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Resources: Course Assignments
Module 10 Assignment: Connection between Confidence Intervals and Sampling Distributions
The purpose of this activity is to help give you a better understanding of the underlying reasoning behind the interpretation of confidence intervals. In particular, you will gain a deeper understanding of why we say that we are “95% confident that the population mean is covered by the interval.”
Click here to open this simulation in a separate tab.
When the simulation loads you will see a normal-shaped distribution, which represents the sampling distribution of the mean (x-bar) for random samples of a particular fixed sample size, from a population with a fixed standard deviation of σ.
The green line marks the value of the population mean, μ.
To begin the simulation, click the very top “sample” button at the topmost right of the simulation. You will see a line segment appear underneath the distribution; you should see that the line segment has a tiny red dot in the middle.
You have used the simulation to select a single sample from the population; the simulation has automatically computed the mean (x-bar) of your sample; your x-bar value is represented by the little red dot in the middle of the line segment. The line segment represents a confidence interval. Notice that, by default, the simulation used a 95% confidence level.
Question 1:
Did your 95% confidence interval contain (or “cover”) the population mean μ (the green line)?
If your confidence interval did cover the population mean μ, then the simulation will have recorded 1 “hit” on the right side of the simulation.
Now, click to select another single sample.
Question 2:
Was your second sample mean x-bar (the new red dot) the same value as your 1st sample mean? (i.e., is it in the same relative location along the axis?) Why is this result to be expected?
Question 3:
A new 95% confidence interval has also been constructed (the new line segment, centered at the location of your second x-bar). Does the new interval cover the population mean μ ?
Notice, under “total” on the right side of the simulation, the number of total selected samples has been tallied.
Now click “ sample 50 ” repeatedly until the simulation tallies a “total” of around 1,000 samples. You will see that the simulation computes the “percent hit” for all the intervals.
Question 4:
What percentage of the many 95% confidence intervals should cover the population mean μ ?
Question 5:
Now let’s summarize some key ideas.
Based on what you’ve seen on the simulation (with the level set at 95%), decide which of the following statements are true and which are false.
1. Each interval is centered at the population mean (μ). 2. Each interval is centered at the sample mean (x-bar). 3. The population mean (μ) changes when different samples are selected. 4. The sample mean (x-bar) changes when different samples are selected. 5. In the long run, 95% of the intervals will contain (or “cover”) the sample mean (x-bar). 6. In the long run, 95% of the intervals will contain (or “cover”) the population mean (μ).
Concepts in Statistics Copyright © 2023 by CUNY School of Professional Studies is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.
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STEP Support - Assignment 10
Introduction
This STEP support module derives some of the compound trigonometry formula ($\sin(\alpha + \beta)$ etc.) and introduces a new function. Previous assignments can be found here , but you can do this one without having done the others first.
STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").
About this assignment
The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.
The warm up for this assignment derives the trigonometrical identities for $\sin(\alpha \pm \beta)$ and $\cos(\alpha \pm \beta)$.
The main STEP question (2005 STEP 2 Question 2) introduces a function which will probably be unfamiliar to you. Don't be put off by this; the question starts by asking you to evaluate the function for some specific values so that you get a feel for what it does.
The final question consists of 4 questions taken from examination papers published more than 150 years ago. Needless to say, calculators should not be used!
For more on logical thinking and "if and only if" see Advanced Problem Solving Module 2 . Some more information on Euler's Totient function can be found on Wikipedia and Wolfram Mathworld (as well as many other places!).
You can explore how modular arithmetic and Euler's Totient function can be used to exchange information securely with this Nrich feature .
Can you use this diagram to derive the compound angle formulae for sine and cosine?
Hints, support and self evaluation
The “Hints and partial solutions for Assignment 10” file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.
Here is a Worked Video Solution to the STEP question from this assignment.
Useful Links
Underground Mathematics: Selected worked STEP questions
STEP Question database
University of Cambridge Mathematics Faculty: What do we look for?
University of Cambridge Mathematics Faculty: Information about STEP
University of Cambridge Admissions Office: Undergraduate course information for Mathematics
Stephen Siklos' "Advanced Problems in Mathematics" book (external link)
MEI: Worked solutions to STEP questions (external link)
OCR: Exam board information about STEP (external link)
AMSP (Advanced Maths Support programme): Support for University Admission Tests (external link)
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Resources: Discussions and Assignments
Module 10 assignment: aging journal.
STEP 1 : Consider the following questions about aging.
- Think of your current lifestyle. What health habits do you need to acquire to live longer and healthier? What can you do to help yourself change these poor habits? Do you have any family history of diseases or early death that would encourage you to do so?
- Think about your parent(s)— their health, children, parents, friends, how they met, marriage, financial problems, disappointments, and vivid memories of good times that happened (ask them about these things if you can). Then consider what you could do differently or the same when you get to be the same age of your parent(s).
- Think about your grandparent(s) and find out what lifestyle they have (ask them about if you can). Do they have enough money to meet their needs? How is their health? What are their relationships like?
- How do you imagine you will handle your successful aging after age 65? Will you gradually withdraw from the world to enjoy peace and solitude do the things you want to do, like read or relax, according to the disengagement theory? Or do you imagine you will want to continue an active life full of socializing, volunteer work, being with family, going on trips, etc., according to the activity theory? Tie your speculations into how you view your personality today.
- Try to paint a picture of your life at 65 or older. What kind of life do you see for yourself after retirement?
STEP 2 : Submit your assignment as either detailed responses to each question, OR as a reflection essay (between 400-600 words) that describes your thoughts on aging.
| Complete response | Writes in clear, descriptive sentences with no or few grammatical errors. Answers responses correctly and appropriately. | Does not provide enough detail in the responses, or responds incorrectly. | Incomplete responses or difficult to understand or follow. | __/5 |
| Personal reflection | Shows personal reflection and includes examples from own experiences. | Demonstrates some personal reflection but is not thorough. | Does not include personal reflection or demonstrate understanding of the material. | __/5 |
| __/10 | ||||
- Assignment: Aging Journal. Authored by : Nancee Ott. License : CC BY: Attribution
- Modification, adaptation, and original content. Provided by : Lumen Learning. License : CC BY: Attribution
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IMAGES
COMMENTS
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Module 10 Assignment: Connection between Confidence Intervals and Sampling Distributions The purpose of this activity is to help give you a better understanding of the underlying reasoning behind the interpretation of confidence intervals.In particular, you will gain a deeper understanding of why we say that we are "95% confident that the population mean is covered by the interval."
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