Conclusions from Data

The goal of every statistical study is to collect data and then use the data to make a decision. The process used to collect the data plays the most important role in the validity of your findings. If the process is flawed, then your decision could be called into question.

Use the following guidelines when designing a statistical study:

Identify the variable(s) of interest and the population of the study
Develop a detailed plan for collecting data. Make sure the data are representative of the population
Collect the data
Describe the data with descriptive statistics techniques
Make decisions using inferential statistics. Identify any possible errors

We will look at several examples here, using all of the information we have learned.

Conclusions from Data Presentation

Now it's time for us to explore some examples.

Central Limit Theorem (CLT) for a Sample Proportion

Central Limit Theorem and Applications

Watch the following collection of videos which explore the central limit theorem.

Video 1 explores the central limit theorem in great detail.

Video 2 explores the central limit theorem in great detail, using samples of various sizes.

Video 3 explores the central limit theorem in more detail.

Video 4 explores the central limit theorem, especially the distributions of the sample means.

Video 5 explores the means, parameters, and probabilities, as well as distributions of the means.

Central Limit Theorem Presentation

It's now time for us to explore and practice problems on the central limit theorem.

Confidence Interval and Margin of Error

The last things we want to look at to help use draw conclusions are confidence intervals and margins of error. A Confidence Interval is an interval for a parameter, calculated from the data, usually in the form estimate ± margin of error. The confidence level gives the probability that the interval will capture the true parameter value in repeated samples.

A Margin of Error is the value in the confidence interval that says how accurate we believe our estimate of the parameter to be. The margin of error is comprised of the product of the z-score and the standard error (or standard error of the estimate), $LaTeX: \frac{\sum}{\sqrt[]{n}}$ equation image indicator . The margin of error can be decreased by increasing the sample size or decreasing the confidence level.

Watch the following collection of videos which explore confidence intervals and margin of error.

In video 1, the confidence interval and the margin of error are explored probability of randomly selected samples is computed. The means and standard deviations are also compared among distributions.

In videos 2, 3, and 4, examples involving confidence intervals and margins of error are explored.

The z-score used in the confidence interval depends on how confident one wants to be. There are a few common levels of confidence used in practice: 90%, 95%, and 99%.

Confidence Level

Corresponding z-score

Corresponding Interval

m ± z × s

90%

1.645

equation image indicator $LaTeX: \mu\pm1.645\times s$

95%

1.96

equation image indicator $LaTeX: \mu\pm1.96\times s$

99%

2.576

equation image indicator $LaTeX: \mu\pm2.576\times s$

Example 1

Suppose that the mean SAT Math score for seniors in Georgia was 550 with a standard deviation of 50 points. Consider a simple random sample of 100 Georgia seniors who take the SAT.

Describe the distribution of the sample mean scores. ___?___
- Sample mean: The distribution, given that there are more than 1000 seniors who take the SAT, should be approximately normal.
- Solution: A statistics is the distribution of values taken by the statistic in all possible samples of the same size from the same population.
What are the mean and standard deviation of this sampling distribution? ___ ? ___
- Solution: The mean is 550 and the standard deviation is 50/(sqrt 100) = 5.
Use the Empirical Rule to determine between what two scores 68% of the data falls, 95% of the data falls, and 99.7% of the data falls.
- 68% of the data is between ___? ___ and ___?___ and is within ___?___ standard deviations away from the mean.
  - Solution: 545, 555, 1
- 95% of the data is between ___? ___ and ___?___ and is within ___?___ standard deviations away from the mean.
  - Solution: 540, 560, 2
- 99.7% of the data is between ___? ___ and ___?___ and is within __?____ standard deviations away from the mean.
  - Solution: 535, 565, 3

For a 95% confidence interval, 95% of all possible samples of size 100 from this population will result in an interval that captures the true mean SAT math score for Georgia seniors. Therefore, the margin of error will be ___? ___ text annotation indicator points.

- - Solution: $LaTeX: \pm5$

Example 2

You are told that a population is distributed normally with a mean, µ= 10, and a standard deviation, σ = 2. You want an 80% confidence interval.

From our discussions about normal curves, you should recognize the situation to be as follows:

normal deviation
.40 on either side of the apex of the curve

The area on each side of the mean = 0.80/2 = 0.40. Use z-scores to find x_min and x_max

What is an 80% confidence interval for this data? ___ ? ___
- Solution: The z-scores are -1.28 and 1.28. Using the formula, the confidence interval is [7.44, 12.56]
What is the margin of error? ___? ___
- Solution: The margin of error is $LaTeX: \pm2.56$

Conclusions from Confidence Interval and Margin of Error with Smaller Sample Sizes

Watch the following collection of videos which explore some examples of confidence intervals and margin of error.

In videos 1 and 2, confidence intervals and margins of error are computed and conclusions are drawn from the data.

In video 3, examples are shown involving confidence intervals and margins of error when sample sizes are small.

Confidence Interval and Margin of Error Practice

It's now time for us to explore and practice working.

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