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Math 3 Module 9: Statistics

9.1 Ready, Set, Go!

Question 1

If the mean is 57 and standard deviation is 2, then one standard deviation above the mean is 59, two standard deviations above is 61, and so on. This video explains it visually: https://www.youtube.com/watch?v=MRqtXL2WX2M

Question 2

If the mean is 126 and standard deviation is 6, then one standard deviation below the mean is 120, two standard deviations below is 114, and so on. This video explains it visually: https://www.youtube.com/watch?v=MRqtXL2WX2M

Question 3

This video shows you how to calculate the standard deviation from a set of numbers: https://www.youtube.com/watch?v=MlkZXiuxodw. Find the standard deviation of each set (A - E) and then put them in order by standard deviation!

Question 4

To calculate the percentage of competitors she beat, you need to write down the values for the fraction (# competitors Robin beat)/(# competitors total) and then convert the fraction into a percentage by multiplying the fraction by 100.

Question 5

Use a calculator to find the mean (average) and standard deviation. This video shows you how to calculate the standard deviation from a set of numbers: https://www.youtube.com/watch?v=5kz16kbOTG0 This shows how to do it by hand: https://www.youtube.com/watch?v=MlkZXiuxodw

Question 6

When you have the values of the mean and standard deviation, you can calculate the value one standard deviation above the mean by addition. E.g. if the mean if 5 and the standard deviation is 2, then one standard deviation above the mean is 7. Similarly, if you want to find the value 3 standard deviations below, you subtract the standard deviation 3 times.

Questions 7 - 12

You have to decide if the distribution in the problem is a "normal distribution". This video talks about normal distributions: https://www.youtube.com/watch?v=xgQhefFOXrM

Questions 13 - 14

For different values of the mean, the location of the center of the curve will be different. The bigger the standard deviation, the broader the curve! This video introduces the Normal distribution: https://www.youtube.com/watch?v=xgQhefFOXrM

Question 15

To find the exact value of the standard deviation from the data on the graph, check out this short video: https://www.youtube.com/watch?v=skJaS_kedR8

Question 16

In this question, you want to find the inverse of the quadratic function. This video has a similar example: https://www.youtube.com/watch?v=8r3XQoQjNTI

 

Question 17

In this question, you want to find the inverse of this linear function. This video has a similar example: https://www.youtube.com/watch?v=j3f88V9M0qM

Question 18

In this question, you want to find the inverse of this square root function. This video has a similar example: https://www.youtube.com/watch?v=01-BrCJn0Xo

Question 19

The easy thing about finding the inverse of function in a table is that you can just flip the columns! Check out this video to find out why: https://www.youtube.com/watch?v=TSztRfzmk0M

Questions 20 - 21

One option is to find the inverse of one of the functions and see if it matches the other function. This video shows you how to find the inverse of linear functions: https://www.youtube.com/watch?v=j3f88V9M0qM

We can also prove that two functions are the inverse of each other by creating a composite function with them, like this: https://www.youtube.com/watch?v=crpeERDwYag

9.2 Ready, Set, Go!

Question 1

If there are six numbers on a "fair" die, then you can expect all numbers to be rolled with the same probability as all others (e.g. 6 rolls gives you one of each number on the die). HOWEVER, when we only roll a few times, the results aren't so accurate. This video talks about "representative samples" and why we need to get lots of samples before our we get results that match the expected probability: https://www.youtube.com/watch?v=rckB8T8WthM

Question 2

If there are six numbers on a "fair" die, then you can expect all numbers to be rolled with the same probability as all others. When we roll a lot of times, we should get a more "representative sample". In this case, 20 out of 50 (or 2/5) rolls show a 5. If it was a fair die, then only 1/6 of the rolls would land as 5, but 2/5 is more than 1/6, so the die mustn't be fair!

Question 3

The core of the question here is - is a group of 100 students a good enough sample for the total group of 729. If so, then 30% of 728 students should also feel that there is enough parking. If this sample isn't representative of the total population, how should that percentage change?

