Section 1.5

Theory Based Approach

Section 1.5 Learning Objectives

• Describe the sampling distribution of a statistic and define the standard error.

• Know when/why simulation and theory will yield different results.

Simulation

So far, we’ve used a simulation-based approach to gather strength of evidence.

• Effective
• Relatively easy

But…

• Can be tedious
• Not always practical in real life
• Enter: Theory!

The Theory-Based Approach

• Used when computers weren’t a thing
• Allows prediction of the null distribution shape (aka, pattern)
• Gives us p-values and standardized statistics
• Should get the same or similar results as simulation

The Null Distribution (in depth!)

• Sometimes, data follows a known distribution
• Commonly, we assume data follows the normal distribution
• The normal distribution is one of many known patterns that data can follow
• We always assume the null distribution is normally distributed

The Null Distribution (in depth!)

Since we know the null distribution is normally distributed, we know the following:

• Bell shaped
• Centered at the hypothesized value for \pi
• Can predict standard deviation
  • Without the Applet!

Example Null Distribution

Why does sample size matter?

If sample size is too small:

• Theory-based approach will not be accurate
• Can’t tell if data is normally distributed

Theory-based “rule of thumb”:

• Need at least 10 successes and 10 failures

To illustrate…

• Both are centered at 0.50
• Left graph n = 30
• Right graph n = 300

Central Limit Theorem

• “Backbone” of statistics

Definition:

• If the sample size (n) is large enough, the distribution of the sample proportions will be bell-shaped (approximately normal), centered at the long run probability (\pi), with a standard deviation:

standard\: deviation\: of\: the\: null=\sqrt{\frac{\pi(1-\pi)}{n}}

Using Theory for Standardized Values

Replace S.D. with theory-based calculation

z=\frac{Observed\: Statistic - Hypothesized\: Value}{Standard\: Deviation\: under\: the\: null}

=\frac{\hat{p}-\pi}{\sqrt{\frac{\pi(1-\pi)}{n}}}

Using the Applet

We can still use the One Proportion Applet for the theory-based approach

• Check “Normal Approximation” box
• Applet fits a normal curve to the data

In more advanced classes, we use calculus to get these values.

Using the Applet

Example: Halloween Treats

Halloween Treats

Let’s identify the following:

• Sample & size

• Variable of interest

• Parameter & symbol

• Observed statistic & symbol

• Hypotheses (in symbols)

Do we meet the criteria for the theory-based approach?

Scenario:
Researchers investigated whether children might be as tempted by toys as by candy for Halloween treats. Test households offered two plates to children: one with candy and one with a small toy. They observed the selections of 283 children that night and found that 148 of the kids chose candy.

Halloween Treats

• Sample & size: n = 283 kids

• Variable of interest: Chose candy or not

• Parameter & symbol: \pi = long run proportion of kids that chose candy

• Observed statistic & symbol: \hat{p} = \frac{148}{283}

• Hypotheses (in symbols):

  • H_{0}: \pi = 0.5
  • H_{A}: \pi ≠ 0.5

Do we meet the criteria for the theory-based approach? Yes! More than 10 failures & successes.

Halloween Treats

Standard deviation using theory:

standard\: deviation\: of\: the\: null=\sqrt{\frac{\pi(1-\pi)}{n}}

=\sqrt{\frac{0.5(1-0.5)}{283}}=0.0297

Halloween Treats

Then, we use the calculated standard deviation to obtain the standardized statistic:

standardized\: statistic\: =\frac{0.523-0.5}{0.0297}=0.774

Halloween Treats

Now we find the standardized statistic with the simulation approach:

standardized\: statistic\: =\frac{0.523-0.5}{0.02}=0.793

Notice, we get nearly the same number!

Halloween Treats Conclusion

Since our standardized statistic using either approach is not outside the twos, we conclude the following:

“We do not have enough evidence to conclude that the long-run proportion of kids who choose candy differs from 0.5.”

Investigation: Tire Story Falls Flat