Final Statistics Review

There's not much to report on today. Students turned in their project papers and there were some interesting  topics that I am looking forward to reading. The review was mostly students working through problems of their choice and me walking around answering questions and providing guidance as needed. Students primarily worked in groups and were conscientiously working together to discuss how to work through problems and compare results. I figured given the compressed nature of the summer semester that allowing students time to work through problems would be the best use of time. While some students still seemed lost, many started to see how straight-forward many of the former quiz and test questions really were.

Statistics Review - Two Sample Hypothesis Tests for Means and Probability

Today was the first of two days of review. The quiz on one-sample hypothesis tests for means indicated that students were still struggling with this idea. A larger segment than I would care to see had found the sample mean and simply compared it to the hypothesized value. There was no referencing the sampling model nor the probability that they would draw a sample with the characteristics they saw.

To start things off, I reiterated the ideas behind statistical analysis and inference. I drew a large cloud on the board and said this was the population we were studying. Inside the cloud I drew a small circle and said this represented the sample we drew. The idea is to draw conclusions about the entire population from the small snapshot that we took via our sample.

I drew more circles throughout the population cloud, some of which overlapped. Each sample we draw provides a different snapshot of the population. We need to account for every possible sample we draw. This is where the sampling distribution model comes into play. The sampling distribution model describes what we should expect given the sample size we have drawn from the population.

We cannot simply compare our sample mean to the hypothesized population mean. Sure, this time it may be greater than our hypothesized value, the next drawn sample could show the mean less or more. Every sample could and probably will be different. We need to take the one sample we drew and use that to draw conclusions about all the possible samples that could be drawn and from this draw a conclusion about the population we are studying.

With that said, we went through the quiz questions, working through results. As students did this, I passed out two sets of die; one of the dice was colored green and the other blue. I asked students to roll both die six times and to count the number of times each colored die won. As soon as we finished going through the quiz problems I collected the die and we started working with the data that was generated.

I described the situation to the entire class, since not every student had a die set passed to them. I described the data collected and asked them what they expected to happen. Several students said they expected the number of wins for each die to be the same. From here I asked them to state null and alternative hypotheses.

H₀: The mean number of blue dice wins equals the mean number of green dice wins (μ_b = μ_g)
H_a: The mean number of blue dice wins does not equal the mean number of green dice wins (μ_b ≠ μ_g)

I then asked the students if the samples we had were independent or not. Here there was some disagreement. One student said that if you knew the number of blue wins then you would also know the number of green wins. Another said that wouldn't necessarily be true since you could have ties. I then brought up that you might not know the exact number of wins but you certainly would know the  maximum number of wins. This indicates that the two samples are not independent of each other. Students need to carefully consider the samples they draw as to whether or not there is any direct connection between the two samples.

I asked students to analyze the results of these samples and draw a conclusion about the data. There was a lot of confusion about how to analyze the data. With a matched pair test, you take the difference in values between the matched numbers and then analyze the differences as a single sample.

Their analysis resulted in a p-value of 0.04. Students were somewhat shocked by this result. The conclusion was to reject the null hypothesis and conclude that the number of wins for the blue die and the green die were different. I told them not to be too shocked as the green die had two number fives on its face and no number two.

Next, I asked several students to roll an individual die 3-4 times and record the values rolled. I made two columns on the board, one for blue die rolls and one for green die rolls. Students recorded their results on the board. I then asked if these two samples were independent. The response this time was a resounding "Yes!"

I asked what an appropriate null hypothesis would be and what should the alternative hypothesis be. Because students now knew that the green die was "unfair" they concluded that the alternative should be the green die rolls would exceed the blue die rolls. We had

H₀: The mean roll of the blue die equals the mean roll of the green die  (μ_b = μ_g)
H_a: The mean roll of the blue die is less than the mean roll of the green die  (μ_b < μ_g)

I asked students to analyze the samples and draw a conclusion. In this situation, students had questions about how to calculate their degrees of freedom. While the text does provide a formula, it is long and complicated and a calculator or computer can easily do the computation for you. I told students this and for the few students who did not have a calculator that would figure the degrees of freedom for them, I told them to simply add the degrees of freedom for the two individual samples, which basically provides an upper-bound to the number of degrees of freedom. For most reasonably large samples the results will not be affected.

