Question

select the correct answer from each drop - down menu. marcus gathered data on the average time it takes for students to get to school each morning. of all the responses, 10 people said 15 minutes, 18 people said 30 minutes, and 12 people said 25 minutes. if the standard deviation of the population is 5.64 minutes, what is the 95% confidence interval for the population mean?

Explanation:

Step1: Calculate sample mean

First, find the sum of all data - points and divide by the number of data - points. The data is \(10\times15 + 18\times25+12\times30\). The sum is \(10\times15+18\times25 + 12\times30=150 + 450+360 = 960\). The total number of people \(n=10 + 18+12=40\). The sample mean \(\bar{x}=\frac{960}{40}=24\) minutes.

Step2: Identify z - value for 95% confidence interval

For a 95% confidence interval, the z - value \(z = 1.96\) (from standard normal distribution table).

Step3: Calculate margin of error

The formula for the margin of error \(E=z\times\frac{\sigma}{\sqrt{n}}\), where \(\sigma = 5.64\) (population standard deviation) and \(n = 40\). So \(E=1.96\times\frac{5.64}{\sqrt{40}}\approx1.96\times\frac{5.64}{6.325}\approx1.96\times0.892\approx1.75\).

Step4: Calculate confidence interval

The confidence interval is \(\bar{x}\pm E\). Lower limit \(=24 - 1.75 = 22.25\) and upper limit \(=24+1.75 = 25.75\). In the form \(\bar{x}\pm E\), it is \(24\pm1.75\).

Answer:

\(24\pm1.75\)