Question

DETERMINE WHETHER THE FOLLOWING STATEMENTS HAVE A KNOWN OR UNKNOWN POPULATION VARIANCE. IDENTIFY ALSO THE FORMULA TO BE USED TO ESTIMATE THE STANDARD ERROR OF THE MEAN.

An SHS teacher claims that the average time it takes a group of students to complete the Statistics and Probability examination is 50.5 minutes with a variance of 17.64 minutes squared. She randomly selected 45 students and found them to have a mean of 52 minutes and a standard deviation of 3.5 minutes. She then used the z-distribution to find out if the group can complete the exam faster than the population.

Answer 1

We are given that,

$\mu=50.5,\space \sigma=17.64$

$n=45,\space \bar{x}=52,\space s=3.5$

Here, the population variance is known since we are provided with the population standard deviation.

The population variance is, $\sigma^2=(\sigma)^2=17.64^2=311.1696$.

Since the population variance is known and the sample size is greater than 30, we apply the formula below to determine the standard error for the mean.

$S E=Z_{\alpha }*(\sigma/\sqrt{n})$ where $Z_{\alpha}$ is the standard normal table value at $\alpha$ level of significance, $n$ is the sample size and $\sigma$ is the population standard deviation.

Therefore, the formula to estimate the standard error of the mean is $Z_{\alpha}(\sigma/\sqrt{n})$