General Resources

• Welcome to the course video
• Course Info (show/hide)

  Welcome students,

  It's my pleasure to introduce myself. My name is Brian Powers, and I'll be your online instructor for this course in Applied Statistics. I've taught statistics in Chicago at University of Illinois at Chicago as well as privately. In 2016 I moved to Scottsdale, Arizona. This is my Nth term teaching MAT240 (I've lost count - how embarrassing!), I'm very excited to be have a new group of students.

  This statistics course will expose you to some of the most important topics and techniques of statistics. We'll be looking at probability distributions and the Central Limit Theorem. These will provide a basis for the estimation and decision making techniques that follow: construction of confidence intervals, and hypothesis testing to name a few. We'll also look at correlation and linear regression. A few things I want to bring to your attention right away:

  1. Please be sure to read the course syllabus.

  It's in the Course Information section on the left. In particular you should all be aware of how your grade for the course is calculated:

  • Weekly work in zyBooks (participation and challenge activities)
  • Discussion Board written assignments (weeks 1, 6 and 8)
  • Assignments (weeks 2, 3, and 5)
  • Projects(weeks 4 and 7)

  2. Have Access to Excel

  In this course we'll be using Excel to do our analysis. It's important that you make sure you have access to Excel (Office 365 is available for free from SNHU). This is a big change from previous terms. I will be sharing resources every week on how to use statistical functions and tools in Excel to help you do your weekly work.

  3. Register your account in zyBooks

  The textbook resource we'll be using is zyBooks - please be sure to carefully read the announcement about registering for this course section in zyBooks. Please note: You must TYPE IN our course section exactly when registering. Please be careful on this step, it will save us all some headaches!

  4. Ask Questions!!

  Please ask questions if you're not sure about something. The General Questions discussion board is a great place to put your questions. I am subscribed to it, and if I don't answer your question right away another student may answer before me! Also other students will benefit from the answers you get. You can also email me - My contact info is under the My Instructor announcement.

  I hope you all get a lot out of this course in Statistics. I will do whatever I can to help the course material be as relevant to you as possible, and to help you get a better understanding of these fundamental concepts in gathering, analyzing and interpreting data, and using it to make sense of the world.

  Dr. Powers

• Student Advice for Success (show/hide)
  Hello class. I surveyed students a few terms ago about how to succeed in this course, and I thought I'd share their responses with you:
  What helped you do well in this course?
  

  • Dr.Powers video’s. Hands down this was the greatest asset in facilitating any of success in this course.
  • To be honest I really struggled with this course, but I am a visual learner and I feel like I would have had a better understanding of the material had I taken it in person. However, your videos that you posted did help a lot.
  • The weekly homework assignments and posted videos that provided further instruction.
  • Watching the videos provided by the teacher
  • The responsiveness from the professor with regards to questions or concerns. The helpful tips in announcements and most of all the videos that guided us through the modules.
  • The video's
  • The instructional videos were absolutely the gateway to this course.
  • Dr. Powers each week provided video examples of how to work through the assignment for that week. That was extremely helpful to have it available at all times to go back on if we needed.

  What advice would you give an incoming student with math/statistics anxiety?
  

  • Do not overthink it. Try to not think about the formulas themselves initially but how they relate to real life scenarios. It will allow for the formulas to THEN make sense.
  • Take your time and reach out if there is something you do not understand.
  • Write down each equation and spend a few hours each week mastering them so you can feel more comfortable.
  • Ask questions!
  • Although this is a statistics class and seems like math is scary, this class is more about understanding the concepts. Most of the math can be done in excel and that's a relief.
  • Just take your time and do the workbook assignments first.
  • Do not be intimidated by the material and terminology, the instructor does a great job of making it comprehendible and relatable.
  • I was the student with math/statistics anxiety and my best advice is to use google and watch videos. There are many videos provided by Dr. Powers and by others on Youtube that give detailed explanations of how to work through assignments. And always ask for help. I never felt intimidated to ask Dr. Powers a question.
  • I suggest to take any and all statistics, finance, math etc classes consecutively. try not to get too behind because it becomes more stressful in the end trying to catch up. and Definitely read through the problems and work through them.

  What would you have done differently from the beginning, knowing what you know now?
  