For this analysis, students calculated that the p-value was 0.056. Using a 5% significance level we would fail to reject the null hypothesis. A student pointed out that this was the wrong decision. For our sample, we committed a Type II error. This error was discussed versus a Type I error.

To confirm that we, in fact, committed a Type II error, I asked students to calculate the expected value (mean) of rolling the green die. This entails constructing a probability model and determining the expected value. Many students struggled with this but were able to complete the task with some assistance. I then asked the class to compute the expected value of the blue die. The values were 4 and 3.5 respectively. So, we did indeed commit a Type II error.

I then asked students to determine the probability of the green die roll exceeding the blue die roll. Students seemed baffled as to how to proceed. I told them they needed to consider all the possible outcomes and which of those met the desired criteria. Listing out the 36 possible outcomes, it becomes readily apparent that there are 18 outcomes when the green roll exceeds the blue roll, resulting in a P(green > blue) = 0.5. Proceeding further I asked what P(green = blue) equals? Students used their outcomes and came to a result of 1/6. This baffled them for some moments as they seemed to expect it would be different.

I asked students to pick 2-3 questions from past quizzes and exams or chapter review problems that they did not know how to complete. We will use these as a basis for further review next class.

I concluded by working with students who had questions about the project reports. Most of the reports that I was shown looked to be on the right track. A few needed to focus more on comparing the two data sets rather than simply viewing them as two distinct, non-related entities. I am looking forward to seeing what they produce for their final versions.

Hypothesis Tests of Means Using Two Samples

The focus today was on testing hypotheses about means when you have two samples. We also had a quiz today covering confidence intervals and hypothesis test of a mean for a single sample. I reviewed material that students had questions about, working through a couple of examples that addressed their questions.

Before taking the quiz, I covered hypothesis testing for paired samples. Paired samples include before and after measurements, working with siblings or spouses, and, as in our first day experiment, comparing things such as dominant and non-dominant measurements.

I used the coin stacking data from our first class, which is readily recognizable as paired data. I asked the class to formulate null and alternative hypotheses for this problem. We discussed a one-tail alternative of the dominant hand performing better and a two-tail alternative of the two hands will perform differently. This totally depends on the question you are addressing. If you are trying to answer the question, "Will your dominant hand stack more coins than your non-dominant hand?" then your alternative hypothesis will be the mean dominant hand stack is greater than the mean non-dominant hand stack, a one-tail upper test. If you ask the question, "Will my two hands stack the same number of coins?" then your alternative hypothesis is that  the mean dominant hand stack is not the same as the mean non-dominant hand stack, a two-tail test.

To perform a paired sample analysis, you take the difference between your individual paired data values and treat these differences as a single sample. All the tests, confidence intervals, and results work exactly the same as a single sample test, except that the results are for the mean of the differences.

After the quiz we covered the situation where you work with two independent samples. In this case the sample sizes may be different and there is no connection between the two samples. The combined sampling model we use is still a t-model with a mean of the difference of the two sample means. The standard deviation is calculated by adding the variances of the individual standard errors and then taking the square root. I showed students how to use their calculator to perform this test. The calculator or software will provide the degrees of freedom. As I told the class, there is a formula to determine the degrees of freedom but it is not something you want to calculate by hand. We worked through an example involving operation times using two different protocols and conducted a hypothesis test and created a confidence interval.

Students are working on finishing their project papers over the next few days and trying to get ready for the final. The project is due next week and the final is one week from today.

Below is an outline of today's lesson with italicized comments enclosed in square brackets [like this].

—Inference for 2 Population Means

   ·         What would you try if you wanted to compare two samples, such as heights of males and females? [reversed the order of presentation since the paired sample was good review for the quiz]

   ·         If the samples are independent we can take the difference between the two

      o   Will still have a t-model

      o   Mean is difference of the means

      o   Standard error is determined by adding variances [due to limited time just stated this result versus having students work through sample data to see that this was the case]

      o   Degrees of freedom—let your calculator or computer figure it out

   ·         Proceed with same process as single sample testing except now you are working with hypotheses about the difference in values

      o   Confidence intervals

      o   p-value and inference

   ·         Practice two independent samples [used example in book for surgery times; samples were of different sizes, which students typically bring into question. The point is were are testing a hypothesis about the difference in the means of each sample, the individual sample sizes will only affect the variances and hence standard deviations when the sampling models are combined.]

