Question
upstudy study bank question image url

Listed below are student evaluation ratings of courses, where a rating of 5 is for "excellent" The ratings were obtained at one university in a state. Construct a confidence interval using What does the confidence interval tell about the population of all college students in the state? \( 3.6,30,4.0,4.6,3.1,4.5,3.4,4.4,4.6,4.0,4.6,3.7,3.0,3.7,40 \) ? What is the confidence interval for the population mean \( \mu \) ? \( \square<\mu<\square \) (Round to two decimal places as needed)

Ask by Ray Newton. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The confidence interval for the population mean \( \mu \) is \( 3.56 < \mu < 4.20 \).

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 construct a confidence interval for the population mean \( \mu \) based on the given student evaluation ratings, we first need to calculate the mean and standard deviation of the sample data. The ratings provided are: \( 3.6, 30, 4.0, 4.6, 3.1, 4.5, 3.4, 4.4, 4.6, 4.0, 4.6, 3.7, 3.0, 3.7, 40 \). However, it seems like the values "30" and "40" are outliers in this context—ratings should generally be between 1 and 5. So let’s focus on the plausible ratings first: 1. Remove the outliers: Ratings used: \( 3.6, 4.0, 4.6, 3.1, 4.5, 3.4, 4.4, 4.0, 4.6, 3.7, 3.0, 3.7 \). 2. Calculate the sample mean (\(\bar{x}\)): \[ \bar{x} = \frac{(3.6 + 4.0 + 4.6 + 3.1 + 4.5 + 3.4 + 4.4 + 4.0 + 4.6 + 3.7 + 3.0 + 3.7)}{12} = \frac{50.6}{12} ≈ 4.22 \] 3. Calculate the sample standard deviation (s): \[ s ≈ 0.40 \quad (\text{using the appropriate formula for standard deviation}) \] 4. Determine the sample size (n = 12) and the critical value for a 95% confidence interval, which for a two-tailed t-distribution with \( n-1 \) degrees of freedom (11 df) is approximately 2.201. 5. Calculate the standard error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{0.40}{\sqrt{12}} ≈ 0.115 \] 6. Construct the confidence interval: \[ \text{Margin of Error (ME)} = t \times SE = 2.201 \times 0.115 ≈ 0.253 \] \[ \text{Confidence Interval} = \bar{x} \pm ME = 4.22 \pm 0.253 \] This results in: \[ 3.97 < \mu < 4.47 \] Thus, the confidence interval for the population mean \( \mu \) is: \[ 3.97 < \mu < 4.47 \] (Rounded to two decimal places as needed.)

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