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Module Five: Normal Distributions & Hypothesis Testing

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Introduction & Goals

This week's investigations introduce and explore one of the most common distributions (one you may be familiar with): the Normal Distribution. In our explorations of the distribution and its associated curve, we will revisit the question of "What is typical?" and look at the likelihood (probability) that certain observations would occur in a given population with a variable that is normally distributed. We will apply our work with Normal Distributions to briefly explore some big concepts of inferential statistics, including the Central Limit Theorem and Hypothesis Testing. There are a lot of new ideas in this week’s work. This week is more exploratory in nature.

Goals:

· Explore the Empirical Rule

· Become familiar with the normal curve as a mathematical model, its applications and limitations

· Calculate z-scores & explain what they mean

· Use technology to calculate normal probabilities

· Determine the statistical significance of an observed difference in two means

· Use technology to perform a hypothesis test comparing means (z-test) and interpret its meaning

· Use technology to perform a hypothesis test comparing means (t-test) (optional)

· Gather data for Comparative Study Final Project.

DoW #5: The SAT & The ACT

Two Common Tests for college admission are the SAT (Scholastic Aptitude Test) and the ACT (American College Test). The scores for these tests are scaled so that they follow a normal distribution.

· The SAT reported that its scores were normally distributed with a mean μ=896 and a standard deviation σ=174

· The ACT reported that its scores were normally distributed with a mean μ=20.6 and a standard deviation σ=5.2.

We have two questions to consider for this week’s DoW:

2. A high school student Bobby takes both of these tests. On the SAT, he achieves a score of 1080. On the ACT, he achieves a score of 30. He cannot decide which score is the better one to send with his college applications.

. Question: Which test score is the stronger score to send to his colleges?

· A hypothetical group called SAT Prep claims that students who take their SAT Preparatory course score higher on the SAT than the general population. To support their claim, they site a study in which a random sample of 50 SAT Prep students had a mean SAT score of 1000. They claim that since this mean is higher than the known mean of 896 for all SAT scores, their program must improve SAT scores.

. Question: Is this difference in the mean scores statistically significant? Does SAT Prep truly improve SAT Scores?

Investigation 1: What is Normal?

One reason for gathering data is to see which observations are most likely. For instance, when we looked at the raisin data in DoW #3, we were looking to see what the most likely number of raisins was for each brand of raisins. We cannot ever be certain of the exact number of raisins in a box (because it varies) no matter how much data we gather. But, we can estimate a likely value or range for the number of raisins and determine the empirical probability that a box of raisins would be in this range.

In Activities A & B, we explore the probabilities seen in a particularly symmetric distribution.

Inv 1, Activity A: Probability in a Distribution

Excerise A1: Relative Frequency

Histograms Relative frequency histograms provide information about probabilities. Consider the following relative frequency histogram for the number of raisins in a 1/2 ounce box of Brand B raisins

(The probability for each interval in the histogram is displayed at the top of the bar for that interval.)

(a) Describe the shape of the distribution.

(b) Suppose I pick a random box of Brand B raisins. Based on this data, what is the probability that the number of raisins in the box is:

· greater than 28?

· less than or equal to 28?

· exactly 28?

· greater than 32?

· between 26 and 30 (inclusive)?

Answers: i) 32%; ii) 69%; iii) 23%; iv) 0%; v) 71%

A common way to "divide up" the histogram is to use the standard deviation as a ruler. We will consider the data in groups that are one, two and three standard deviations from the mean. Let's start by looking at the group of data that is within one standard deviation of the mean: One Standard Deviation from the Mean:

We can add these values to our graph to see the group within one standard deviation of the mean:

Exercise A2: Use the above histogram to:

(a) determine the probability that a random box of brand B raisins would fall between 25.8 and 29.4 (within one standard deviation of the mean)

(b) calculate the values that are two standard deviations away from the mean. Sketch them on the graph.

(d) Repeat steps b and c for raisin boxes that fall within three standard deviations from the mean.

(e) Summarize your work from this exercise in a table like the one shown below:

Percentage of Observations	Brand B Raisins
Within 1 standard deviation of the mean
Within 2 standard deviations of the mean
within 3 standard deviations of the mean

Exercise A3: The following distribution shows a relative frequency histogram of students scores on a Math placement exam. Like the Brand B Raisins distribution, it is a roughly symmetric, mound-shaped distribution.

