Question
upstudy study bank question image url

Suppose a random sample of size 50 is selected from a population with \( \sigma=8 \). Find the value of the standard error of the mean in each of the following cases. (Use the finite population correction factor if appropriate. Round your answers to two decimal places.) (a) The population size is infinite. (b) The population size is \( N=50,000 \). (c) The population size is \( N=5,000 \). (d) The population size is \( N=500 \).

Ask by Hanson Little. in the United States
Mar 16,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The standard error of the mean is approximately 1.13 for parts (a), (b), and (c), and 1.07 for part (d).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To calculate the standard error of the mean (SEM), we can use the formula: \[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \text{ for an infinite population}, \] and for a finite population, we use: \[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \cdot \sqrt{\frac{N - n}{N - 1}}, \] where: - \(\sigma\) is the population standard deviation, - \(n\) is the sample size, - \(N\) is the population size. (a) For an infinite population size, \(N\) is effectively infinite. Therefore: \[ \text{SEM} = \frac{8}{\sqrt{50}} \approx \frac{8}{7.071} \approx 1.13 \text{ (rounded to 2 decimal places).} \] (b) For population size \(N = 50,000\): \[ \text{SEM} = \frac{8}{\sqrt{50}} \cdot \sqrt{\frac{50000 - 50}{50000 - 1}} = \frac{8}{7.071} \cdot \sqrt{\frac{49950}{49999}} \approx 1.13 \cdot 0.999 = 1.13. \] (c) For population size \(N = 5,000\): \[ \text{SEM} = \frac{8}{\sqrt{50}} \cdot \sqrt{\frac{5000 - 50}{5000 - 1}} = \frac{8}{7.071} \cdot \sqrt{\frac{4950}{4999}} \approx 1.13 \cdot 0.999 = 1.13. \] (d) For population size \(N = 500\): \[ \text{SEM} = \frac{8}{\sqrt{50}} \cdot \sqrt{\frac{500 - 50}{500 - 1}} = \frac{8}{7.071} \cdot \sqrt{\frac{450}{499}} \approx 1.13 \cdot 0.9487 \approx 1.07. \] To summarize: (a) SEM = 1.13 (b) SEM = 1.13 (c) SEM = 1.13 (d) SEM ≈ 1.07

Related Questions

Latest Statistics Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy