<< Chapter < Page Chapter >> Page >

Two Column Step by Step Example of a Sampling Distribution for a Mean

A study involving stress is done on a college campus among the students. The stress scores follow a uniform distribution with the lowest stress score equal to 1 and the highest equal to 5.

Using a random sample of 50 students, find the probability that the mean stress score for the 50 students is less than 2.65.

Guidelines Example
Plan: State what we need to know. We are asked the probability that the mean stress score of the sample of 50 students is less than 2.65.
Model: Think about the assumptions and check the conditions.

Randomization Condition: The problem states that the sample was random.

Independence Assumption: It is reasonable to think that the stress scores of 50 randomly selected students will be independent of each other. (But there could be exceptions, such as , if they were all from the same class that was having a major exam the next day.)

10% Condition: I assume the student body is over 500 students, so 50 students is less than 10% of the population.

Sample Size Condition: Since the distribution of the stress levels is uniform, a larger sample size is needed. I believe that my sample of 50 students seems large enough.

State the parameters and the sampling model. Since the individual stress scores follow a uniform distribution, X ~ U(1,5) where a=1 and b=5 (See Continuous Random Variables for the uniform).
µ X = a + b 2 = 1 + 5 2 = 3
σ X = ( b - a ) 2 12 = ( 5 - 1 ) 2 12 = 1.15
Plot: Make a picture. Sketch the model and shade the area we’re interested in.

Mechanics: Let x = the mean stress score for the 75 students. Then,
X ~ N ( 3 , 1.15 50 )
  • where n = 50 .
Find the resulting probability from a table of Normal probabilities, a computer program or a calculator.
A z-score will be used:
z = x - μ σ = 2.65 - 3 0.1626 = -2.15

Look this up in the table or use a computer. P(z<-2.15) = 0

Conclusion: Interpret your result in the proper context, and relate it to the original question. The probability that the mean stress score for the 50 college students is less than 2.65 is about 0.

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics using spreadsheets' conversation and receive update notifications?

Ask