   ·         What if two samples are not independent, such as coin stacking?

      o   Take differences in values since they are paired

      o   Now have a one-sample situation

      o   Proceed as if dealing with a single sample and infer about difference

         §  The differences should be roughly symmetrical, the raw data doesn’t matter

   ·         Practice paired sample [Used coin stacking from the first day; this makes a nice closure to the entire semester.]

Statistical Testing Errors

The focus today was on statistical testing errors, specifically the nature and consequences of making Type I and Type II errors.

The class was asking good questions about hypothesis testing, so I took some time to go through these. I also discussed the differences between creating confidence intervals and conducting hypothesis tests. For confidence intervals, we are interested in estimating a value. We use our level of confidence to establish the range of values that "make sense" based upon the sample data that we collected. For hypothesis testing we are simply giving a yeah or nay to a specific value (the hypothesized population parameter). If our well-collected data is consistent with the null hypothesis it's a yeah vote, if our data is inconsistent with the null hypothesis it's a nay vote.

I used a problem from the last lesson regarding average apparel expenditures to illustrate this idea. We worked through the hypothesis test and rejected the null hypothesis. Since the null hypothesis was tossed out, it is only natural to wonder what a more reasonable value for the population mean would be. A confidence interval is used to establish a new range of possible values with our best guess simply being the mean of our sample.

From here I moved to the issue of statistical testing errors. I have a chart that I can use to discuss the different errors and how values like the level of significance and power of the test relate.

We spent quite a bit of time discussing these and the ramifications of these errors within the context of a problem. I like to use a drug test as an example. Would the drug manufacturer rather see a Type I or Type II error made and why? What about the drug manufacturing regulator? Consider the scenario of a company evaluating a sales training program: what happens if they make a Type I error and what happens if they make a Type II error? These questions and discussions really help bring meaning to the errors.

After this, I pointed out graphically the relationship between significance level and power. Basically if the level of significance is reduced (making the value smaller) the power is also reduced and vice versa. The only way to reduce the level of significance without affecting power is to increase sample size.

We then looked at non-parametric testing methods for when you have small sample sizes that are skewed. This was to make students aware that other methods are available for non-conforming data sets.

I spent some time showing students how a calculator assists in calculating confidence intervals, t-scores, and p-values. We practiced using the calculator on a couple of different problems; one we had worked before and one new one.

I also passed out sample project reports and rubrics and gave students time to look through the samples to see how items in the report represented the requirements of the rubric.

Finally, it was course evaluation time. I expect that I will be scored low on a number of fronts as many students have struggled and their grades are lower than they would like.

Below is an outline of today's lesson with italicized comments enclosed in square brackets [like this]

·         Test errors – may be wrong and won’t know because usually won’t know population values

   o   Show diagram of correct/incorrect hypotheses versus decisions and errors

   o   Discuss alpha, beta and power

   o   Show chart connecting relationship between alpha and beta

   o   Discuss what happens if sample size increases and its impact on alpha, beta, and power

·         What if you have a small sample that is skewed?

   o   Use the Wilcoxon signed-rank test

   o   Explained in section 9.6 of the book [provided a slide with example to describe the process and how to calculate the test statistic]

Hypothesis Test for One-Sample Mean

Today we transitioned from creating confidence intervals to conducting hypothesis tests. While mechanically these are closely related, there are a few significant points to keep in mind:

1. Confidence intervals are based upon a model developed using our sample statistics. We use our sample data to estimate a probable value for the true population mean.
2. Hypothesis tests are based upon a hybrid model that is centered on an assumed value for the population mean but making use of our sample standard deviation to create a standard error for the model.
3. Confidence intervals are equivalent to a two-tailed hypothesis test, we are excluding extremely small and extremely large values.
4. Hypothesis tests can be one-tailed or two-tailed, there is not an equivalent confidence interval for a one-tailed test.

Will get into test errors and their meanings next class. Will also provided project write-up examples so students better understand the end product they need to produce. Ideally I would have a range of samples and have students rank sort the papers and we would discuss if they were A-, B-, C-level or worse. Unfortunately, I only have high quality examples, so I'll have them read through and look for characteristics from the scoring rubric that are demonstrated in the papers.

Below is the outline of today's lesson with italicized comments enclosed in square brackets [like this].