(a) Determine the probability that a random student's score would fall within one standard deviation of the mean; within two standard deviations of the mean; and within three standard deviations of the mean.

(b) Add your findings to the table you started in Exercise A2, by adding a column for Math Placement Exam Scores:

Percentage of Observations...	Brand B Raisins	Math Placement Exams Scores
within 1 standard deviation of the mean
within 2 standard deviations of the mean
within 3 standard deviations of the mean

Our findings illustrate a “big idea” called the Empirical Rule:

THE EMPIRICAL RULE (68-95-99.7%) In "special" symmetric, mound-shaped distributions, about 68% of the observations fall within one standard deviation, about 95% fall within two standard deviations, and about 99.7% (nearly all) all within three standard deviations.

Exercise A5: Practice with the Empirical Rule Each of the distributions shown below is a symmetric, mound-shaped distribution with a mean of 0 and a standard deviation of 1. Use the empirical rule to determine the percentage of observations represented by the shaded area on each distribution.

(answers follow)

1. Between -2 and 2	2. More Than 2 or Less Than -2

3. Greater Than -1	4. Greater than 1

Answers: (1) 95%, (2) 5%, (3) 84%, (4) 16%

Inv 1, Activity B: The Normal Distribution

The "special" symmetric, mound-shaped distributions that follow the Empirical Rule (like the ones you looked at in investigation 1 are called Normal Distributions. Normal distributions are a family of distributions with very specific properties, though the way most of us think of them is as a "bell curve." The key properties we think of for a normal distribution are:

· One peak in the middle (mean)

· Symmetric about the mean

· Follows the Empirical Rule: 68-95-99.7

These are not the only defining characteristics of a normal distribution. Normal distributions are defined by equations, which dictate specifics about the shape of the mound, how it curves, and the areas beneath the different sections. However, for most situations you will encounter, mound-shaped symmetric distributions can be considered to be nearly normally distributed. Why are they called normal? For starters, many variables in many different contexts follow the normal distribution, making this distribution the typical, expected or "normal" pattern. When we talk about skewed distributions, we usually mean "skewed from the normal." The samples you worked with in Investigation 1 (the raisin boxes or the math placement scores) are displayed with histograms. The histograms are roughly symmetrical and mound-shaped. They approximate the theoretical normal curve that represents the entire population. The graph below shows the theoretical normal curve superimposed on the distribution of the Brand B raisin data:

Notice that this sample of 22 raisin boxes nearly fills the area beneath the theoretical curve. The normal curve models what we think the true population distribution should be.

The histogram shows the distribution of the actual sample data . As such, its mean and standard deviation are statistics, represented by μ and s, respectively.

· The normal curve shows the distribution of the theoretical population , so its mean and standard deviation are parameters, represented by μ and σ.

Complete the following activities about Normal Distributions: Exercise B1 Watch the video clip entitled, " 7. Normal Curves " in the Annenberg Series Against All Odds. As you watch, take notes on the following questions:

· What is a density curve? How is it related to a histogram or other display of a distribution?

· What properties does a Normal Curve have?

· What are some variables that are normally distributed? What is it about them that makes them normally distributed?

· Why are the Normal Curves considered to be a family?

· What is the formula for standardizing an observation? Why would you want to standardize?

Inv 1, Activity C: Applying the Empirical Rule

Normal distributions are entirely defined by the mean and standard deviation of the distribution. When we know these two values, we have the full picture of the distribution of a variable for the population. From this, we can sketch the distribution and determine the likelihood of specific events occurring (using the empirical rule and other methods.) This is the purpose of the following example: Examples: The Scenario At Lesley Middle School, all students were timed when they ran a 100m dash. It was found that the data was normally distributed with a mean of 17.2 seconds and a standard deviation of 2.1 seconds.

· What range of times would you expect to be typical for the 100m dash at this school? (let’s say that “typical” would mean at least 95% of students would be in this range)

· A student ran the 100 m dash in 10.9 seconds. How likely is it that a student could run the 100 m dash in 10.9 seconds or less?

First, Picture it In this scenario, we know everything we need to know to get a picture of this data. So, let's make the graph. We know it is normally distributed - this gives us the general shape The graph need not be perfectly to scale, but it should show the inflection points – places where the curvature switches. Recall from the Annenberg video that inflection points are found 1 standard deviation from the mean.

