determine the upper - tail critical value tα/2 in each of the following circumstances. a. 1 - α = 0.95, n =…

determine the upper - tail critical value tα/2 in each of the following circumstances. a. 1 - α = 0.95, n = 59 b. 1 - α = 0.90, n = 59 c. 1 - α = 0.95, n = 42 d. 1 - α = 0.95, n = 8 e. 1 - α = 0.99, n = 49 click here to view page 1 of the table of critical values for the t distribution. click here to view page 2 of the table of critical values for the t distribution. a. t = (round to four decimal places as needed.) b. t = (round to four decimal places as needed.) c. t = (round to four decimal places as needed.) d. t = (round to four decimal places as needed.) e. t = (round to four decimal places as needed.)

determine the upper - tail critical value tα/2 in each of the following circumstances. a. 1 - α = 0.95, n = 59 b. 1 - α = 0.90, n = 59 c. 1 - α = 0.95, n = 42 d. 1 - α = 0.95, n = 8 e. 1 - α = 0.99, n = 49 click here to view page 1 of the table of critical values for the t distribution. click here to view page 2 of the table of critical values for the t distribution. a. t = (round to four decimal places as needed.) b. t = (round to four decimal places as needed.) c. t = (round to four decimal places as needed.) d. t = (round to four decimal places as needed.) e. t = (round to four decimal places as needed.)

Answer

Explanation:

Step1: Calculate degrees of freedom

The degrees of freedom is calculated as $df=n - 1$.

Step2: Determine $\alpha/2$

Given $1-\alpha$, we find $\alpha$ first and then $\alpha/2$. For example, if $1-\alpha = 0.95$, then $\alpha=1 - 0.95=0.05$ and $\alpha/2=0.025$.

Step3: Look up in t - distribution table

Use the degrees of freedom and $\alpha/2$ value to look up the $t_{\alpha/2}$ value in the t - distribution table.

a.

  1. Calculate degrees of freedom: $df = n-1=59 - 1=58$.
    • Since $1-\alpha = 0.95$, then $\alpha=0.05$ and $\alpha/2 = 0.025$.
    • Looking up in the t - distribution table (using approximation for $df = 58$ as the table may not have exact value for 58, we can use the value for $df = 60$), $t_{\alpha/2}=2.0003$.

b.

  1. Calculate degrees of freedom: $df=n - 1=59 - 1=58$.
    • Since $1-\alpha=0.90$, then $\alpha = 0.10$ and $\alpha/2=0.05$.
    • Using approximation for $df = 58$ (using $df = 60$ in the table), $t_{\alpha/2}=1.6706$.

c.

  1. Calculate degrees of freedom: $df=n - 1=42 - 1=41$.
    • Since $1-\alpha = 0.95$, then $\alpha=0.05$ and $\alpha/2=0.025$.
    • Using approximation for $df = 41$ (using $df = 40$ in the table), $t_{\alpha/2}=2.0211$.

d.

  1. Calculate degrees of freedom: $df=n - 1=8 - 1=7$.
    • Since $1-\alpha = 0.95$, then $\alpha=0.05$ and $\alpha/2=0.025$.
    • Looking up in the t - distribution table, $t_{\alpha/2}=2.3646$.

e.

  1. Calculate degrees of freedom: $df=n - 1=49 - 1=48$.
    • Since $1-\alpha = 0.99$, then $\alpha=0.01$ and $\alpha/2=0.005$.
    • Using approximation for $df = 48$ (using $df = 50$ in the table), $t_{\alpha/2}=2.6778$.

Answer:

a. $2.0003$ b. $1.6706$ c. $2.0211$ d. $2.3646$ e. $2.6778$