Questions 4 - 6

This video shows you how to put the 68-95-99.7 rule to practice with a few examples: https://www.youtube.com/watch?v=2MgYDrGcn6c

This video goes over the 68-95-99.7 rule from the beginning (super helpful if you haven't heard of it or need a refresher!): https://www.youtube.com/watch?v=cgxPcdPbujI

Questions 7 - 14

This video goes through the log rules that you'll need to solve this problem: https://www.youtube.com/watch?v=bPLyCH1WaEY

9.3 Ready, Set, Go!

Question 1

The percentage of students that prefer school lunch is (# of people that prefer school lunch)/(total # people). Then, multiply this fraction by 100 to find the percent.

Question 2

Go through and calculate the percentage that prefer school lunch for each grade level (each column). The percentage of students that prefer school lunch is (# of people that prefer school lunch)/(total # people). Then, multiply this fraction by 100 to find the percent. Are there any grades that are more than 50%? Why is that important?

Is that because there are only 2 options? If more than 50% like something, then it's the preferred option.

Question 3

To find the "overall" answer, you can look at the data in the last column to figure out whether more people prefer the school cafeteria or eating out because this includes all students.

Question 4

This video shows you how to find the z-score: https://www.youtube.com/watch?v=1o-t_mVDDYQ

Question 5

What percent of the time does the company make more than $29,500? What about less? Think about where this monthly income stands in the normal distribution. The percent above it and the percent below it are the answers to this question. Watch this video to help think about your approach: https://www.youtube.com/watch?v=OhRr26AfFBU

Question 6

Essentially, this question is asking about where this range of money lies in terms of the standard deviations from the mean. The standard deviations away this range is from the mean will have a percent value associated with it, and that's your answer. Watch this video to get a broader understanding: https://youtu.be/MRqtXL2WX2M

 

Question 7

To make profit, they just need to make more than the break even point. So, this question is really asking, how likely is it that the company makes more than $16,400 in a month. Pay attention to the standard deviations and their associated percentages.

Question 8

You have been told the average, the standard deviation, and the average range. Use this to find how many people are average. Here's a very similar problem: https://www.youtube.com/watch?v=cgxPcdPbujI

Question 9

Using what you already know about this normal distribution, think about how many people score 135 or higher. This video will help: https://www.youtube.com/watch?v=cgxPcdPbujI

Question 10

Woah Einstein! Genius or genius? Watch this video that shows how to calculate a z-score: https://www.youtube.com/watch?v=Wp2nVIzBsE8

Question 11

Einstein had a score of 160. Where does this lie on the normal curve? What are the chances of someone with a score higher than 160? This video will help: https://www.youtube.com/watch?v=cgxPcdPbujI

Question 12

This video has a similar example with a 4th degree polynomial: https://www.youtube.com/watch?v=cBCVOPPXHWw

Questions 13 - 14

In this question you have a 7th degree polynomial. This video shows you how to plot higher order polynomials: https://www.youtube.com/watch?v=CHEtGgTexHI

This is a pretty long video so skip to 15:15 to see a similar example of graphing a polynomial: https://youtu.be/q6vgQEZuNKk?t=916

Question 15

Check out this video about the graphs of cubic functions: https://www.youtube.com/watch?v=-ErUrKEDzUI

9.4 Ready, Set, Go!

Question 1

This video has an intro to 2-way frequency tables: https://www.youtube.com/watch?v=97SZ1JR4rzA

This video works through an example using two-way frequency tables: https://www.youtube.com/watch?v=l5MrtV7ZN88

Question 2

Once you have the data in the frequency table, you want to calculate the relative frequency, like in this video: https://www.youtube.com/watch?v=l5MrtV7ZN89. Essentially, calculate (number passed)/(total) for each class and for all the students.

 

Question 3

Use the first table with all the scores in it and combine them together so you just have a long list of scores. You should create buckets for different ranges of scores (I'd just do 40 - 49, 50 - 59, etc) and count how many scores fall into each bucket!