Next, label the mean and the values of the points that are 1 standard deviation above and below the mean of the graph. Lastly, label the points that are 2 and 3 standard deviations from the mean. Then, Apply the Empirical Rule to answer the questions:

1. The Empirical Rule states that 95% of the data will lie within 2 standard deviations of the mean. For this scenario, that would be roughly between 13 second and 21.4 seconds.

So we can say that a typical range of times for middle schoolers at Lesley Middle School is between 13 and 21.4 seconds, because 95% of all students would have times in this range. 2. Times that are as fast or faster than 10.9 seconds would be more than 3 standard deviations from the mean. From the Empirical rule, we know that 99.7% of the data is within 3 standard deviations of the mean. This means that 0.3% is within the two tails. So the likelihood of having a time that is 10.9 second or less would be HALF of 0.3%, or 0.15%. This is very unlikely – Perhaps this student is a truly exceptional athlete; or perhaps there was an error in recording this time? A probability this low warrants extra attention.

Exercise C1: SAT scores from DoW #5: In DoW #5, we are given that the SAT (Scholastic Aptitude Test) scores are normally distributed with μ=896 and σ=174.

(a) Sketch a normal curve for this distribution, with three standard deviations from the mean marked out.

(b) What range of scores is "typical" for this test (with “typical” meaning 95% students score in this range)?

(d) What is the approximate probability that a student would score between 1200 and 1400? Explain how you found this answer.

Exercise C2: ACT scores from DoW #5: In DoW #5, we are given that ACT (American College Test) scores are normally distributed with μ=20.6 and σ=5.2.

(a) Sketch a normal curve for this distribution, with three standard deviations from the mean marked out.

(b) What range of scores is "typical" for this test? (with “typical” meaning 95% of students score in this range)

(d) What is the approximate probability that a student would score between 15 and 30? Explain how you found this answer.

Exercise C3: Bobby’s Test Scores: In DoW #5, we learn that Bobby scored 1080 on the SAT and 30 on the ACT. Consider your work in Exercises C1 and C2. Which score do you think Bobby should send to his colleges? Why? Post your response to Exercise C3 to the Discussion Thread “ DoW#5: Bobby's Scores” by Tuesday, 10 PM EST. Review the posts of others and make at least two follow-up posts by Thursday, 10 PM EST Do Not Read On until you have completed these exercises: the answer to Exercise C3 is discussed in Activity D.

Inv 1, Activity D: Beyond the Empirical Rule

In Exercise C3, Bobby’s scores were not an even number of standard deviations away from the mean. This makes comparing them (and determining the probabilities associated with those scores) difficult (though not impossible). There are ways of determining the probabilities when the desired values are not 1, 2, or 3 standard deviations from the mean. We will explore one such approach in this investigation. Recall that all normal curves are members of the same family. By changing the mean, you can alter the location of the curve (shift it left or right). By changing the standard deviation, you can alter the height and width of the curve. These properties allow us to compare values from different curves by standardizing the data. Consider the graphs below, showing the normal curves for the SAT and ACT from the last investigation. Added beneath each is a new scale, showing the number of standard deviations from the mean.

This rescaling allows us to compare values for each of the distributions by looking at how many standard deviations the values are from the mean. Now, we need to calculate how many standard deviations Kathy's score and Bobby's score are from the mean of its distribution. We do this using the standardization formula:

Where x is a specific observation, μ is the population mean and σ is the population standard deviation.

This shows that Bobby’s ACT score is a greater number of deviations from the mean. On a standardized normal curve (where μ=0 and σ=1), we can compare the two scores:

Two ways you can think of this are:

· The percentage of students scoring Bobby’s score (or higher) on the ACT is less than the percentage of students scoring Bobby's score (or higher) on the SAT.

· Bobby’s ACT score is higher than Bobby's SAT score, relative to their respective means.

So, the ACT score is the stronger college test score!

Please watch Normal Calculations (Video Unit 8), also from the Annenberg series.

Z-scores are useful for comparing data; in addition, they are useful for finding probabilities that we cannot find with the Empirical Rule alone. Suppose we wanted to know the probability that a student would score as high as Bobby, or higher:

From the Empirical Rule, we know that 47.5% of students would score between 20.6 and 31.0 (2 standard deviations). So, we could estimate the percentage above Bobby’s ACT score of 30 to be a little more than 2.5% (50% - 47.5%).