  • Taken my advice from above.
  • I would have taken the time to use my resources for help in understanding the material.
  • To be honest not really. I was pretty happy with how well the course was taught and the knowledge I retained, primarily concerning excel and more advanced statistics.
  • Dedicated more time to the material
  • I am glad I set aside time everyday to focus only on this class. I took my time with the modules in the Zybook and that allowed me to really concentrate on this class and do my best.
  • Do the workbook assignments first.
  • not have worried
  • Because of my statistics anxiety, I started the work at the beginning of the week and sometimes i would finish in a day or two, sometimes it took me all week. Only thing i would say is to prioritize the work especially if you're taking more than 1 class. The work isnt too difficult but it is extremely stressful starting on the assignment in the middle of the week.
  • I would have kept up with the assignments and turn them in on time.

  How would you suggest ordering your work for the week?
  

  • Read through all the announcements pertaining to that module. Glance over the zybooks to get familiarized with what you’ll be learning. Then go and watch Dr.Powers videos. At that point I suggest re-reading zybooks, and completing the participation and challenge activities.
  • The ordering was great! You should do your reading of the course work first and do the participation and challenge activities while your reading so that you do not get confused going back to the problems. Then complete any extra assignments. I would probably save discussion posts for last so that the information for the assignments and projects are still fresh in your head.
  • Began each week with a check list of assignments that are due and slowly progress and edit your work until you’re finished.
  • Read announcements, watch videos, Zybooks material, assignments
  • This is a more advanced class, I would recommend working on this one first as it can take up more time than planned.
  • Workbook assignments first then go to the discussion post and assignments.
  • I would suggest ordering the work the way each module suggests.
  • Didn't realize this was a question but i've answered above. Definitely starting the work at the beginning of the week. Extremely less stressful getting statistics homework finished in the middle of the week vs 10PM on Sunday when everything was due.
  • I would read announcements then go into the module zybooks text and then complete the projects due.

Module 1: Descriptive Statistics

• Module Info(show/hide)

  Module 1 is concerned with descriptive statistics, data visualizations and summaries. A statistic is any measurement or calculation resulting from some data. This definition is quite vague, of course. There are some useful statistics and some not-so-useful (depending on the data and context). Descriptive statistics describe the sample data, while inferential statistics attempt to infer something about the larger population. When we say "population" in this context, we do not mean the population of human inhabitants of the U.S. (for example). In statistics, "population" refers to the realm of possibilities from which we sample data (which, of course COULD be the U.S. population).

  Descriptive statistics of "central tendency" (mean, median and mode) describe the "average", "middle" or "most common observation" in our data. Statistics of "spread" or "dispersion" (range, variance, standard deviation) describe how much variability there is in our data.  Visualizations are charts and graphs that are useful visual summaries of the data. Histograms and box plots will be useful for visualizing numerical (quantitative data) while bar plots (and once in a rare while pie charts) will be useful for categorical (qualitative) data.

• Discussion Board Assignment(show/hide)
  Hi everybody. I thought I'd just write this up as a little clarification of the discussion assignment.

  The principal wants to get a sense of the general preference of the total population of the students at the high school. She'll sample the student population and base the decision on this.

  Grade Level	Number of classrooms	Total Students	Males	Females
  9th Grade	10	200	100	100
  10th Grade	12	250	150	100
  11th Grade	13	250	100	150
  12th Grade	15	300	150	150
  Totals	50	1000	500	500

  As you can see from the numbers, there are 1000 total students at the school, half male and half female. There are slightly more in the 12th grade and slightly fewer in the 9th grade.

  Suppose the sample will consist of opinions from 80 students from the school (you can, of course, use a different number in your post). For the sample to be 'representative' of the population, it should be like a mini version of the population. So you can infer the total student population opinion by just scaling up your sample data. If the sample is not representative of the population, then it has a bias.  

  The idea of a 'simple random sample' is that all students in the school have an equal likelihood of being chosen for the sample. One way to do that is to put all 1000 student names in a hat and pull out 80 names and ask them their opinion.

  A stratified sample attempts to model the population by controlling the sample. For example, the principal could ensure that the sample consists of

  	male	female
  9th grade	8	8
  10th	12	8
  11th	8	12
  12th	12	12

  This will match the population demographics. The drawback - it's a little more difficult to carry out such a sample (but probably not TOO difficult). Is it worth it? 

