Find the margin of error.
-The principal randomly selected six students to take an aptitude test. Their scores were: $\begin{array} { l l l l l l } 77.9 & 89.1 & 80.7 & 78.6 & 74.4 & 82.0 \end{array}$
Determine a 90 percent confidence interval for the mean score for all students.
A) $84.54

Question

Find the margin of error.
-The principal randomly selected six students to take an aptitude test. Their scores were: $\begin{array} { l l l l l l } 77.9 & 89.1 & 80.7 & 78.6 & 74.4 & 82.0 \end{array}$
Determine a 90 percent confidence interval for the mean score for all students.
A) $84.54 < \mu < 76.36$
B) $84.64 < \mu < 76.26$
C) $76.36 < \mu < 84.54$
D) $76.26 < \mu < 84.64$

Quizplus · Accepted Answer

The margin of error (E) is calculated using the formula $E = t_{\alpha/2} \times \frac{s}{\sqrt{n}}$, where $t_{\alpha/2}$ is the t-score corresponding to the desired confidence level (in this case, 90%), $s$ is the sample standard deviation, and $n$ is the sample size. First, calculate the mean ($\bar{x}$) and standard deviation (s) of the sample scores. The mean score is $\frac{77.9 + 89.1 + 80.7 + 78.6 + 74.4 + 82.0}{6} = 80.45$. The sample standard deviation (s) can be calculated using the formula $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$, which gives $s \approx 5.187$. For a 90% confidence interval and a sample size of 6, the degrees of freedom (df) is $n-1 = 5$. The t-score for a 90% confidence level and df = 5 can be found in a t-distribution table or using statistical software, which is approximately 2.015. Plugging these values into the margin of error formula gives $E \approx 2.015 \times \frac{5.187}{\sqrt{6}} \approx 4.09$. The confidence interval is then $\bar{x} \pm E = 80.45 \pm 4.09$, which gives the interval $76.36 < \mu < 84.54$.

Find the Margin of Error 77.989.180.778.674.482.0\begin{array} { l l l l l l } 77.9 & 89.1 & 80.7 & 78.6 & 74.4 & 82.0 \end{array}77.9​89.1​80.7​78.6​74.4​82.0​

Find the Margin of Error $\begin{array} { l l l l l l } 77.9 & 89.1 & 80.7 & 78.6 & 74.4 & 82.0 \end{array}$