A Collection of Demonstration Applets for New Teachers

A Collection of Demonstration Applets for Introductory Statistics

Following is a description of a collection of applets that help demonstrate some of the important ideas in introductory statistics. Most applets can be used for simple in-class demonstrations in classrooms with computer projection capability. In addition to the descriptions, teaching strategies have been suggested for some applets. Three of the applets have an associated investigation activity that can be completed by students either in a lab setting or working on their own.

Histograms and Bin Widths

http://www.rossmanchance.com/applets Click on Histogram Bin Width

Pick a data set from the drop down menu. Use the red slider to manipulate the bin width. Notice how changing the bin width can result in substantially different looking histograms.

New data can be graphed by clicking the Edit/Paste Data button, pressing delete to clear the existing data, and entering new data values one-by-one.

http://www.stat.sc.edu/rsrch/gasp/ Under Educational Procedures, click on A Histogram Applet

Use the slider to manipulate the bin width for a histogram displaying Old Faithful eruption length data.

Dotplots and Summary Statistics

http://www.rossmanchance.com/applets Click on Dotplot Summaries

Begin with the first exploration activity to get information about how the applet works. Students can estimate location of mean and median and the size of the standard deviation and interquartile range and then observe the actual values. Points can be dragged to observe the effects of extreme values on summary statistics.

New data can be graphed by clicking the Edit Data button.

Least Squares Regression Demonstration

http://www.dynamicgeometry.com/javasketchpad/gallery/pages/least_squares.php

This applet can be used to demonstrate the least-squares criterion and to help students visualize how only one line “best fits” the data. The six points labeled P1 through P6 represent data points. A line is drawn through the points, and from each data point a vertical segment is drawn to the line and a square is constructed. The red square in the lower right corner represents the total area of the six squares (i.e.,

Michael Legacy, NCSSM Summer Writing Project 2007 revised by Gloria Barrett, NCSSM, January, 2013 the sum of the squared vertical distances.) By clicking on the red dots labeled “y-intercept” and “slope”, the student can adjust the y-intercept and slope of the line in an attempt to find the one that minimizes the sum of the areas of the squares. The line that yields the absolute minimum is the least squares regression line for the data. (This applet does not have the option to display the true least squares line.)

It is also possible to move the points P1 – P6, by clicking on the red dot on the square.

Regression by Eye

http://onlinestatbook.com/stat_sim/reg_by_eye/index.html In general, you can get to this and other demonstration applets by going to http://onlinestatbook.com/rvls

This applet lets you estimate the regression line by placing the mouse at a starting position, holding it down and drawing a line. When you release the mouse button, the mean square error MSE (which is the average squared deviation of points from the line) is displayed. You can draw another line and see if you can lower the MSE. Each line you draw is in a different color, as is the matching MSE. At any time, you can see the MSE for the best fitting line, by selecting the "Show Minimum MSE" option.

To see the best fitting line displayed (by the criterion of least squares), select the "Draw regression line" option. This line will be in black.

Five possible values for the correlation are listed. Students can guess which one of them is the correlation for the data displayed in the scatterplot. To see the correct value, click on the "Show r" button.

Click the "New Data" button to try again with a new data set.

Teaching strategy: Let students attempt to draw the least squares line; often they will begin by drawing a line that approximates the major axis of the ellipse formed by the data. Repeated efforts demonstrate that making the line less steep so that it better approximates the mean y-value for each x-value reduces the MSE.

Demonstrating Influential Points

http://www.stat.sc.edu/~west/javahtml/Regression.html

The applet is designed to help students visualize how adding (or moving) a point can affect the regression line. Original points are given along with the resulting least-squares regression line and the correlation (in black). A new point may be added to the plot by clicking the mouse button somewhere within the graph space. Resulting changes to the line, the regression equation, and correlation are given in red.