  A cluster sample is done by by sampling everybody in a few locations. For example, because each classroom has roughly 20 students in it (you can gather than by the numbers in the table: 10 classrooms house 200 9th graders, for example) She could just pick 4 random classrooms - one for 9th, 10th, 11th and 12th grade and just survey everybody in those 4 classrooms.

  A systematic sample is like a simple random sample, but people aren't chosen randomly exactly. She could just sort the 1000 student names alphabetically and then pick every 12th or 13th student from the list. She'd get a sample of about 80 students doing it that way.

  A convenience sample would be what's easiest - She could just go to the cafeteria at 12:30 and ask the first 80 students she sees. The problem of course is that this sample probably won't represent the population well. 

  She could post a survey and ask everybody to write their response at their leisure - one problem is that not everybody will respond and the results will be biased towards those people who did respond. Maybe she wants to focus the survey on the people in the school who would be more likely to attend the movie in the first place, since their opinions arguably count more.

  There are a lot of considerations you could make. You should think about the time and energy to conduct the survey. Can she ensure that she doesn't have bias in her results (respondent bias, for example)? How big of a sample should be made? Should she ask people to give ranked choices or just their top choice? How much time should be allocated to analyzing the results? I'm sure the principal doesn't want this to turn into a giant project.
• Module 1 Challenge Activities

Module 2: Linear Regression

• Module Info(show/hide)

  This week we attempt to estimate how related two THINGS are. When we talk about "related" in this context, we are talking about a linear relationship. A linear relationship means that if X increases by 1 unit, then Y should increase (or decrease) by some amount of units. You may have at some point seen some of these ideas - scatterplots, linear regression, line of best fit. We'll be looking at all of these and of course we will use Excel tools to make the work a bit easier (so we don't have to stress out about math formulas). 

  A linear relationship between variables X and Y should follow a linear equation Y=β₀+β₁X+ε, here β₀ is the intercept (sometimes called the intercept coefficient), β₁ is the slope coefficient, and ε represents some random "noise". We are not interested in estimating the noise, but what we are interested in estimating is the intercept and the slope. So We're going to be interested in estimating the expected value of Y (E(Y), the average value of Y) for a given X value. E(Y)=β₀+β₁X

  Notice that the epsilon (the noise) is gone - because its average value is zero.

  The linear relationship between two variables is called correlation. The term is used loosely in our vernacular, but for a statistician it is important that we only talk about correlation when there is a linear relationship. Whether two variables are positively correlated or negatively correlated, whether the correlation is strong, weak or nonexistent - these are all important questions that we're interested in. But you should always remember - correlation does not mean causation. Just because a linear relationship exists, it does not mean that the one variable CAUSES the response in the other. There are a number of reasons for this - and determining causation is another challenge altogether.

• Module 2 Assignment(show/hide)

  Hello everybody. This week you have your first assignment due. When you are writing it up, please bear in mind these tips which will help you get the best grade possible:

  • Remove bracketed text before submitting - when you download the Assignment 2 Template you'll see some [bracketed text]. Be sure that all bracketed text is deleted or replaced with your own writing when submitting. You should have 5 sections in your final assignment: Introduction, Representative Data Sample, Data Analysis, Scatterplot, The Pattern
  • Write in paragraphs! I get irritated when students submit a report and instead of paragraphs they have a bulleted list (like this one). 
  • Write as if you're submitting a report to your supervisor who doesn't understand technical jargon. It's ok to use statistical terminology if you clarify exactly what it means to someone who is not a statistician.
  • Read through the rubric so you know exactly what I'll be grading
  • Create a truly random sample! Use the RAND() function in excel! See the resources in my weekly announcement to learn how to get a truly random sample from the data.
  • Be sure the X variable and Y variable are correct. The X variable must be 'median square fee' and the Y variable must be 'median listing price'.
  • You'll probably need to reorder data columns when creating your scatterplot - the X variable column must be to the LEFT of the Y variable column. 
  • Get the linear regression equation in excel for the prediction - don't just eyeball it. When you add the trendline to your graph, check the option to display the regression equation too. This will allow you to give a precise estimate for the 1200 square foot house.

  These should help you do really well!