—Hypothesis Test for One Population Mean

   ·         Compare hypothesis testing to trial

    ·         Provide hypotheses mentor texts [mentor texts are problem statements along with appropriate hypothesis statements that show what these hypothesis statements look like. Four examples were provided that included one-tail upper, one-tail lower, and two-tailed alternative hypotheses.]

      o   Use always, sometimes, never [In the context of the problem statement, what do you always see, sometimes see, and never see in a hypothesis statement.]

      o   Discuss what was seen [At this point students have a better sense of what is being tested]

   ·         Hypotheses are statements about population parameters

      o   Provide examples for H₀ and H_a  [Examples included commentary on hypothesis statement structure and content.]

   ·         Significance level and alpha values

      o   Discuss picking a probability for which you would reject your hypothesis—alpha level

      o   Connect to critical values and significance level

   ·         Usually don’t know the population standard deviation so use t-test [Just stated this was the case and we were working with t-model]

      o   Works the same way

      o   Must include degrees of freedom so know which t-model was used [The degrees of freedom (df) specifies the exact t-model being used from family of all possible t-models]

      o   Can use with moderate to large samples, even if data is not symmetric [The t-test is a robust test that works reasonably well with even relatively small samples that are somewhat skewed. Provided students a rule of thumb for different sample size ranges but basically said unless the sample is small and highly skewed to not worry about it.]

   ·         Discuss meaning of p-value

      o   Conditional probability—given the null hypothesis is true, what is the probability of seeing the random sample that was drawn?

      o   The more unusual the sample the smaller the p-value—it’s not likely to be seen

      o   Conclusion: either we drew a bad sample or the null hypothesis is wrong

      o   If we followed good data collection procedures than the conclusion must be the null hypothesis is incorrect

   ·         Use sibling data and have students calculate a p-value

      o   What if our class is viewed as a sample; does sample support claim that the mean number of siblings in the US is 1.86?

   o   Work through problems in the book

Confidence Intervals, Margin of Error and the t-model

We continued looking at confidence intervals today, specifically focusing on critical values, margin of error and the t-model. We used the class sibling data set and found critical values and confidence intervals for 80% and 98% confidence intervals. We then used these intervals to discuss margin of error. We were able to use the formula for margin of error to then see what sample size was needed in order to obtain a margin of error that was roughly half of the current margin of error.

From here we moved to the t-model. This is typically what you end up working with because you don't know the mean nor the standard deviation of the population. It is natural to replace the population standard deviation with the sample standard deviation. However, this introduces more variability into the model, specifically, every sample size results in a slightly different standardized model. This family of models is know as t-models and the specific family member used is determined by the sample size. Specifically, the degrees of freedom (df) is one less than the sample size, i.e. df = n - 1.

We worked through the same examples we used before but no longer assumed the population standard deviation and and sample standard deviation were the same. In this case, since we are using the sample standard deviation, s_x, to estimate the standard deviation, we distinguish this by calling the t-model's standard deviation a standard error. Otherwise, t-models behave and are used similar to a normal model.

We were in the computer lab today, so I was able to show students how to conduct their analyses using the Minitab software. Although we won't cover hypothesis testing until next class, there was enough foundational pieces in place that students could follow the general idea and how they could proceed with their projects.

Below is an outline of the lesson along with italicized comments enclosed in square brackets [like this].

·         Margin of error

   o   Length of ci/2 or value of what is being added and subtracted to mean [actually looked at value being added/subtracted first and then mentioned the interval length divided by 2 gives same value]

   o   Calculate margin of error for two practice confidence intervals

·         Estimate sample size needed

   o   Solve algebraically starting with ME value [worked through a couple of problems using different confidence levels and margin of errors]

·         Typically don’t  know the population standard deviation, just like we don’t know the population mean – what can we do

   o   Use sample standard deviation for population standard deviation

   o   This introduces more error

      §  No longer have standard deviation have standard error

·         t-model

   o   looks like normal model, same basic properties

   o   as sample size increase looks more and more like a normal model

·         t-table

   o   in book and handout [book did not include a t-table that provides df and t-score and then shows percentage in upper tail]

·         Confidence intervals

   o   Same as before except use t-score and t-table instead of z-score and normal table

   o   Practice problem as before but assume don’t know population standard deviation

   o   Use calculator [introduced this after making use of tables on several intervals]

·         Sample size for t-model – worst case is to use z-score since don’t know sample size

   o   Can get a better estimate after by recalculating [didn't get into this, just have them estimate using z-score]

Confidence Intervals for Population Means Assuming Known Variance

Today we transitioned from looking at the sampling model of sample means to working with this model to construct confidence intervals. I first went through looking at subjective confidence intervals. This allows students to see that as the gain confidence their interval widens to capture more values. It also allows me to communicate how confident we are that the true mean lies within the interval created.