 

This video uses different data, but it's otherwise the same process you need to use to answer this question: https://www.youtube.com/watch?v=AndS0RLdxtk

Question 4

This question is asking whether the 2nd hour is a representative sample of the other two hours. Check out this video on representative sampling: https://www.youtube.com/watch?v=rckB8T8WthM.

Question 5

In this problem, you want to figure out how many standard deviations from the mean each athlete is in their own category and then rank them according to who is the greatest number of standard deviations ABOVE the mean.

Question 6

Here's a video which explains a problem that's pretty much the same! https://www.youtube.com/watch?v=tqZApkDgRqQ

Question 7

They might have made a typo on this one because it's not a log or exponent problem at all 🤣. Solve it for practice as it is!

 

Question 8

This video has a really similar example: https://www.youtube.com/watch?v=oM25leefVRQ

Question 9

The example they work through in this video is similar: https://www.youtube.com/watch?v=fzVEBUHK7-o

Question 10

This example isn't exactly the same, but a lot of the ideas you need to use are the same: https://www.youtube.com/watch?v=59j0ALU3N7k

Question 11

The fraction might make this problem look really confusing but you can multiply both sides by the denominator to get ln(x + 7) = ln(2x - 3) and solve from there like this: https://www.youtube.com/watch?v=-qBTKWrHHko

Question 12

The fraction might make this problem look really confusing but you can multiply both sides by the denominator to get log(4x + 2) = log(15) and solve from there like this: https://www.youtube.com/watch?v=-qBTKWrHHko

Question 13

The fraction might make this problem look really confusing but you can multiply both sides by the denominator to get log₃(3x + 6) = log₃(81) and solve from there like this: https://www.youtube.com/watch?v=-qBTKWrHHko

Questions 1 - 4

In this problem they use the word "explains" but the word we usually use is "correlation". This video talks all about the difference between correlation and causation to help you figure out which is which: https://www.youtube.com/watch?v=U-_f8RQIIiw

Questions 5 - 7

The parameter is the value they want to find. This video shows you a similar example worded problem where you need to find the population and sample: https://www.youtube.com/watch?v=VPM84_yfx5Q

Questions 8 - 13

Is this sampling method is representative of the population? Check out this video on representative sampling: https://www.youtube.com/watch?v=rckB8T8WthM.

Question 14

This video shows you how to find the amplitude, period, and shift of cosine curves: https://www.youtube.com/watch?v=KPvon2ufyNs. This one shows you how to find the endpoints of the first period (or "primary interval"): https://www.youtube.com/watch?v=ijTIr-aykUk

Question 15

This video shows you how to find the amplitude, period, and shift of sine curves: https://www.youtube.com/watch?v=KPvon2ufyNs. This one shows you how to find the endpoints of the first period (or "primary interval"): https://www.youtube.com/watch?v=ijTIr-aykUk.

Question 16

The sine graph that they're asking you to graph has been transformed a few times. This video has a similar example: https://www.youtube.com/watch?v=6SvQt8vHARI

9.5 Ready, Set, Go!

9.6 Ready, Set, Go!

Questions 1 - 8

This video shows you how to find probabilities from a two-way table: https://www.youtube.com/watch?v=kgs2Jzb_dwM

Question 9

We know 100 boys were surveyed and 68 of them preferred dogs, so if this sample is representative, approximately how many boys prefer dogs in the school? The question is really, is a random 100 boys a representative sample of all the boys in school?

Question 10

For this probability, you want to find the number of all students that prefer cats and calculate the percentage with that number and the total number of students. What probability did you already find that can help you answer this?

Question 11

This video discusses different types of survey sampling techniques: https://www.youtube.com/watch?v=rASK8PpqakM

Question 12

Because the boys and girls were "randomly sampled" and quite a lot of students were survey, the data is probably a representative sample of the larger school population, so having more data (which would be very similar) would probably not impact percentages by a lot.

Questions 13 - 16

This video does a good job explaining the difference between surveys, observational studies and experiments: https://www.youtube.com/watch?v=Z_OJzgkKe2A

Question 17

In a survey you can ask participants for lots of different information. In this case, it might make sense to ask students whether they do homework or whether they think the homework is good practice for tests. She could also ask about their test scores to check for a correlation.