• Module 2 Challenge Activities
• Module 2 Assignment
• Module 2 More Regression Practice
• Excel: SLOPE and INTERCEPT functions

Module 3: Linear Regression (Continuted)

• Module Info(show/hide)

  This week we'll be looking at the assumptions of linear regression and the conditions you need to look for in the data. We have a few tools we can use to check those assumptions to see if they are violated by the data. One tool is the residual plot. If we take all of the residuals and plot them (as 'y' variables) against the X variables in a scatterplot, we've created a residual plot. To see how we use this residual plot, let's take a quick look at the simple linear regression model:

  Y = β₀+β₁X+ε

  The first assumption, of course, is that the variables Y and X have a linear relationship. You can check this assumption by looking at the shape of the residual plot. Here are two examples:

  

  The residual plot on the left looks fine - this is the kind of shape you'd see when X and Y have a nice linear relationship. The residual plot on the right is indicative of a CURVED relationship between X and Y. When X and Y are related in this way, then simple linear regression is not appropriate.

  Another assumption is that the variation around the regression line doesn't depend on the value of X. Take a look at these three residual plots

  

  The first one looks fine- the spread around the 0 line is about the same all across the residual plot. In the second residual plot the spread is noticeably less in the middle (maybe 1/3 the size) of the spread on the ends. The residual plot on the right also shows changing variation of residuals - the variation decreases as the X values increase. Residual plots 2 and 3 exhibit heteroscedasticity, which means different variation. That's a violation of this assumption, and it means that simple linear regression is not appropriate for this data.

  We'll look at two new statistics used to measure how well the linear model fits the data. The first is the correlation coefficient, we denote r, and it ranges from -1 to 1. The closer to 0 it is, the weaker the linear relationship. As you may guess, this number will also tell you whether the relationship is positive or negative. 

  The second is the coefficient of determination, (aka R squared) and it's denoted R². It is literally r*r, and as such ranges from 0 to 1. This statistic tells you what proportion of the variation of Y is explained by variation in X. So if R2=.423, you can say that 42.3% of the variation of Y is explained by the variation in the X variable.

• Module 3 Assignment(show/hide)

  As you work on Module 3 please keep the following in mind:

  1. For the linear regression equation, be sure to use the values of the slope and intercept that you calculate for your data, not just a generic equation.
  2. Please include all tables and charts in the body of your summary report. No need to include the data sample in your summary report, but if you feel the need to include it, please add it as an appendix at the end.
  3. Please include section headers, but write up each section in paragraph form. Please do not submit a report full of bullet points. This is a report, not a powerpoint slide!
  4. Since this is a report, you should treat it as though you are reporting to your boss who is not a statistician. When you report your statistics you must explain how to interpret them to a person who is not a stats expert. 
  5. Be sure to use Square Footage as the X variable and Listing Price as the Y variable. This is especially important for when you predict the median listing price for a county where the median square footage is such-and-such.
  6. Please DISREGARD the question where you're asked to find the value of the land. You can only use a linear model of housing prices to predict housing prices when the square footage is within the range of the data. There are no houses in the data with a square footage so low as zero, so it is bad practice to assign meaning to the intercept. It only makes sense to interpret the intercept by itself if the X data range includes zero.
  7. Be sure to read the rubric thoroughly so you know exactly what I'm grading. In particular, you are asked to INTERPRET R². R² is not the proportion of dots that fall on the line, be sure to use the correct interpretation of this statistic.

  Please use the General Questions forum if you have questions on this assignment! I will answer your question and everybody else in the class who might have the same question will benefit from you asking!

• Module 3 Challenge Activities
• Module 3 Linear Regression

Module 4: Probability and the Gaussian Distribution

• Module Info(show/hide)

  Here we will develop our understandings of probability. We first need to understand the concept of a "probability distribution". Essentially, if we consider some numerical variable that has an uncertain value (tomorrow's temperature, how many emails I get today, how long it takes to drive to work, how much I will win on a scratch ticket) and we somehow assign probability to each of the possible values, we have just created a probability distribution. If there are only a few possible values (a scratch ticket could be $0, $5, $50 or $1000, or I could get 0,1,2,3,4,... emails) then we are talking about a discrete random variable. If the value could be any number in a range (the temperature, the length of commute time) it is known as a continuous random variable, and has a continuous probability distribution.

  As a rule of thumb: discrete random variables are counted, continuous random variables are measured.

