Weekly Math Problem! #42

Sampling Distribution. This week's problem is inspired a student I'll be working with this week. Statistics presents itself in various subjects outside of mathematics. A common topic of students learning statistics is computing and using a z-score. A z-score helps to standardize data so that probabilities related to that data can be found...at least that's how my brain 🧠 thinks about it. Age, salaries, test scores, etc. If it can quantified and certain other information (mean and/or standard deviation) about the data is known, then you may be able to find and use z-score. ...And you know what that means...word problems! 😩

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to calculate the z-score
        ✔️     How to use a Normal probability table

Here is WMP! #42:

Happy solving!

Check back on Friday, March 19th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Alright, alright. Because word problems can be annoying, I always advise to identify the information in the problem that you will be using to solve it. So, here is my color-coded version of the problem, highlighting the important information:

Next, we'll use the highlighted information, above, in the z-score formula. The only way we'll be able to find the probability of interest is to standardize the values of interest. Since there are two values of interest, we compute two z-scores. This will allow us to use a Normal probability table.  

 

Now that we have the two z-scores we need, we can find probability of interest. You can definitely use a wide variety of software and probability calculators to obtain the answer. I used a couple and will share those screen shots soon. First, let's use a Normal probability table. There are many to choose from. The table I linked has a simple display, but you still need to understand the symmetrical aspect of a Normal curve to use it.

The problem asked for "the probability that the average blood pressure...will be between 118 and 124 mmHg." Because we standardized the data, we can look at this as the probability between the z-scores -0.4 and 0.2

One of our z-scores is negative and the other is positive. The Normal table being used here only shows positive z-scores. Here is where the beauty of the symmetrical nature of the Normal curve is helpful. I will "switch" the negative z-score with its positive counterpart so that I can use the probabilities from the table.

As promised, here are a few screen shots of online probability calculators I used that confirmed the answer above:

**This plot was generated using Geogebra's Classic Probability Calculator. Click image to enlarge.

**This plot was generated using Online Stat Book's Area Under Normal Distribution Calculator. Click image to enlarge.

**This plot was generated using MathCracker's Normal Probability Grapher. Click image to enlarge.

So to answer the question... The probability that the average blood pressure of this sample will between 118 and 124 mmHg is 0.2347. 👍🏿

 

▪️ How did you go about finding the probability?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next WMP! #43. 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem! #38

Linear Regression. I scanned my current problems list and said, "Hmm...🤔... I don't see STATISTICS on here." I thought it would be fun to do a little linear regression so, here we are. I think STATISTICS is cool, however, there are many instances when performing the calculations can make you feel a little dizzy. 😵 This is why we're grateful for software. 🙃 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to find the regression line using the Least Squares method
        ✔️     How to use the regression line
        ✔️     How to interpret the slope regression line

You'll have to lookup the formulas and so will I. Perform the calculations by hand and use software to check. I know the numbers a little big but, we'll be okay. (You may want to do the calculations in stages, if doing it in one sitting is too much.) Alright, here goes WMP! #38:

Happy solving!

Check back on Friday, February 19th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Hopefully, you're not mad at me for these calculations. 🥴 Somehow, I found this to be a little, key phrase "a little", therapeutic. Yeah, I know, I know, I'm weird...but who isn't? 🤷🏿‍♀️ I will acknowledge that mistakes can easily be made, especially if you're not paying attention. I had to check my work...a must. I entered the data in Google Sheets to do so. Let's get started shall we...

a. To find the equation of the regression line, certain calculations are needed. We'll start with squaring and multiplying the data points, as needed, organizing them in chart. That information will, then, be used to calculate the intercept and the slope of the regression line. Finally, the equation of the regression line can be written.

   

b. To estimate the blood pressure, we'll use the equation of the regression line from part a. How? Substitute the age 50 in place of x, then perform the indicated operations. The answer you get will be the estimated blood pressure, assuming no computation errors were made.

c. In our situation, the residual is the difference between the blood pressure of a 42-year old as observed in the data and the estimated blood pressure of a 42-year old as computed from the regression line equation.

d. To interpret the slope, keep in mind the context of the problem and think about how the blood pressure number changes, when the age of a person increases by one year.

I hope you found this helpful. I sure did! It was super helpful for me, actually. Part d. was my favorite part of the problem because, it required to me explain the meaning of the slope. Interpreting/explaining numbers reminds me that it isn't always about what the number is. Just as important is what the number means. This is how reports get written. This helps with decision making in certain sectors. 

But before we go, I just have one more thing to share...a scatter plot of the data from the problem with the fitted regression line.

*This plot was created using Google Sheets.*

▪️ Did you find this problem hard, annoying, or something else?

▪️ How did you feel about STATISTICS, in general?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

L👀ks like WMP! #39 is quickly approaching.

Cheers!

The Younge Lady

Weekly Math Problem! #12

Solving A STATISTICS Word Problem. One of my favorite sections of math is stats! 😍 I really like it because it was the first time I really noticed math that I learned in school all around me--in advertisements, commercials, in education, in finance, in the medical field, pharmaceuticals, etc. This is one area in mathematics that I want to be so much more comfortable with and better at. I have a few books that I've purchased over the years from the time I first started learning and understanding stats in undergrad until now. Two of those titles include The Humongous Book of Statistics Problems by W. M. Kelley and R. A. Donnelly Jr., and Statistics in a Nutshell, Second Edition by Sarah Boslaugh. Of course there are also electronic textbooks and stats PDF files in my possession. **Does anyone else scour the internet for PDF files? 👀 Asking for a friend. 😅** Also, because I tutor statistics, I make sure to have resources on hand to refer to, so that the people I work with get the best help they can get from me. I have also purchased and read Numbers Rule Your World by Kaiser Fung. I need to read this book again. I enjoyed it the first time I read it and definitely wouldn't mind reading it again.

Even though I really like statistics, word problems can definitely be annoying. If you're like me, you need to read word problems at least three times before feeling comfortable enough to start solving. 

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Working with normally distributed data

    ✔️  Working with "small" sample sizes

    ✔️  How to conduct a t-test

    ✔️  Hypothesis testing 

    

Here is WMP! #12...

Happy solving! 

Check back on Friday, July 24th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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I'm working with some students this summer and after their review of pertinent information concerning methods to solve specific types of word problems, they freely expressed how much word problems gave them a problem. I totally get it. Word problems definitely make me sigh quite a few times while reading them. So, I decided to color-code this week's word problem with the information and formula used to come to the appropriate conclusion.

**The critical value above came from this t-table.

GeoGebra Classic  has a cool probability tool that shows you the graph of some discrete and continuous probability distributions. It also generates some statistical results of a few different types of statistical tests that are typical when learning statistics. (I encourage you to check it out.) Here are the graph and results of the t-distribution with information from the word problem:

◾️ How do you feel about statistics?? 

◾️ Were there any differences between how you solved the word problem and what I presented above?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

See you next week for...WMP! #13 👩🏿‍🏫

Cheers!

The Younge Lady