MATH 1040 - Introduction to Statistics

Practice

A professor randomly surveys 41 students about how many hours they studied for the last exam. The sample mean is 14.2 hours, and the sample standard deviation is 4.6 hours. Construct a 95% confidence interval for the true mean number of hours students studied for the exam.

Solution

From the problem we get the following information:

• A confidence level of 95%
• Sample Size is \(n=41\)
• The sample mean is \(\bar{x} = 14.2\)
• The sample standard deviation is \(s=4.6\)

First, verify the central limit theorem

• Is the sample random? Yes (states in the problem)
• Is the sample large enough? Yes (sample size is at least 30)

The Central Limit Theorem holds.

Now, we find the critical value. Since we only have a sample standard deviation, we need to use the t-score with \(DF = n-1 = 40\) degrees of freedom. Doing this, we get \(t_c = 2.021\).

\[E = t_c\frac{s}{\sqrt{n}} = 2.021\frac{4.6}{\sqrt{41}} = 1.45\] \[\bar{x} + E = 14.2 + 1.45 = 15.65\] \[\bar{x} - E = 14.2 - 1.45 = 12.75\]

Solution:

We are 95% confident that the true mean for students’ study time is between 12.75 and 15.65 hours.

Return back to Lesson 18.5