# High School: Statistics and Probability

### Making Inferences and Justifying Conclusions HSS-IC.B.6

6. Evaluate reports based on data.

Students should understand that even though statistics has more charts, tables, and numbers than your students might think is humanly possible, that's not all statistics encompasses. There are meanings behind the numbers. Yes, we're talking about context.

Students must evaluate their knowledge and understanding of statistical principles in the proper context. Students must be able to read an analysis or report and actually decipher the intent, the meaning, and the significance of what's in front of them.

It sounds easy, but critical thinking isn't! It can only occur once the fundamentals are mastered and knowledge can be used to actually critique something. Here are some good questions to ask the class and facilitate understanding.

• What is the purpose of the study?
• How did the authors go about collecting data?
• Were the collection methods appropriate for this particular study?
• Were they randomized?
• Were the significance levels appropriate?
• What was the null hypothesis?
• What was the alternative hypothesis?
• Were the correct statistical tests used for the data?
• Was the data fully presented?
• Who were the study participants?
• Who funded the study? Can bias arise from that?
• What were the results?
• Why are the results significant even if there were no significant findings?

Aside from asking questions about already completed studies, students can come up with hypothetical studies to help develop a deeper understanding of the core concepts. That way, students can choose topics that interest them and think about how they would go about testing aspects of these topics to ensure statistical validity.

#### Drills

1. A recent study in a well-known peer-reviewed publication has shown that the average rate at which a person text-messages is about 10 words per minute. (Can you beat that?) The standard deviation in the study was 9.3 words per minute. Which of the following is least likely to be the cause of this large standard deviation around the mean?

Improper calculation

This question forces you to think about why the study results are the way they are, since journalists don't often reveal much more than what is written in the question. In a well-known peer-reviewed publication, the math is typically well checked and is usually not the answer to why the standard deviation is so large. Different phones and test conditions may make texting easier or more difficult, and the age of the participants may also cause the standard deviation to increase. That means (A) is the least likely.

2. Two studies were conducted independently and simultaneously on the same population pool. One study's outcome estimated the population average to be 3.4 while the other estimated it to be 8.6. What can account for such a radical difference?

All of the above

This is a bit of a tricky question. (But Trix aren't just for kids, silly wabbit! They can fool adults too.) Answer (B) is obvious: the study could have been biased in its sampling methodology, causing the major difference. A large standard deviation can justify such a difference as well. Finally, even proper randomized sampling regardless of standard deviation can, by chance, produce "biased" looking results.

3. You log into your Facebook account after posting a picture of your trip to find that there are 30 comments on it from you friends, all of them very positive. From a statistical validity standpoint, what can you conclude from these comments?

Nothing because this is a biased data pool that doesn't represent the overall reality

While (B) is nice to think about, the reality is that your friends are much more likely to be biased in your favor. Your friends might give you a mighty fine ego boost, but their opinions and Facebook comments don't represent a situation that has statistically significant results.

4. A study samples 100 Coca-Cola drinkers and finds that 99 of them really dislike the taste of the new cola drink. What inference can be drawn from this?

Most people who drink Coca Cola would dislike the new cola

First, you must understand that this is a biased sample. If you understand that then you know that the only general inferences you can draw is between this sample and a similar population (in this case, only Coca-Cola lovers). This immediately eliminates (B) and (D) (and (A) is just silly). Only (C) is correct because it is the population from which the sample was drawn.

5. A surfing magazine conducts a study about which surfboard is the best and it is sponsored by one particular surfboard manufacturer whose board is in the study. Out of 100 different surfboards, the surfboard manufactured by the sponsor came out on top. What can we conclude from the study?

This study should be scrutinized from every angle very carefully

It's unclear how biased the study is based on the question, but we definitely want to make sure we investigate it very carefully and not believe everything we hear until we do so. The manufacturer may be just so confident their board is much better that they're willing to sponsor a study to do it, or they may be doing something behind the scenes that we're not aware of. We can't be certain the study is biased, but we can do our best to check everything in order to make sure it's not.

6. A business decided to evaluate whether or not it sells more video games when it advertises than when it doesn't in order to cut back on any unnecessary costs. It decided to conduct a study concerning revenue with and without advertisement. What would be the null hypothesis in this study?

This is how we phrase a null hypothesis. We assume no difference between our two groups. All other answers are simply not correct. Answers (C) and (D) may be either true or false, but they are not valid hypotheses that can be tested statistically.

7. If a study on how best to study for a test leaves out certain participants from its final calculations, which of the following might you assume?

All of the above

All of the above answers are valid responses. The only way to figure out which one is the case is if the researchers mention why they pulled data out. (If they don't mention why, you really have to question their motives.)

8. An amazing study decided to measure if pigeons in France are smarter than pigeons in the U.S. The study found no significant difference between the pigeons in the U.S. and the pigeons in France. Is this study significant?

The study found no significant difference, but the result is still significant

The majority of studies published in peer-reviewed publications are those that have results. (This is done many times for marketing and revenue related issues. Because studies showing no difference or no "real result" are sort of downers, they are many times left out of publications unless the studies contradict a previous study on the same matter.) But even a study that has no statistically significant differences says something about the topic being studied. Therefore, (B) is correct because any result that comes from a valid study is still significant.

9. A study is done to determine which steroid cream is more effective for bug bites. If the only bug bites treated in the course of the study were mosquito bites, which of the following is true?

The study will only be able to produce results concerning the effect of the steroid creams on mosquito bites

Since the sample survey is no longer randomized, it could be considered biased and all the results from the study will only apply to the effects of the creams on mosquito bites. We have yet to obtain any data, so answers (A) and (B) are incorrect because these steroid creams may or may not work at all. Answer (D) is incorrect because the study is not an observational one.

10. Why would a study that focuses on lung cancer rates on people who have smoked a long time be more appropriate than a study that focuses on lung cancer rates in young people who just started smoking?