Answer to Question #141764 in Statistics and Probability for Nana Yaw

Question

Researchers are concerned about the impact of students working while they are enrolled in classes, and they’d like to know if students work too much and therefore are spending less time on their classes than they should be. First, the researchers need to find out, on average, how many hours a week students are working.
EV(6 marks)
3
They know from previous studies that the standard deviation of this variable is about 5 hours.
(i) A survey of 200 students provides a sample mean of 7.10 hours worked. 95% confidence interval based on this sample is 6.41 to 7.79.
Do you agreed with the researcher that the 95% confident interval of the unknown population mean of the students worked hours lies between 6.41 and 7.79?
Indicate whether the researchers are right or wrong, giving reasons for your answer.

Answer 1

We need to construct the 95% confidence interval for the population mean $\mu.$

The following information is provided:

Sample Mean $\bar{x}=7.1$

Population Standard Deviation $\sigma=5$

Sample Size $n=200$

The critical value for $\alpha=0.05$ is $z_c=z_{1-\alpha/2}=1.96.$  

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{x}-\dfrac{z_c\times\sigma}{\sqrt{n}},\bar{x}+\dfrac{z_c\times\sigma}{\sqrt{n}})=$$

$$=(7.1-\dfrac{1.96\times5}{\sqrt{200}},7.1-\dfrac{1.96\times5}{\sqrt{200}})=$$

$$=(6.407,7.793)\approx(6.41,7.79)$$

The researchers are right.