From here I moved to asking students to consider how we could make use of the central limit theorem and the sampling model. I used the class's sibling data and assumed that the class standard deviation was in fact equal to the population standard deviation. We created the sampling model and I drew a graph of this model on the board with the mean and standard deviations labeled above and below the mean. I then asked students what would be the 68.26% confidence interval? Once students grasped that it was just the interval from one standard deviation below to one standard deviation above the mean, they were much quicker about determining the 95.44% confidence interval. I hadn't labeled the graph beyond two standard deviations, so I asked them what the 99.74% confidence interval would be? Most students found the new end points although a few still had questions.

I pointed out that what we had done to construct the confidence interval was to add or subtract a integral multiple of the sampling model standard deviation away from the mean, i.e. μ ± Nσ where N = 1, 2, 3. This is all well and good but saying we are 95.44% confident seems a bit much; it would be nicer to have our confidence intervals at integral values rather than the multiples of the standard deviation.

I asked students what z-scores would result in a 95% confidence interval rather than a 95.44% confidence interval. This threw many of them for a loop. It was obvious that they still were not comfortable working  with normal models. After a bit more guidance students determined that z-scores of -1.96 and 1.96 are what were needed. I told the class the value of 1.96 was called a critical value as it was the z-score value that was needed to construct a 95% confidence interval.

For homework, I asked students to determine the critical values for a 90% and 99% confidence interval.

Below is the outline of today's lesson with italicized comments enclosed in square brackets [like this].

—Confidence Interval for One Population Mean

   ·         Confidence questions

o   Want students to realize as they gain confidence the spread of values increases

   ·         Take a sample, what is estimate of mean?

o   Use CLT to say best estimate is mean of sample

   ·         How can you account for sample variation?

o   Can the normal model help?

   ·         Every sample creates a different confidence interval [showed graphs of 20 confidence intervals developed from 20 samples, able to indicate that the interval may not contain the true population mean]

·         Calculating confidence intervals

o   Find z-scores, multiply by standard deviation, add and subtract from sample mean [focused on just adding and subtracting integral values and had students use practice problem below to first construct a 95.44% confidence interval]

o   Use calculator [decided to hold off on this until next class]

   ·         Practice – age of civilian workforce

o   Calculate 90% and 95% confidence intervals [used 95.44% confidence interval and then asked students to find 95% critical value, will continue this next class]

Building understanding of the Central Limit Theorem

Today I focused on building understanding of the Central Limit Theorem and how it can be used. I used a series of problems to help students work directly with sampling distributions and then moved to using the results of the central limit theorem.

To start things off, we had a data set of 5 basketball players and their heights. We looked at all possible samples of two individuals. Students found the mean height for each of the 10 samples. They calculated out the mean and standard deviation of their 10 samples and we compared that to the mean and standard deviation of the population. We also constructed a histogram of the sample mean distribution. We then compared these results to the theoretical model. In this case, the means aligned and the standard deviation was off slightly but it was close. As I explained to the class, models are useful to help explain behavior but they may not be accurate. We also calculated the probability of sample mean equaling the population mean and the probability of the sample mean being  within 1.0 inches of the sample mean.

Next, we looked at a couple of situations, discussed the population and variable of interest and then compared the sampling distribution models for two different sample sizes. This helped students to get comfortable with specifying the sampling distribution model we were using.

Finally, we looked at two problem situations that assumed specific population parameters and then asked what percent samples of a certain size would fall within given ranges. This basically gets back to finding z-scores and working with a normal model. Students still wanted to use the population standard deviation when determining probabilities, but with some reinforcement, most students seemed to understand why the sampling distribution had a different standard deviation.

Afterward, we went through a review activity that I use often and described in a previous post. The second mid-term is tomorrow, after which we will begin developing the concept of confidence intervals.

Sampling Distributions and the Central Limit Theorem

Today we worked through some sample problems using the normal model. I also showed students how to find probabilities using their calculators and how to find a z-score from a percentile on their calculator.