  This week we will be looking at a very important probability distribution: The (so-called) Normal Distribution. This is an important continuous probability distribution for a few reasons. First of all, many natural phenomena have normal (or close to normal) distributions:

  • Height & weight of individuals
  • Standardized test scores
  • Sizes of components manufactured by machines
  • Blood pressure of individuals
  • Errors in measurement

  For instance, take a look at this histogram. You will see the bars form a bell curve shape.

  

  (https://help.plot.ly/excel/histogram/)

  Secondly, the normal distribution is important in statistics because of the Central Limit Theorem, which says essentially that when we try to learn something about the mean of a population by sampling, we can use the normal distribution even when the population is NOT normally distributed. This is part of the magic of random sampling!

  This week you will be working a lot with the normal distribution. In your problem set, two problems will be calculating normal probabilities, and calculating normal percentiles. The questions will be like this:

  • Suppose X is normally distributed...what is the probability that X is greater than 4?
  • Suppose X is normally distributed...what is the 93rd percentile (a certain value such that there is a 93% chance that X is less than this value)?

  Something I'd like you to all know ahead of time: The "Normal distribution" is called normal through an accident of history. It is not "normal" in the sense of ordinary or typical. A normal distribution means bell-shaped and symmetric. You could also call it "Gaussian" or "Bell-Curve" - please don't get confused on this point.

  We'll also look at the T distribution, which looks like the normal distribution but with fatter tails (and is always centered at zero). The T distribution is important for statistical inference, but we'll get into that next week. For now you'll need to get some practice calculating probabilities with these distributions.

• Project One(show/hide)

  Hi everybody. On Sunday your Project One is due. It combines the stuff you've been doing in modules 2 and 3. Here are some tips to make sure you get the best score possible

  • Download and use the Project One Template. Be sure to remove ALL bracketed text before submitting.
  • You should be sure to keep the section titles. The sections are:
    • Introduction
    • Data Collection
    • Data Analysis
    • The Regression Model
    • The Line of Best Fit
    • Conclusion
  • Write up your report in paragraphs and proofread! If I don't understand what you're trying to tell me, I will get frustrated and I won't be able to give you a good grade.
  • Be sure to select 50 counties from the entire US (not just one region) in a RANDOM sample. 
  • Once again, use (median) square feet as the X variable and (median) listing price as the Y variable. You may have to move the columns so that square feet is to the LEFT of listing price.
  • You are asked to give both the correlation (r) and the coefficient of determination (R²) AND to interpret them. Whenever you give summary statistics like this, you should explain what they mean and what they tell you about the data and the research question at hand.
  • You are asked to make a prediction for a listing price - the square footage can be anything you want EXCEPT you should be careful not to extrapolate beyond the range of the data. Do you know what I mean?
  • Read through the rubric so you know exactly what I'll be grading.
  • If you have any questions, post them in the General Questions forum. I'll be answering the questions as soon as I see them.
• Module 4 Normal and T Probabilities in Excel
• Module 4 Challenge Activities
• Module 4 Project One

Module 5: Hypothesis Testing (Mean)

• Module Info(show/hide)

  So far in this course we've covered some of the fundamental ideas about probability and uncertainty, studied the normal distribution and now we have two of the most important applications of this distribution (due to the Central Limit Theorem) - statistical estimation and hypothesis testing. When estimating we spend time on interval estimating - confidence intervals. A confidence interval is a good way of saying "Based on the data, I'm pretty confident that the population mean is between X and Y". The primary objective of testing is answering a questions such as "Everybody says the population mean is X. I think it is greater than X. What conclusion should I reach based on the data?"

  When we wish to make some decision about the world by looking at only a small part of it, we have to make a conclusion knowing that we could be wrong. In statistical inference this is a theme we will never stop seeing - living in harmony with uncertainty. 