Teaching strategy: Let students investigate adding points (or you add points if you are doing this in a classroom setting) and have them summarize the results (effect on slope and correlation) when points are added near the center of the data range versus when points are added far from the center in the x-

Michael Legacy, NCSSM Summer Writing Project 2007 revised by Gloria Barrett, NCSSM, January, 2013 direction. The students should see that adding a point near the regression line barely changes the existing line. Now add a point near the center of the data range (in the x-direction), but fairly far above the existing line. Notice that the line moves upward but stays somewhat parallel to the existing line. Now add a point at the edge of the data (in the region of minimum and maximum x-values) and far from the line. This additional point can substantially change the slope of the regression line.

Also have students add a point in line with the existing regression line, but outside the data range. Does the correlation increase or decrease with the addition of this point? Does the slope of the regression line change substantially? [The slope should not change much but the correlation should increase.]

Probability (long-run relative frequency)

www.whfreeman.com/tps4e Click on Statistical Applets and then Probability

When you toss a coin, there are only two possible outcomes, heads or tails. In this applet you can toss a coin multiple times and observe the long run behavior of the proportion of heads. Set the number of tosses and click the Toss button to randomly toss the coins. The vertical heads/tails bar shows you the cumulative proportions. The graph shows the cumulative proportion of heads after each toss.

Notice how the proportion of tosses that produce heads (or tails) can be quite variable at first, but will eventually converge to the true probability. Note as well that there can be long runs of one outcome. For example, it would not be all that unusual for a long run of heads to occur in 1000 tosses of a coin. You can also change the probability of getting heads by entering a new value and clicking Reset.

Conditional Probability

www.stat.tamu.edu Search on conditional probability applet

Students can manipulate a Venn diagram and observe how a variety of probabilities change.

Using a Normal Approximation for the Binomial Distribution

http://www.whfreeman.com/tps4e Click on Statistical Applets and then Normal Approximation to Binomial Distributions

As n increases, the binomial distribution with n trials and probability of success p can be better and better approximated by a normal distribution. This applet uses sliders to change both n and p. You can click and drag a slider with the mouse.

The histogram shows the binomial probabilities. The vertical black line marks the mean μ = np. The red curve is the normal density curve with the same mean and standard deviation as the B(n,p) distribution ( m = np and s =np(1 - p ) ). Use the slider to set lower and higher values of p.

Michael Legacy, NCSSM Summer Writing Project 2007 revised by Gloria Barrett, NCSSM, January, 2013 Teaching strategy: Let students investigate various values for n and p. What values of n and p seem to yield a binomial distribution that can be reasonably approximated by a normal distribution? Students should note that the binomial probability distribution becomes more skewed as p moves farther away from 0.5. In order for the binomial probability histogram to look more like a normal curve, students should discover the need to increase n as p becomes more extreme.

This activity helps demonstrate why the conditions check on large sample size for inference for proportions involves verifying that np >10 and n(1- p ) > 10 . Note that some books suggest checking against 5 or 15.

Investigating Sampling Distributions

This investigation directs students to explore various aspects of sampling distributions using the Rice Virtual Lab in Statistics simulation at www.onlinestatbook.com/rvls. See activity titled Investigations of sample statistic distributions using an on-line simulator.

Investigating a Sampling Distribution of Sample Proportions

http://www.rossmanchance.com/applets Click on Reeses Pieces

This applet lets you build a sampling distribution of sample proportions using Reeses Pieces. The parameterq represents the proportion of orange candies in the population in the candy machine. To start, set q =.6, a sample size of n =10 , and the num samples to be drawn at 1. Click the Animate box and then click the Draw Samples button. Ten candies will fall from the machine into the bins below sorted by color. The red pˆ is the proportion of the sample that is orange. The proportion of orange candies in the sample is also shown on the dotplot. Click Draw Samples a few more times to begin building the sampling distribution of sample proportions. (The use of the Animate button slows this procedure down for students to actually see the process of building the distribution.) Change the num samples to 10 and click Draw Samples. Once students are clear on the process, deselect the Animate button and change num samples to 50. Also select the plot normal curve box. Click Draw Samples several times. How closely is the sampling distribution of sample proportions approximated by the normal curve?

