Question

if n = 17, $\bar{x}$(x - bar)=35, and s = 4, construct a confidence interval at a 99% confidence level. assume the data came from a normally distributed population. give your answers to one decimal place. < $mu$ <

Explanation:

Step1: Determine degrees of freedom

The degrees of freedom $df=n - 1$. Given $n = 17$, so $df=17 - 1=16$.

Step2: Find the t - value

For a 99% confidence level with $df = 16$, from the t - distribution table, the two - tailed t - value $t_{\alpha/2}=2.921$.

Step3: Calculate the margin of error

The formula for the margin of error $E=t_{\alpha/2}\frac{s}{\sqrt{n}}$. Substitute $t_{\alpha/2}=2.921$, $s = 4$, and $n = 17$ into the formula. $\sqrt{n}=\sqrt{17}\approx4.123$, $\frac{s}{\sqrt{n}}=\frac{4}{4.123}\approx0.97$, $E = 2.921\times0.97\approx2.83$.

Step4: Construct the confidence interval

The confidence interval for the population mean $\mu$ is given by $\bar{x}-E<\mu<\bar{x} + E$. Given $\bar{x}=35$, so $35-2.83<\mu<35 + 2.83$, which is $32.2<\mu<37.8$.

Answer:

$32.2<\mu<37.8$