We can determine a more exact percentage using the graphing calculator. The graphing calculator can calculate the probability that a z-score is between two values on a standard normal curve using the function normalcdf:

Press [2nd][VARS] and select Normalcdf(

To determine the probability that a student scored higher than Bobby on the ACT (above 1.81 standard deviaions) we are looking at the part of the graph between 1.81 and positive infinity. So, enter:

normalcdf(1.81, 99999) = 3.5%

Notice that the second value, 99999, is an arbitrary large postive number. Since the area under the curve becomes so small as the z-scores get larger and larger, this value will give an accurate percentage. We have used the calculator to determine that the probability that a student would score as high as Bobby on the ACT is 3.5%. In other words, P(z>1.81)=3.5%

An alternate option to the graphing calculator is this online convertor .

Since we are only interested in greater than choose one tailed. Insert 1.81 for the z score. Click submit and you will get the same answer.

Exercise D1. Use the graphing calculator to determine the probability that a student would score as high as Bobby on the SAT.

Exercise D2: Use the graphing calculator to determine the following normal distribution probabilities:

(a) P(z<1.28)

(b) P(z>1.28)

(d) P(-1.1<z<1.1)

Answers: D1) 14.5

Answers: D2. a) 90.0% ;b) 10.0%; c) 1.2%; d) 72.9%

The calculator function normalcdf( ) will also calculate probabilities without the use of a z-score. Let's revisit Bobby’s ACT score one more time. He scored a 30 on the ACT, which has m=20.6 and s=5.2. We can use the graphing calculator to find the probability that a student would score 30 or higher on the ACT without first finding a z-score. To do this, enter:

normalcdf( minimum, maximum, m, s)

normalcdf( 30, 99999, 20.6, 5.2)

This gives the same percentage we found earlier using Bobby’s ACT z-score (3.5%).

Another option for this is to use this link .

You will insert 20.6 for the mean, 5.2 for standard deviation, and 30 for x. This will give you a cumulative probabilty of .96467. The percent above this is nearly the same answer of 3.5%

Exercise D3: Use the graphing calculator to recalculate Bobby’s SAT probability, as we just did for his ACT probability.

Investigation 2: Testing Hypotheses

The second part of DoW #5 asks us to evaluate a claim based on sample data: is the sample of scores from SAT Prep students different from other SAT scores, in a way that is more than just random variation. This question is asking us to take what we know about SAT scores – they follow a normal distribution, with a mean of 896 and a standard deviation of 174 – and infer whether or not a sample mean SAT score of 1000 is “not normal”. Certainly, students score higher than 1000 all the time…but the question is,

“How likely is it that a whole group of 50 students would have an average score that is a full 100 points above the mean of 896? Couldn’t this just be due to normal variation?”

This investigation looks at how to address this question statistically.

Inv 2, Activity E: Sample Means are Normal

Exercise E1: Consider the question posed above: “How likely is it that a whole group of 50 students would have an average score that is a full 100 points above the mean of 896? Couldn’t this just be due to normal variation?”

What are your initial thoughts on this question? Do you think SAT Prep’s results support their claim that they raise SAT scores? Record your initial thoughts in your journal.

In the SAT Prep claim, they refer to the mean of a sample of 50 scores. It is hard to compare a mean of 50 scores to the mean for the whole population. To better understand this situation, we need to look at LOTS of samples of 50 scores from the whole population, and get an idea of what their means look like.

Exercise E2: Complete the Central Limit Theorem Central Limit Theorem - Alternative Formats

document. In this exercise, you will run a virtual experiment to collect random samples of 50 SAT scores and calculate their means. This will produce a distribution of sample means, that you can compare to the sample mean for SAT Prep.

Exercise E3: In Exercise E2, you answered the question, “do you think there sufficient evidence to conclude that SAT Prep improves SAT scores, as they claim?”

Post your response to this question to your group’s DB for DoW #5:SAT Prep by Friday, 11:59 PM EST. Review the responses of your group. Consider your work in the remainder of this investigation. Post at least three meaningful responses to your group by Sunday, 11:59 PM EST.

