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 ⬇️.

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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! #39

Contingency Table Probabilities. A contingency table is a nice way to summarize frequency related data for categories that have some of the data in common. You've probably seen one before. In the past I have helped some students prepare for an SAT exam and, without fail, there was at least one contingency table on every practice exam. You'll find that when a contingency table is presented, probabilities are asked for...especially conditional probabilities. That's what we're working on this week.

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

        ✔️     How to complete a contingency table with given information
        ✔️     How to use a contingency table to find probabilities of interest

Definitely have a calculator on-hand, in case you're interested in converting the probabilities to decimal format. (The question image is linked to the original source. I told y'all that I search the internet for all sorts of materials and things. 😁) WMP! #39 says the following:

Happy solving!

Check back on Friday, February 26th 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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🏊🏿‍♀️ I'm jumping straight in. 

a. Let's organize the data in a contingency table! This part of the question has us placing the given information in the table so that we can see which cells need to filled-in. We're also finding the row and column totals. Totals are super important, as that information will be needed for the remining questions. The following are my partially filled table, calculations, and completed table:

Now that we have a completed table, let's move on the rest of the questions. Note: Make sure that 👏🏿 every 👏🏿 row and 👏🏿 column 👏🏿 total 👏🏿 check 👏🏿 out. If not, than any probability you pull from the table will be off or just incorrect.

b. The key word in this question, as small as it is, is the conjunction "and". "And" denotes an intersection, which is represented by '∩'. What does that mean for us? That means in order to find the probability that a randomly selected reader is female and [simultaneously] prefers Snapchat, we need to zero in on the cell that is in the Female row and Snapchat column; it's the cell where those two categories meet, or intersect. The cell contains 81. The reader being chosen at random is being chosen out of the 578 readers that responded to the survey. The probability is found by dividing the number of people in the event by the total number of people who participated in the survey. Here is the probability with proper notation:

c. This question threw us a little curve ball. Were you able to recognize that this probability is a conditional probability? 🤷🏿‍♀️ Conditional probability is represented by '|', which separates the category or event of interest and the condition. In effect, the condition changes the total to be considered. Instead of the total being all participants, it will be the total of the specified condition. What's the condition? In our case, the condition is "of the respondents who prefer Snapchat". This phrase is telling you to forget about everyone and just focus on those who prefer Snapchat--199 readers. How many of those 199 Snapchat folks are Female? 81. Just like the previous question, we'll divide but, this time, we're dividing the total number of Snapchat Females by the total number of people who prefer Snapchat. Here is the probability with proper notation:

d. Here's the second curve ball. What happens when you switch the categories in a conditional probability question? Let's find out because, that's what happened here. "Suppose a female reader is selected at random" is indicating to us that we're choosing from the Females only--362 of them. So, out of all 362 Females, how many of them prefer Snapchat? 81. Here is the probability with proper notation:

And we're done. YES! I actually like contingency tables. My favorite part about the data being organized in a contingency (or two-way) table is that I didn't need any formulas to answer the questions. 😁 The more I work with them, the more confident I become answering them and explaining them to other people. 

▪️ Were you able to find the probabilities?

▪️ How did you feel about PROBABILITY, 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!

And now we move on to WMP! #40.

Cheers!

The Younge Lady