reliable, so the extra tail heaviness diminishes. In the limit, the distribution converges to the standard normal distribution. This is why the t-distribution is described as approaching the Z-distribution as degrees of freedom grow. Why the other options don’t fit: the Z-distribution is the normal distribution with known population standard deviation and does not account for extra uncertainty from estimating variability; the chi-square distribution is skewed to the right and not symmetric; the uniform distribution is flat, not bell-shaped.

This distribution is used when you’re estimating a population mean but the population standard deviation is unknown. Because you’re substituting the sample standard deviation for the true one, there’s extra uncertainty, and the resulting distribution has heavier (fatter) tails than the normal distribution. It remains symmetric and bell-shaped, like the normal, but with these fatter tails to account for that extra variability.

As the degrees of freedom increase (which happens with larger sample sizes), the estimate of the standard deviation becomes more reliable, so the extra tail heaviness diminishes. In the limit, the distribution converges to the standard normal distribution. This is why the t-distribution is described as approaching the Z-distribution as degrees of freedom grow.

Why the other options don’t fit: the Z-distribution is the normal distribution with known population standard deviation and does not account for extra uncertainty from estimating variability; the chi-square distribution is skewed to the right and not symmetric; the uniform distribution is flat, not bell-shaped.