  A hypothesis test essentially comes down to these components:

  1. State a null hypothesis (our status quo, or what we ASSUME to be true unless we can prove otherwise)
  2. State the alternative hypotheses (our hunch, where the burden of proof rests)
  3. Collect some data and calculate a test statistic
  4. Decide, based on the sampling distribution of the test statistic, if it sufficiently supports the alternative hypothesis or not (i.e. reject the null hypothesis or don't)

  For example, suppose a marketing company wants to decide if the average price a customer would pay for a product is $14.50 or if the average price they'd pay is less than $14.50.  The null hypothesis is the premise where we specify exactly what the average is: "The average acceptable price is $14.50" (it could even be more, but in this case we simplify the null to be equal) and the alternative is "the average acceptable price is LESS THAN $14.50." Formally, using typical notation, we write

  H₀: μ = 14.50  (some textbooks use μ ≥ 14.50)

  H₁: μ < 14.50

  Suppose they collect good quality survey data from 100 people and of the sample the average price people say they'd pay is $14.37 with a standard deviation of $0.72. We calculate a test statistic, which we call "t" because it (theoretically) has a t distribution sampling distribution.

  

  We use μ₀=14.50 because in the null hypothesis we assume that μ = 14.50.

  Finally we determine if this test statistic would be unusual if the null hypothesis WERE true. We can determine this by either

  a) calculating a rejection region and rejecting the null hypothesis if t is in this region or

  b) calculating a "p-value" for this test statistic and reject the null hypothesis if this "p-value" is smaller than the "significance level" for this test.

  In this case, because the alternative hypothesis is "μ < 14.50 ", we call this an lower-tailed test. The p-value is P(T>t) for a T distribution with n-1=99 degrees of freedom, i.e. 

  P(t<-1.8056) = 0.037 . The p-value is the probability of observing a test statistic like we did IF the null hypothesis is true. We usually specify some significance level like .05 or so. This .037 p-value means the data collected is not plausible if the null hypothesis were true.

  A rejection region for a significance level of .05 would be any value t less than t*, a critical value such that P(T<t*)=.05 exactly. For this distribution (a t distribution with 99 degrees of freedom) this is -1.6604. This picture shows the rejection region as well as the test statistic. You can see that our test statistic is IN the rejection region:

  

  Our conclusion: We reject the null hypothesis. We would end our test with a concluding sentence, something like this: "From the data collected, we have substantial evidence that the true average price that people would pay for this product is less than $14.50."

• Module 5 Assignment(show/hide)

  The assignment this week is worded very strangely. So I want to try to clarify the premise.

  One of the company’s Pacific region salespeople just returned to the office with a newly designed advertisement. It states that the average cost per square foot of his home sales is above the average cost per square foot in the Pacific region. He wants you to make sure he can make that statement before approving the use of the advertisement. The average cost per square foot of his home sales is $275.

  So let's take it as a given that this particular agent's average cost per square foot (CPSF) is $275 exactly. What's unknown to us is the (population) average CPSF for the Pacific region, μ. His claim is that $275 is above the average (i.e. $275 > μ)

  We are used to putting μ on the left side. So this claim can be stated as μ < $275. Because this claim is the inequality, it is the ALTERNATIVE hypothesis. Thus the null hypothesis is that μ = $275. So I'll restate:

  H₀: μ = 275

  H_a: μ < 275

  I hope this helps clarify!

  Other tips:

  • Remember to read the rubric well so you know exactly what I am going to be grading
  • This week we're looking at the cost per square foot variable. We will NOT be looking at listing price or square foot. Be sure you're working with the right variable!!
  • Use the module 5 assignment template, and remove all bracketed text. Be sure to keep the following section titles: Introduction, Setup, Data Analysis Preparation, Calculations, Test Decision, Conclusion
  • You will be using the entire dataset of 1001 house listings in the Pacific region for your assignment. There is no need for using the RAND() function in excel
  • For your hypothesis test you can do it a number of ways - you can use the "Data Analysis" method where you let Excel do all of the work, you can calculate your T statistic and p-value yourself - either one is fine.
  • Be sure to explain in your conclusion whether the sales rep is justified in making the claim that his "Cost per square foot is above average for the Pacific Region" in his advertisement.
• Module 5 Challenge Activities
• Module 5 Assignment

Module 6: Confidence Intervals (Mean)

• Module Info(show/hide)

  Last week you got some practice in statistical inference, doing hypothesis tests. The tasks of statistical inference (inferring something about a population from just a sample) broadly fall into two categories: estimation and decision making. Right now we are concerned with estimation. If you are given the task of estimating what mean number of recharge cycles of laptop batteries is, how would you go about doing this? Certainly taking a large sample of laptop batteries is a good start, and calculating the sample mean is probably a good estimate.

