Question

You want to compute a $90\%$confidence interval for the mean of a population with an unknown population standard deviation. The sample size is $30$. What critical value should you use for this interval?

a. $1.645$

b. $1.699$

c. $1.697$

d. $1.96$

e. $2.045$

Short answer

The critical value that should be used in this interval is$1.699$

The required correct option is $\left(b\right)$

Step-by-step solution

Given information

The standard deviation of the population is unknown.

Sample size $\left(n\right)=30$

Confidence level $=90\%$

The objective is to find out the critical value that should be used in this interval

The level of significance is:

Level of significance $\left(\alpha \right)=1-$Confidence level

$$\begin{gathered} =1-0.90 \\ =0.10 \end{gathered}$$

The t-critical value must be calculated here.

The degree of freedom is calculated as follows:

localid="1654412235779" $$\begin{gathered} (df)=n-1 \\ =30-1 \\ =29 \end{gathered}$$

The critical value at $29$degrees of freedom and a $10\%$significance level is calculated as follows:

$$\begin{gathered} t_{\frac{\alpha }{2},df}=t_{\frac{0.10}{2},29} \\ =1.699 \end{gathered}$$

Therefore, the correct option is$\left(b\right)$