In general, we use randomization to get a sample that's representative of the total population instead of selecting people that are all similar (which can happen intentionally or by accident in different situations). This video explains more: https://www.youtube.com/watch?v=QmgnQ48RsMM

Question 18

In observational studies, you watch how people behave and try to draw conclusions about how certain behaviors are correlated to, or sometimes cause, other behaviors. Watch the first couple minutes of this video for an explanation: https://www.youtube.com/watch?v=n1xSeGPGQEg

Question 19

For this type of design, you're testing a hypothesis and changing some variables. This video explains the structure: https://www.youtube.com/watch?v=NqQaV2DfVAI. This video has a clear explanation of randomized trials (the type of experiment they're talking about in this problem): https://www.youtube.com/watch?v=fkOCYov1p-o

Question 20

Here's something to ponder: an observational study could probably be used to track which students are doing their homework and which aren't, BUT the group that aren't doing homework might all be students who are already struggling, so their test scores could be low but it doesn't necessarily tell us if other students who are doing better in the class benefit from doing homework. Is there another strategy that could be better?

Question 21

This video shows you how to draw a normal distribution when you're given the mean and standard deviation: https://www.youtube.com/watch?v=qEGYkkif6xU

Question 22

In this question you need to find the percentage of people that have resting heart rates in the range 55 - 80. This video shows you a similar example: https://www.youtube.com/watch?v=_86q-hn_3DQ

Question 23

80 beats per minute is 1 standard deviation above the mean. If it's one standard deviation away and everything above it is unhealthy, how many people have an unhealthy heart rate?

9.7 Ready, Set, Go!

Question 1 

You need to create a frequency table. This video explains how: https://www.youtube.com/watch?v=lyRbCwDDnJo

 

For this particular situation, if I flipped a coin 5 times and 3 of those flips landed on heads, I'd write a tally in the 60% table row. Now I flip the coin 5 times again and only 1 time was heads, so I tally in the 20% row. Follow that process until you have a list of 20 tallies. Then, count up the number of 0s in your list and put it first open row of the Frequency column, count of the number of 1s in the list and put it in the next open box, etc.

Question 2

This video shows you how to make a histogram: https://www.youtube.com/watch?v=gSEYtAjuZ-Y

Question 3

The prompt should really say, flip the coin 20 times and put a tally in the appropriate row for how many heads you get. Repeat the process 5 times. This is similar to question 1.

Question 4

This video shows you how to make a histogram: https://www.youtube.com/watch?v=gSEYtAjuZ-Y

Question 5
Think about where the means are located and how much variance each one has, which is represented by its spread.

Question 6
As you get more data, your sample of flips gets closer and closer to the "ground truth" about how many flips should give heads and how many should give tails. Based on the two histograms you made above, what do you make of all this information?

Question 7
This video talks about this EXACT problem (and the TV show haha) https://www.youtube.com/watch?v=4Lb-6rxZxx0

Question 8

You just need to fill in the table with the tallies from data you get playing the game "Stick or Switch"

Questions 9 - 10

You can find conditional probability from a two-way table like this: https://www.youtube.com/watch?v=VduDWrFCqZk

 

Question 11

Do the same thing as in questions 8 - 10, just with 100 games instead of 20.

Question 12

The prompt should really say, flip the coin 20 times and put a tally in the appropriate row for how many heads you get. Repeat the process 5 times. This is similar to question 1.

Questions 13 - 17

In this question, you need to use the Venn diagram to calculate probabilities, like this: https://www.youtube.com/watch?v=ErJ2F8lWJKc

Question 18

This video has an example of independent events: https://www.youtube.com/watch?v=7QlZjoLmg3I&t=45s. After watching it, what do you think?

This video is kind of long, but covers everything about independent and dependent events through lots of examples and I highly recommend it! https://www.youtube.com/watch?v=lWAdPyvm400

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