  Likely the mean in your sample is not EXACTLY the same as the population. If it's not exact, then how far off is it likely to be? That's what margins of error and confidence intervals are all about. A confidence interval may be reported something like this:

  A 95% confidence interval for the mean number of recharge cycles of these batteries is 324.2 to 376.7.

  We are saying that there is a 95% chance that the range from 324.2 to 376.7 contains the true population mean. If you'd like a tighter interval you have two options: 1) Take a bigger sample or 2) be more willing to be wrong (i.e. lower the confidence level from 95%). 

  We'll typically use the t distribution for the sampling distribution of x-bar, as we did with hypothesis testing. There are some very specific situations where it's ok to use a standard normal (but honestly, they're completely unrealistic)

• Module 6 Discussion Assignment(show/hide)

  This week you have a discussion post due. You're asked to consider what sample size and resulting margin of error would be best for the situation. Specifically, you're imagining that you are an analyst at a data science company. The B&K real estate company wants to get an estimate for the median listing price for the Northeast region. You have different options for sampling. The discussion prompt gives only two options: n=100 for $2000 or n=1000 for $10000.

  I would like you to consider this expansion. Suppose your company offers three data collection/analysis packages:

  I was thinking to really get students to make considerations about the cost of the sampling, if they offered tiered packages, something like this:

  Bronze Package: $2000 for 100 sample size + $1000 for each additional 100

  Silver Package: $10000 for 1000 sample size + $1000 for each additional 200

  Gold Package: $25000 for 4000 sample size + $1000 for each additional 250

  Suppose the population standard deviation is approximately $125,000. You can calculate the approximate margin of error for any sample size using: ME=1.96 × 125000/sqrt(n). For instance, the margin of error for n=100 would be about $24,500, the margin of error for n=1000 would be about $7,750 (these numbers are more realistic than the ones in the prompt, I'd like you to use these numbers instead).

  You are a consultant trying to explain the pros and cons of a larger sample size. Come up with a recommended sample size for the B&K company and explain why this is the right choice for them.

  THERE IS NOT JUST ONE RIGHT ANSWER. Try to act in an impartial way, even though in this scenario you're acting as a salesperson as well as an analyst.

• Accuracy vs Precision(show/hide)

  Hi everybody. When making your discussion post, I want you to think about the two concepts of accuracy vs precision.

  If I am predicting with 90% accuracy, then there's a 90% chance that I am correct. If I guess a person's age to be between 34 and 37, and I have 90% accuracy, then there's a 90% chance that I'm right. The precision of my estimate is how wide of a margin of error I have. If I guess someone else's age to be between 55 and 65, that's a far less precise estimate, though it may be just as accurate, if there's a 90% chance that it's correct.

  A 95% confidence interval has 95% accuracy regardless of how wide it is - it's going to correctly guess the population parameter 95% of the time; but it's precision depends on the sample size. The larger the sample size, the more precise of an estimate is possible.

  Hope this helps!

  Dr. Powers

• Module 6 Challenge Activities

Module 7: Inference on Population Proportions

• Module Info(show/hide)

  So far we've looked at the two main branches of statistical inference - decision making and estimation - as they relate to the population mean. Now we will turn our attention to a different population parameter, the population proportion. For example, suppose a marketing company wants to decide if more than 65% of the public would purchase a new product. The null hypothesis is "no, only 65% of public would purchase it" (it could even be less, but we simplify the null to be equal) and the alternative is "yes, more than 65% would purchase it." Formally, using typical notation, we write

  H₀: p = .65
  H₁: p > .65
  

  Suppose they collect good quality survey data from 100 people and of the sample 68 people say they would purchase it. The sample proportion then would be denoted p̂=0.68. We calculate a test statistic, which we call "z" because it (theoretically) has a standard normal sampling distribution.

  

  We use p₀=.65 because in the null hypothesis we assume that p=.65.

  Finally we determine if this test statistic would be unusual if the null hypothesis WERE true. We can determine this by either

  a) calculating a rejection region and rejecting the null hypothesis if z is in this region or
  b) calculating a "p-value" for this test statistic and reject the null hypothesis if this "p-value" is smaller than the "significance level" for this test.
  

  In this case, because the alternative hypothesis is "p > .65", we call this an upper-tailed test. The p-value is P(Z>z) i.e. P(Z>.629) = .2647.