Next, we looked at the idea of sampling distributions and the Central Limit Theorem (CLT). This took about 45 minutes to go through and hopefully established some foundation of the concept and form of sampling distribution models. Next class we'll work through some problems and then review for the upcoming mid-term.

Below is the outline of the lesson with italicized comments enclosed in square brackets [like this].

—Sample Distribution of Sample Mean

·         Take random samples of 3 students and calculate mean height [Took 7 random samples and compared means to the class mean.]

   o   What happens? [Did any sample have a mean the same as the class? Were the means all the same?]

   o   Sampling error – each sample yields a different mean [Each sample differs, this is sample variability.]

·         What would you expect to happen if took larger samples? [Students thought spread would decrease; continue to increase sample until sample entire population then how much variation-none!]

   o   Mean stays the same but the spread decreases

·         What will the mean of the means be? [Mean will be close to population mean, more samples of larger size provides even closer results.]

   o   Population mean

·         What will the standard deviation be?

   o   Population standard deviation / sqrt(n) [Students realize variability decreases, just now how.]

·         What is distribution of the sample means?

   o   Create a histogram [Created histogram of the seven sample means, unimodal, slightly skewed but symmetric enough-call this nearly normal. A normal model can be useful in helping to understand behavior.]

·         CLT

   o   Discuss mean and standard deviation of sampling distribution

·         A look at sampling distributions [Showed different sample populations, graphs of large number of samples for various sizes; all look normally distributed but the standard deviation gets smaller.]

·         Calculate for siblings

   o   Use sample size of 5 then 10 then 20 to demonstrate law of diminishing returns [Started by having students calculate population (the class) mean and standard deviation. Showed that the sampling distribution model would be of the form N( mean, std dev / sqrt(sample size) ). Next asked students to calculate the sampling distribution standard deviation for a sample size of 5, then 10, then 20. Showed that doubling size each time has less and less effect on decreasing the size of  the  sampling model standard deviation. This is the law of diminishing returns. Students wondered why the standard deviation of the sampling model was so much smaller than the standard deviation of the population. It was a good opportunity to remind students we are looking at the distribution of means calculated from drawn samples. It turns out that the standard deviation of this sampling model is connected to the standard deviation of the population but as the samples we select increase in size the variation we see in the means of the samples decrease until we sample the entire population. At this point every sample will be the same and there will be no sample variation in the sample means.]

z-scores and the standard normal model

Today focused on finding z-scores and working with the standard normal model. This lesson went pretty much as planned. The only thing I didn't cover was how to find normal probabilities on the calculator. We'll do this at the start of next class. The news that the calculator could calculate values was greeted by good-natured groans and curses.

Below is an outline of the lesson with italicized comments enclosed in square brackets [like this].

—Normal Model

·         Which is more unusual?

   o   Discuss as class, how could these be compared? [Compare someone walking into the room who is 82" tall or with a show size of 4.]

·         Quick investigation [Went as described, allows to see students where z-score is derived.]

   o   Pick 5 different values

   o   Calculate the mean and standard deviation

   o   Now, add a value of 9 to each original value

      §  What happens to mean and standard deviation

   o   Next, subtract a value of 9 from each original value

      §  What happens to mean and standard deviation

   o   Multiply each original value by 9

      §  What happens to mean and standard deviation

   o   Divide each original value by 9

      §  What happens to mean and standard deviation

·         What will the mean and standard deviation be if you subtract x-bar from each datum

   o   Mean is zero, standard deviation unchanged

·         What will the mean be if you divide transformed data by the standard deviation

   o   Mean is zero and standard deviation one

   o   What are the units of this transformed data [Students need to realize that z-scores are unitless.]

   o   This is known as a z-score

·         What is the meaning of the z-score

   o   Tells how many standard deviations a value is away from the mean

·         Going back to opening question, what if we calculate the z-score for both items? [Didn't have data, used their 5 numbers that they picked.]

   o   Can compare to see which is more unusual

·         Work with z-scores on worksheet

·         The distribution of z-scores is known as the standard normal model [Used 15 z-scores from their randomly selected numbers and created a histogram. Students described distribution as unimodal with a slight left skew.]

   o   Mean is zero and standard deviation is one

·         Calculate areas under curve [Worked through several examples using a z-table.]

   o   Using table

   o   Using 68.26, 95.44, 99.74 rule

·         Practice with normal model worksheet