In these exercises, you created a portion of a Sampling Distribution for the Means: a distribution of the means from lots of different samples. The true sampling distribution for the means would have EVERY mean from EVERY possible random sample of 50 test scores. The distribution in Part II, with 355 scores, is very close to the true sampling distribution for the means:

You probably noticed that this distribution has nearly the same mean as the SAT scores population mean score (896). However, its spread is MUCH MUCH smaller. In fact, the standard deviation of the sampling distribution is a fraction of the original standard deviation (174). This should make sense – in the population, individual students will have scores that are very high and very low. But, within a sample of 50 students, those highs and lows will “average out” bringing the overall mean of the sample closer to the true population mean. This observation reflects a major theorem in statistics – the Central Limit Theorem.

The Central Limit Theorem states that if you have sample sizes larger than 30, the sample means will be nearly normal distributions.

· The mean of the sampling distribution will be the same as the population mean.

· The standard deviation of the sampling distribution will be population standard deviation divided by the square root of the sample size.

Note: One powerful part of the Central Limit Theorem is that the population does NOT have to be normally distributed itself to have a sampling distribution that is normally distributed

For our example, the SAT Prep sample is larger than 30 (it is 50). By the Central Limit Theorem, all samples of size 50 will have means that follow a normal distribution. This normal distribution will have a mean of 896 (the same as the mean for the population of SAT scores). BUT, the standard deviation will be much much smaller: 174/sqrt(50) = 24.6. This tells us that the mean scores for groups of 50 students will not vary much - they will be very close to the center of the distribution.

Population of SAT Scores m = 896 s = 174

Sampling Distribution of Means from samples of 50 Scores

s = 24.6

With the SAT Prep sample, we can now calculate the probability of a random sample of 50 SAT scores having a mean of 1000:

Normalcdf( 1000, 99999999, 896, 24.6) = 0.00001

This is an incredibly low probability – it is highly unlikely that the sample of SAT Prep students is “the same” as other samples of 50 students.

Inv 2, Activity F: Hypothesis Testing

The process of using statistics to test the validity of claim is part of a branch of statistics called Inferential Statistics. Up to this point, our course has focused primarily on Descriptive Statistics.

Descriptive Statistics are used to summarize and describe the data. Tools like histograms, box plots, mean, median, and standard deviation all describe and summarize. Descriptive statistics allow us to describe the distribution and make comparisons among distributions.

Inferential Statistics is a use of statistics to make an inference beyond what is immediately known from the data. Sometimes this involves estimating the true value of a population parameter (like estimating the true mean number of raisins in a ½ oz box of raisins). It also involves using statistics to support or disprove a hypothesis, as we did with the claim by SAT Prep.

The process of evaluating a hypothesis is called Hypothesis Testing. In Activity E, you looked at an overview of the process: we started with a claim, we found a way to calculate the likelihood of the claim, and we used this to determine whether or not the claim was probable. The actual process of Hypothesis Testing is more technical. We will touch the surface of this concept in the next two activities.

Exercise F1: Watch the video 20. Significance Tests in the Annenberg Series, Against All Odds. As you watch, take notes on:

· The Steps of hypothesis Testing

· What is a p-value?

· What is meant by statistically significant?

· What is the difference between a one-tail test and a two-tail test.

· Optional Information on Tailed tests

The Steps of a Hypothesis Test:

· State a Null Hypothesis: H0 This is generally a statement that is assumed to be true or based on a known truth.

· State the Alternative Hypothesis: HA This is the statement you are trying to support

· Determine the desired level of significance a (alpha) This is the probability level that would allow you to reject the Null Hypothesis with Confidence.

· Calculate the p-value This is the probability of your observed results

· Compare the p-value to the level of significance a If the p-value is below the level of significance, then the H0 is rejected in favor of the HA; if not, H0 remains the operating hypothesis. It is important to note the H0 is never PROVEN. It can be supported by the data or discredited by the data. Likewise, HA is never PROVEN. It can become the new hypothesis, because the old hypothesis is discredited. This is not proof of certainty.

Example: Perform a hypothesis test for our work with SAT Prep in DoW #5

· H0: The sample mean score of the SAT Prep students is the same as the population mean score for all SAT students. H0:

· HA: The sample mean score of the SAT Prep students is greater than the population mean score for all SAT Students HA: Note: We had three choices for the HA. The sample mean ( ) could be greater than the population mean, less than the population mean, or just not equal to the population mean. We chose greater than here, because the claim is that SAT Prep improves the scores.