  The p-value is the probability of observing a test statistic like we did IF the null hypothesis is true. We usually specify some significance level like .05 or so. This .26 p-value means the data collected is quite plausible if the null hypothesis were true (if the null hypothesis were true, about 1 out of 4 samples would give data like we see, that's not unusual at all).

  Our conclusion: We do not reject the null hypothesis. Even though we observed .68 (which is more than .65 from the null hypothesis) from a sample size of 100 it's simply not big enough to conclude that the population proportion is more than .65!

  For confidence intervals, because there's no hypothesis, we use the sample proportion for calculating standard error (and thus the margin of error). The formula for confidence interval is just like for the population mean:

  The z^* is the critical z value for the confidence level that we are using. For example, a 95% confidence interval will use z^*=1.96. As you can see, the larger the sample size, or the more extreme the sample proportion, the smaller the margin of error will be. But if you increase your confidence level, the interval will widen.

• Project Two(show/hide)

  Here are some tips to help you get the best grade on this project:

  • As always, use the template, and remove all bracketed text.
  • Include RELEVANT tables of sample statistics and charts in the body of your paper - but don't include irrelevant statistics. 
  • Read the rubric and be sure to discuss all of the items that you're asked for. 
  • In particular, be sure to give proper attention to the discussion of confidence intervals and how they relate to the second inference question (regarding square footage)
  • Be sure to properly conclude your tests with a statement that ties back to the original research question - just saying "I reject the null hypothesis" is not enough
  • While we're on the subject, remember we either reject the null hypothesis in favor of the alternative because you have sufficient evidence for the alternative, or we DON'T reject the null hypothesis, because we lack sufficient evidence. We don't accept the null - it's assumed true when doing the test, but we're not in the business of proving the null hypothesis to be true. If we don't reject it, we retain it as the assumed hypothesis, but that doesn't mean it's true and it doesn't mean we've proved it.
  • While we're at it, remember that we never know the truth of the null and alternative hypotheses for certain! We can only make our conclusions with some degree of certainty, but not 100% certainty. Please bear this in mind when you discuss your conclusions.
• Module 7 Challenge Activities
• Module 7 Project Two

Module 8: Two Sample Inference

• Module Info(show/hide)

  This week's module continues with the ideas of hypothesis testing, now extending it to comparing two populations or groups. Remember that we use the term "population" very broadly. In the statistical sense, a population could be anything you want to make inferences about. Two sample hypothesis tests are used all the time, especially in clinical testing. Say we want to decide whether a new allergy medicine is better than the leading brand (whatever that happens to be) or a placebo. Specifically, we want to decide between two hypotheses:

  Null Hypothesis: The new drug is essentially the same as the leading brand (or placebo)

  Alternative Hypothesis: The new drug is better

  We get two groups of people, Group A (takes the new drug) and Group B (takes the leading brand or placebo). (You may see 1 and 2 used instead of A and B, by the way). We want to conclude whether the new drug results in fewer allergy attacks (on average) than the leading brand. Let μA represent the average number of allergy attacks for people on the new drug, and μ_B the average number of allergy attacks for people using the leading brand. Now we can state these hypotheses more specifically:

  H₀: μ_A = μ_B
  H_A: μ_A < μ_B
  

  Although you may equivalently state it like this (you will see both forms):

  H₀: μ_A - μ_B = 0
  H_A: μ_A - μ_B < 0
  

  As with hypothesis testing in general, we collect some data, calculate a test statistic, and decide (based on the sampling distribution of the test statistic) if it is significant enough to reject the null hypothesis. In this case the test statistic will be based on the difference between two sample means, what we might call x̄_A and x̄_B

  There are essentially THREE types of two sample inferences we will be looking at:

  • Paired T test: Comparing the average change before and after some treatment. Here we're concerned with μ_d, the mean difference. Once we look at all the differences, this is actually just a 1 sample T test!
  • Independent Sample T test: Comparing the difference of two population means. An independent sample from two populations is taken and we are trying to infer something about μ₁-μ₂. The calculation of the T statistic follows a formula, as does the calculation of degrees of freedom.
  • Independent Sample Proportion Test: Comparing the difference of two population proportions. An independent sample from two populations is taken and we are trying to infer something about p₁-p₂. We calculate a z statistic using a formula, and use the standard normal distribution to calculate p-value.
• Module 8 Challenge Activities