Margin of error is computed by multiplying the standard error by a critical value from distributions such as the Z or t distributions. Which options best describe the source of the critical value?

Margin of error is computed by multiplying the standard error by a critical value from distributions such as the Z or t distributions. Which options best describe the source of the critical value?

• A. Normal (Z) distribution
• B. t-distribution
• C. Both Z and t distributions
• D. Chi-square distribution

Answer: (C)

The margin of error comes from multiplying the standard error by a cutoff value that comes from a sampling distribution. Which distribution provides that cutoff depends on what we know about the population standard deviation and the sample size. If the population standard deviation is known (or the sample size is large enough to treat the population as effectively known), the standard normal distribution (Z) provides the critical value for the desired confidence level. If the population standard deviation is unknown and we’re estimating it from the sample, the sampling distribution of the sample mean follows a t distribution with degrees of freedom n−1, so we use the corresponding t critical value. As the sample size grows, the t distribution resembles the normal distribution, so using Z becomes a good approximation even when sigma is unknown. The chi-square distribution, by contrast, is used for confidence intervals about a variance or standard deviation, not for the mean’s margin of error. So the critical value can come from either the normal or the t distribution depending on the context, which is why both are correct sources.

The margin of error comes from multiplying the standard error by a cutoff value that comes from a sampling distribution. Which distribution provides that cutoff depends on what we know about the population standard deviation and the sample size. If the population standard deviation is known (or the sample size is large enough to treat the population as effectively known), the standard normal distribution (Z) provides the critical value for the desired confidence level. If the population standard deviation is unknown and we’re estimating it from the sample, the sampling distribution of the sample mean follows a t distribution with degrees of freedom n−1, so we use the corresponding t critical value.

As the sample size grows, the t distribution resembles the normal distribution, so using Z becomes a good approximation even when sigma is unknown. The chi-square distribution, by contrast, is used for confidence intervals about a variance or standard deviation, not for the mean’s margin of error. So the critical value can come from either the normal or the t distribution depending on the context, which is why both are correct sources.