Chapter 7 [7.1, 7.2, 7.3]

Chapter 7

[7.1, 7.2, 7.3]

STAT 109 MSU SPRING 2022

7.1 Learning About the World Through Surveys

7.1 Learning About the World Through Surveys

7.2 Measuring the Quality of a Survey

When assessing the accuracy of survey results, we have to analyze the methods used to obtain those results.

- What estimators were used, and what were the resultant estimates?

- ​Does the methodology maximize accuracy AND precision?

- Accuracy refers to a single estimate's error from the true

population parameter

- Precision refers to the variation among several estimates​

7.2 Measuring the Quality of a Survey

7.2 Measuring the Quality of Surveys

​(a) High degree of precision and accuracy

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(b) High degree of precision, but accuracy

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(c)​ Some accuracy, but very little precision.

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(d) Very little accuracy and precision.

7.2 Measuring the Quality of Surveys

Since statistics vary from sample to sample, our goal is to design surveys and use statistical methods that maximize accuracy and precision. We hope that our estimates are not only very close to the true population parameter they are meant to estimate, but also that they are close to one another.​​

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Since we use random samples, a sample statistic is itself the random outcome of a probability experiment! This means that probability experiment with each of its outcomes being a sample statistic has a probability distribution. It's called a Sampling Distribution.​

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The Sampling Distribution of P-Hat (p̂)

A probability distribution that represents the probabilities of obtaining every possible sample proportion (p̂) from a population with population proportion (p).

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A sampling distribution for p̂ is accurate if its mean is close to p and it is precise if it has a small standard deviation. How can we improve accuracy and precision in sampling methods?

7.2 Measuring the Quality of Surveys

A sampling distribution is achieved when every possible sample of size n is taken from a population. Recall that a probability distribution must represent all possible outcomes - that means this one represents every possible sample and its resultant sample statistic.

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But, if we could take every possible sample from a population, we could just gather data from the entire population. The sampling distribution is theoretical. Theories regarding a sampling distribution help us to build inferences about a population, make estimates, and understand the level of error associated with those estimates.

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7.2 Measuring the Quality of Surveys

7.3 The Central Limit Theorem for Sample Proportions

Conditions of the Central Limit Theorem: Consider a sample of size n and a sample proportion p̂​...

1. If the sample is random and observations within the sample are independent from one another,

2. If the sample size is large (enough to expect 10 success and 10 failures) based on the known/estimated population proportion,

3. If ​the population is at least ten times larger than the sample...

7.3 The Central Limit Theorem for Sample Proportions

7.3 The Central Limit Theorem for Proportions

Multiple Select

Suppose a candy company produces 6 colors of candy coated chocolate and we believe that 30% of the candy produced is orange. Suppose we collect an independent random sample of 100 candies and are interested in the sample proportion of orange candies.

Check that each of the conditions of the central limit theorem are met.

• The sample of candies is random and observations are independent of one another.

• The sample size is large enough to expect at least ten successes and at least 10 failures.

• The population is at least ten times as large as the sample.

Multiple Choice

Using what we know about the sampling distribution of sample proportions, what can we say about a package of 100 randomly selected candies from this company?

• We expect 30% of the candies in the package to be orange, give or take 4.6%.

• Exactly 30% of the candy in the package will be orange.

• Exactly 4.6% of the candy in the package will be orange.

• We expect 4.6% of the candy in the package to be orange, give or take 30%.

Using the Central Limit Theorem to Analyze Sample Proportions:

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Of particular interest is the question, "what is the probability that a sample has a sample proportion of __ when the population proportion is __?"

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Sample proportions are just data values within a larger distribution, so we can use the Normal distribution and the associated Z-Tables to calculate probabilities associated with particular values.​

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7.3 The Central Limit Theorem for Proportions

7.3 The Central Limit Theorem for Proportions

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7.3 The Central Limit Theorem for Proportions