Q 8.143. - Elementary Statistics

Question

Table IV in Appendix A contains degrees of freedom from $I$ to $75$ consecutively but then contains only selected degrees of freedom.

a. Why couldn't we provide entries for all possible degrees of freedom?

b. Why did we construct the table so that consecutive entries appear for smaller degrees of freedom but that only selected entries occur for larger degrees of freedom?

c. If you had only Table IV, what value would you use for $t_0$ os with df $=87$ with $\mathrm{df}=125?$ with $\mathrm{df}=650?$ with $\mathrm{df}=3000$ ? Explain your answers.

Step-by-Step Solution

Answer

Part (a) the $t$ table does not include $t$ values for all degrees of freedom.

Part (b) For larger degrees of freedoms consecutive $t-$values are (approximately) same.

Part (c) There is no difference between $t-$value and $z-$value.

Part (a) Step 1: Given information

A contains degrees of freedom from $1$ to $75$ in order, but only chosen degrees of freedom after that.

Part (a) Step 2: Concept

$t-$value and $z-$ value concept used.

Part (a) Step 3: Explanation

Because we can utilize the linear interpolation method or technology to find the $t-$values for those degrees of freedom that are not shown in the table. Additionally, for degrees of freedom greater than $2000$, the $t$ and $z$ values are nearly equivalent. As a result, the $t$ table does not include $t$ values for all degrees of freedom.

Part (b) Step 1: Explanation

Because consecutive entries, i.e. consecutive $t-$values, are significantly different for fewer degrees of freedoms, but (roughly) the same for bigger degrees of freedoms.

Part (c) Step 1: Calculation

We would use the value of $t_{0.05}$ corresponding to $d f=85$ i.e., $1.292$

Because, $85$ is the nearest to the $d f 87$

$\therefore$ For $df=87, t_{0.05}=1.292$

Similarly, for $t_{0.05}$ with $d f=125$ we would use the value of $t_{0.05}$ with $d f=100$ i.e., $1.660$ $\therefore$ For $df=125, t_{0.05}=1.660$

For $t_{0.05}$ with $d f=650$, we would use the value of $t_{0.05}$ for $d f=600$ or $700$ , because both are same.

$\therefore$ For $df=650, t_{0.05}=1.647$

We'd use $z_{0.05}$, or $1.645$, for $t_{0.05}$ with $d f=3000$ because for degrees of freedom more than $2000$, we'd use $z_{0.05}$, or $1.645$

There is no difference between $t-$value and $z-$value.

$\Rightarrow \text{ For }df=3000, t_{0.05}=1.645$