Teaching Strategy: Vary the investigation with different values of n andq . Click Reset each time you want to clear the sampling distribution dotplot. Have students determine what values of n and q yield a sampling distribution that is approximately normal.

Michael Legacy, NCSSM Summer Writing Project 2007 revised by Gloria Barrett, NCSSM, January, 2013 Understanding Confidence Level and More (why we need t when s is unknown)

http://www.rossmanchance.com/applets Click on Simulating Confidence Intervals for a Population Parameter

This applet constructs confidence intervals using several methods to estimate a population mean (µ) or a population proportion (π).

Confidence intervals for a population proportion - (Note that the Wald method is the one taught in AP Statistics.) Set the value of the population proportion π and the sample size n with values you choose. Set the initial confidence level at 80%. Enter 1 next to Intervals; then click the Sample button for a single SRS. Note that the upper right graph shows a bar chart with the sample results and pˆ value. The pˆ value is displayed in the lower right graph, and the confidence interval based on this pˆ is shown in the graph on the left. The dot marks the sample proportion, and the lines on each side of the dot span the confidence interval. Click the Sample button a few more time to observe that once in a while an interval “misses” the true proportion π. Misses are shown in red. A running total of the “success rate” is displayed in the lower left corner. The effect of changing the confidence level on the size of the interval can be demonstrated by entering a different confidence level and clicking the Recalculate button.

Click on Reset, enter 50 next to Intervals, then click the Sample button. The dotplot in the lower right corner displays the 50 pˆ values. Red dots are displayed for sample proportions that result in unsuccessful confidence intervals. Press the Sample button a few more times so that students can see that the running total is very close to the confidence level. For example, when the confidence level is set at 80%, in repeated sampling about 80% of the samples drawn will yield an interval that contains the value of the population proportion.

Students can investigate the effect of increasing the confidence level to 90%, 95%, and 99% by changing the confidence level, clicking the Recalculate button, and observing how the lengths of the confidence intervals and the success rate change.

Confidence interval for a population mean – Choose Means and “z with sigma” (in the upper left corner). As an example, we can sample from the population of professional basketball players. Choose µ = 73.5 and σ = 2.3 (the mean and standard deviation of the player heights) and set n = 5. Begin with one interval at a time so that students can observe the sample values, sample mean, and confidence interval. Then change the number of intervals to 50 and repeat the steps followed above. It is valuable for students to observe the red dots associated with unsuccessful samples on the dot plot. In this situation, all misses are associated with exceptionally low or high sample means.

Now change the computation to “z with s.” Again, begin with 90% confidence level and one interval at a time so that students can observe that the lengths of these intervals are no longer uniform. Why? Then change the number of intervals to 50 and press Sample several times. Note that the success rate does not seem to be approaching 90% - it is much lower. Students may also observe that some of the misses are not associated with exceptionally low or high sample means. Finally, change the computation to “t with s” so that students can verify that the success rate does match the confidence level when we make the adjustment to use a t-distribution critical value rather than a z-distribution critical value.

Michael Legacy, NCSSM Summer Writing Project 2007 revised by Gloria Barrett, NCSSM, January, 2013 Investigating Errors and Power in Significance Tests for Means (s is known)

Starting with a problem situation about a packaging machine, students simulate the probability of Type I and II errors by using a power applet at http://wise.cgu.edu/power/power_applet.asp. In addition, students investigate the power of a test by changing the a -value, the sample size, and the difference between the hypothesized mean and the true mean. See the activity Investigating Errors and Power in Significance Tests for Means (s is known).

Sampling Regression Lines Activity

http://www.rossmanchance.com/applets

This investigation lets students examine the sampling distribution of sample slopes for a linear regression. The applet allows the student to first look at possible sample slopes one at a time. The student then draws 100 samples to examine the sampling distribution of sample slopes. The student varies sample size, the population standard deviations , and the spread of the x-values, to determine their effect on the variability of the sample slopes. See the activity Sampling Regression Lines Using an On-Line Simulation.

Michael Legacy, NCSSM Summer Writing Project 2007 revised by Gloria Barrett, NCSSM, January, 2013