· The level of significance can vary from situation to situation. A strong level of significance is ( 0.5%). In this setting, we mean that IF the probability of the sample of 50 SAT Prep students having a mean score of 1000 is LESS THAN 0.5% under H0, we would consider it to be statistically significant and that H0 would be rejected.

· Calculate the p-value for the sample mean. We did this when we talked about Central Limit Theorem earlier. Recall the sampling distribution:

s = 24.6

Based on this, we calculate the probability of getting a sample mean at or above 1000:

p = Normalcdf( 1000, 99999999, 896, 24.6) = 0.00001

This means that if the sample of 50 SAT Prep scores truly were the same as every other sample of 50 scores, the likelihood of getting a sample mean as high as 1000 would be 0.00001or .01% – highly unlikely!

· Since the p-value (.0001) is well below the level of significant (0.5%), we reject the null hypothesis in favor of the alternative hypothesis. We now assume that the mean score of the SAT Prep students is higher than the mean SAT score for the population.

Inv 2, Activity H: Hypothesis Testing

Evaluating a hypothesis test can be done using the graphing calculator or online tools. There are two hypothesis tests to be familiar with for testing sample means. This Activity introduces you to these two tests. Exercise H1: Complete a hypothesis test comparing the Sample Mean for SAT Prep to the known population mean for the SAT using Hypothesis Test Mean

. This video describes how to do the T-test on a TI-84 or 83.

This link provides a calculator for the Hypothesis Test Mean.

http://easycalculation.com/statistics/hypothesis-test-population-mean.php Exercise H2: Complete a hypothesis test comparing two sample mean SAT Scores using Hypothesis Test 2 Means

. This video describes how to do the T-test for two means on the TI-84 or 83.

Submit Problem Set #2

Please submit your solutions to problem set #2 via the the hyperlink in the title of this item by Sunday. You were introduced to this assignment in Module #3.

Comparative Study Progress

You should be gathering data for your Comparative Study. Next week , you will share your progress with your group through a progress summary. Your summary should include:

· The results of your data collection efforts. Did your collection tool work as you expected? What surprises or challenges did you encounter? Are there are sources of bias in your data? What might you do differently “next time”?

· An initial analysis of your data. This need not be complete; it should give the group an idea of what you have done, what you intend to do, and any questions or concerns you have about the analysis.

· Specific questions or topics you would like feedback on.

Post your summary to your group's discussion board NO LATER THAN Week 6, Friday, 10PM EST.

Week 5 Journal Reflection

To access your Journal, please click on the Journal link on the left side navigation bar. This journal will be a private document where you will communicate your thoughts to the instructor. Only you and the instructor can see what you write.

Each week, you will conclude the week’s activities with a formal reflection of your work. You should not just repeat things from previous discussions. Focus on what you learned that made an impression, what may have surprised you, and what you found particularly beneficial and why. Specifically:

· What did you find that was really useful, or that challenged your thinking?

· What are you still mulling over?

· Was there anything that you may take back to your classroom?

· Is there anything you would like to have clarified?

Your Weekly Reflection will be graded on the following criteria for a total of 5 points:

· Reflection is written in a clear and concise manner, making meaningful connections to the investigations & objectives of the week.

· Reflection demonstrates the ability to push beyond the scope of the course, connecting to prior learning or experiences, questioning personal preconceptions or assumptions, and/or defining new modes of thinking.

Your reflection should be about 250 words and no more than 500 words. This should be done by Sunday, 10 PM EST.

Problem Set #3 (Due in Module 8)

Review and download Problem Set #3 Problem Set #3 - Alternative Formats

. Be prepared to submit your solutions in Module 8.

Checklist

Investigations & Other Learning Activities

· Complete Investigation 1

· Complete Investigation 2

Assignments

· Begin to gather & analyze data for Final Project. (Progress summary post due Week 6, Friday)

· Begin work on Problem Set #3 (due Week 8)

· Submit Weekly Reflection by Sunday

· Submit Problem Set #2 by Sunday, 10 PM EST

Discussions

· Discussion: Exercise C3: Bobby's Scores Post by Tuesday, 10 PM EST

· Respond by Thursday, 10 PM EST

· Discussion: Exercise E3: SAT Prep Post by Friday, 10 PM EST

· Respond by Sunday, 10 PM EST