Q. 57 - The Practice of Statistics for AP

Question

57. Critical values What critical value t* from Table B would you use for a confidence interval for the population mean in each of the following situations?
(a) A $95\%$ confidence interval based on  $n=10$ observations.
(b) A $99\%$ confidence interval from an SRS of $20$ observations.

Step-by-Step Solution

Answer

(a) The critical value for 95% confidence interval based on n=10 observations is $t^{*}=2.262$.

(b)The critical value for 99% confidence interval based an SRS of 20 observations is $t^{*}=2.861$.

Part (a) Step 1: Given information

When the confidence level is $95\%$ and the population mean is $10$, the critical value $t*$ is calculated.

Part (a) Step 2: Explanation

Utilize the formula $df=n-1$ to estimate the degree of freedom. Where, the population mean is $n=10$.
$df=n-1$
$=10-1=9$
The row in table $B$is represented by the degree of freedom $df$.

Convert the confidence level  $95\%$ into decimal.
$\frac{95}{100}=0.95$
Determine the column:
$$\begin{gathered} \frac{1-c}{2}=\frac{1-0.95}{2} \\ =0.025 \end{gathered}$$
Using the table$B$, to determine the critical value $t*,$ for row $9$ and the column $0.025$:
$t^{*}=2.262$

Therefore, the critical value is $t^{*}=2.262$.

Part (b) Step 1: Given information

When the confidence level is $99\%$ and the population mean is $20$, the critical value $t$  is calculated.

Part (b) Step 2: Explanation

Using the formula $df=n-1$, to determine the degree of freedom. Where, the population mean is $n=20$.
$df=n-1=20-1=19$.

Convert the confidence level $99\%$ into decimal as:

$\frac{99}{100}=0.99$

Determine the column by:

$$\begin{gathered} \frac{1-c}{2}=\frac{1-0.99}{2} \\ =0.005 \end{gathered}$$

Using the table B, determine the critical value $t^{*}$, for row $19$ and the column $0.005$:

$t^{*}=2.861$

Therefore, the critical value is $t^{*}=2.861$.