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This document linked from Means (All Steps)

]]>Here is an interactive demonstration which allows you to choose the population proprtion and sample size. You can generate a single sample at a time or have it generate a large number of samples to simulate the entire sampling distribution.

This document is linked from Sampling Distributions.

]]>You can choose which event to shade and move the events around. Pretty neat!

This site uses a different notation instead of “AND” and “OR.” The U = OR, two events with nothing between them = AND, and the c as a superscript (in the exponent location) is for complement.

This document is linked from Basic Probability Rules.

]]>Here is an interactive demonstration which lets you build the histogram and compare the mean and median for various shapes.

This material is from Interactivate.

This document is linked from Measures of Center.

]]>Link to Reading (≈ 2500 words)

This document is linked from Errors and Power.

]]>We will start with two examples of using the applet and then ask a few questions. The applet has changed slightly and does not look exactly the same in the link above as our images below but the processes are the same.

We are interested in studying whether the mean IQ score among children with high blood lead levels is lower than the population average (which is 100). We will assume that the standard deviation of the population is 16. (Read more: A related article ≈ 4200 words.)

Our hypotheses are:

**Ho:** μ = 100 (mu = 100)

**Ha: **μ < 100 (mu < 100)

We want to be able to detect a difference of 5 points. In other words, if the true mean IQ among children with high blood lead levels is 5 (or more) points lower than 100, we want to have a good chance to detect that difference and reject the null hypothesis.

The difference of 5 points represents the effect size of interest in this problem. It represents the difference between the true mean and the null value that we would like to be able to detect.

We would like a power of around 80% and need to decide on a sample size for our study.

Using the interactive applet we can easily calculate and visualize the power of this test:

If **n = 2** (this is the smallest possible sample size available, and much too small)

This is actually a fairly good chance considering we only used a sample size of 2, but this is not nearly enough for our target. Notice** the probability of a Type II error is 1 – 0.111 = 0.889.**

If** n = 25** (this is still a relatively small sample)

By increasing the sample size to 25, we have increased the power of our test to 46%. We have a 46% chance of rejecting the null hypothesis when we take a sample of size 25 and the true population mean is 95 (5 points lower than 100). To hit our target will still need a larger sample.

It is not clearly illustrated, however, if you look at the x-axes you will see that the variability displayed in the distributions is decreasing as the sample size increases. This is the result seen in Module 9, as the sample size increases, the spread of the sampling distribution decreases.

It is this decrease in the variability of x-bar that is causing the increase in power in this example. We are not “moving” the center of the distributions, they are simply becoming less variable so that they overlap less as indicated in the image below.

See if you can find the answer. Using the applet, enter the values we have above for the null and alternative hypotheses, the standard deviation, and the alt. mean. You should not need to change the significance level but it should be set to 5%.

The example above illustrates the first factor affecting power discussed earlier – **increasing the sample size results in an increase in the power of the hypothesis test** when all else remains the same. This is a direct result of the fact that the variation of the statistic (in this case, x-bar) decreases as the sample size increases.

Now that you have learned to use this tool, we want to use it to illustrate two other factors affecting power. In the following activity we will illustrate:

- If the true difference (often called the “effect size”) increases, the power of the hypothesis test increases.

- If α (alpha) decreases, Power = 1 – β = 1 – beta also decreases.

This document is linked from Errors and Power.

]]>For this discussion, we suggest you specify the hypotheses and the true population mean *μ* (mu), use the New Sample button to create a random sample from the population and display the sample mean from this one sample.

**Repeat this process to get an idea of how often you get the wrong answer.**

Try this both when Ho is true (to illustrate a Type I error) and when alternatives are true (to illustrate a Type II error).

If when you load the page, the picture does not look the same as those below, you may need to close your browser and reopen the page. We have had this problem when working with this tool in Firefox.

In the section, we provide two examples of using a similar applet. In both cases we consider IQ scores. Our null hypothesis is that the true mean is 100. Assume the standard deviation is 16 and we will specify a significance level of 5%.

This document is linked from Errors and Power.

]]>This document is linked from Hypothesis Testing.

]]>As you will remember from a previous activity, the applet shows a normal-shaped distribution, which represents the **sampling distribution of the mean** (x-bar) for random samples of a particular fixed sample size, from a population with a fixed standard deviation (σ, sigma). The green line marks the value of the population mean (μ, mu).

To begin the simulation, click **“sample 25”** button. You have used the simulation to select 25 samples from the population; the applet has automatically computed the sample means and the corresponding confidence intervals.

Notice, along the left of the applet, that you can change the confidence level. Do this, and watch what happens to the intervals as the confidence is changed among all the levels available.

This document is linked from Population Means (Part 2).

]]>The applet has a new look and different ways of entering the information but otherwises, works the same and still illustrates the same concepts.

Transcript – Video of Confidence Interval Applet in Action

When the applet loads, you see a normal-shaped distribution, which represents the **sampling distribution of the mean** (x-bar) for random samples of a particular fixed sample size, from a population with a fixed standard deviation of σ (sigma).

The green line marks the value of the population mean, μ (mu).

To begin the simulation, click the **“sample”** button. You will see a line segment appear underneath the distribution; you should see that the line segment has a dot in the middle.

You have used the simulation to select a single sample from the population; the applet has automatically computed the mean (x-bar) of your sample; your (x-bar) value is represented by the dot in the middle of the line segment. The line segment represents a confidence interval. Notice that, by default, the applet used a **95%** confidence level.

If your confidence interval **did** cover the population mean μ (mu), then the applet will have recorded 1 “hit”.

Now, click to select another single sample.

Notice, under “total” on the right side of the applet, the number of total selected samples has been tallied.

Now, click “**sample 25**” repeatedly, until the applet tallies a “total” of around 1,000 samples. You will see that the applet computes the “percent hit” for all the intervals.

Based on what you’ve seen on the applet (with the level set at 95%), decide which of the following statements are true and which are false.

Note that in actual scientific practice, we only select one single sample and therefore we can only see the one corresponding interval, out of the potential thousands that are displayed using the applet. Give the best interpretation of a single 95% confidence interval as follows, based on what you’ve learned from the simulation:

Fill in the blanks in the following statement in next set of questions:

We are ___ confident that our interval is one that ____ cover _____

This document is linked from Population Means (Part 1).

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