Question

question 2 2 pts which of the following statements are not true? the confidence intervals based on the t distribution are always the same as the confidence intervals based on the z distribution. the degrees of freedom are needed to determine the critical value based on the t distribution. the degrees of freedom for estimating the population mean based on the t distribution is n - 1. when the population mean is unknown, the t distribution should be used instead of the z distribution.

Explanation:

• For the first option:
• The \(T\) - distribution has heavier tails than the \(Z\) - distribution, especially when the sample size \(n\) is small. The formula for the confidence interval using the \(Z\) - distribution is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\) (when \(\sigma\) is known), and for the \(T\) - distribution is \(\bar{x}\pm t_{\alpha/2,n - 1}\frac{s}{\sqrt{n}}\) (when \(\sigma\) is unknown and \(s\) is the sample standard deviation). Since \(t_{\alpha/2,n-1}

eq z_{\alpha/2}\) for \(n<\infty\) (except in the limit as \(n
ightarrow\infty\) where \(t\) - distribution approaches \(Z\) - distribution), confidence intervals are not always the same.

• For the second option:
• The critical value of the \(T\) - distribution \(t_{\alpha/2}\) depends on the significance level \(\alpha\) and the degrees of freedom \(df\). The formula for the critical value selection for \(T\) - distribution is based on \(P(-t_{\alpha/2,df}<T<t_{\alpha/2,df})=1 - \alpha\), so degrees of freedom are needed.
• For the third option:
• When estimating the population mean \(\mu\) using the sample mean \(\bar{x}\) and sample standard deviation \(s\) (since \(\sigma\) is unknown), the test statistic \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) follows a \(T\) - distribution with \(n - 1\) degrees of freedom. This is because we lose one degree of freedom when we estimate the population standard deviation \(\sigma\) with the sample standard deviation \(s\).
• For the fourth option:
• The \(Z\) - distribution \(Z=\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}\) is used when the population standard deviation \(\sigma\) is known. When \(\sigma\) is unknown (which is more common in practice), we use the \(T\) - distribution \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) where \(s\) is the sample standard deviation.

Answer:

The confidence intervals based on the T distribution are always the same as the confidence intervals based on the Z distribution.