Students will be able to:

• Calculate and interpret the mean and standard deviation of the sampling distribution of a sample proportion
• Determine if the sampling distribution of is approximately Normal
• Calculate and interpret the mean and standard deviation of the sampling distribution for a difference in sample proportions,
• Determine if the sampling distribution of a difference in proportions is approximately Normal
• Use a Normal distribution to calculate probabilities

Recall from Chapter 2, we had two main calculator functions that were used:

• normalcdf(lower bound, upper bound, mean, st dev): used to find the area under the Normal distribution between two bounds
  • Example: IQ scores are normally distributed with mean 100 and a standard deviation of 15. Calculate the proportion of individuals that have IQ scores above 135.
  • Solution: normalcdf(135, ∞, 100, 15) ≈ 0.0098 --> approx 1%
• invNorm(area under the Normal distribution to the LEFT of a z-score ("percentile"), mean, st dev): used to find the z-score or observation in a Normal distribution that is located at the nth percentile. 
  • Example: IQ scores are normally distributed with mean 100 and a standard deviation of 15. Mr Mills claims he scored in the 98th percentile on an IQ test. What is his IQ score?
  • Solution: invNorm(0.98, 100, 15) ≈ 130.81

Answer the following questions below using your knowledge from Chapter 2

While normal distributions provide good models for some distributions of real data, the distributions of some the common variables are usually skewed and therefore distinctly non-normal. Examples include economic variables such as personal income and total sales of business firms, the survival times of cancer patients after treatment, and the lifetime of electronic devices. It is risky to assume that a distribution is normal without actually inspecting the data, so it is important to check a distribution for normality. You can assess the normality of a distribution by plotting the data using a dotplot, stemplot, or histogram or checking whether the data follow the 68–95–99.7 rule. However, just because a plot of the data looks normal, we can't say that the distribution is normal. For a better assessment of whether a data set follows a normal distribution, we can use a normal probability plot.

To assess the normality of a distribution from its normal probability plot, look at the plotted points, and see how well they fit the normal line. If they fit well, you can safely assume that your process data is normally distributed. If your plotted points don't fit the line well, but curve away from it in places, you may have a non normal distribution.

Follow the steps below to create a Normal Probability Plot! Answer the questions as you go through each step.  

Let's switch gears and look at sample proportions. The main question will investigate on this page: 

When can we approximate the sampling distribution for proportions as Normal? 

The model below allows you to select a sample size of your choice and set the population proportion to anything you want. We can use this model to evaluate claims about sample proportions in any context.  You may want to drag the slider at the top to generate your samples more quickly. Scroll down to see instructions and questions.

IMPORTANT - PLEASE READ

As you saw on the previous slide, the sampling distribution for \(\hat p\) depends on \(n\) and \(p\) . When \(p\) is closer to 0.5, and \(n\) is larger, the sampling distribution looks more Normal.

Here's a summary: Choose an SRS of size \(n\) from a population of size \(N\) with proportion \(p\) of successes.  Let \(\hat p\) be the sample proportion of successes. Then, the following is true for the sampling distribution of \(\hat p\) (as long as the conditions are met):

We will check these 3 conditions EVERY TIME we do a sampling problem about proportions.