1. Sampling Distributions Chapter 18

2. Sampling Distributions A parameter is a number that describes the population. In statistical practice, the value of a parameter is unknown. (µ, σ, and now p or π) A statistic is a number that can be computed from the sample data without making use of any unknown parameters. We often use a statistic to estimate an unknown parameter. (, s x, and now )

3. Sampling Distributions A proportion is not something we just compute from a set of categorical data. We now see it as a random quantity that has a distribution. We call that distribution the sampling distribution model for the proportion.

4. Sampling Distributions Sampling variability is the concept that in repeated random sampling, the value of the statistic will vary. This makes sense, the proportions vary from sample to sample because the samples are composed of different values.

5. Sampling Distributions To describe sampling distributions, use the same descriptions as other distribution: overall shape, outliers, center, and spread.

6. Sampling Distributions The term bias has been used to suggest that a sample technique favors a certain outcome. When we use the term bias in relation to a sampling distribution, it is the idea that the center of the sampling distribution is not that of the population. A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

7. Sampling Distributions The variability of a statistic (σ) is described by the spread of its sampling distribution. The spread is determined by the sampling design and the sample size. Larger samples give less variability.

8. Sampling Distributions of Proportions Sampling Distribution of a Sample Proportion – Categorical Data Choose an SRS of size n from a large population with population proportion p having some characteristic of interest. Let be the proportion of the sample having that characteristic. Then the sampling distribution of p is approximately normal as long as the conditions on the following page are met.

9. Sampling Distributions of Proportions Conditions: 1) Randomization. The sample should be a simple random sample (SRS) of the population. (This is often difficult to achieve in reality. We at least need to be very confident that the sampling method was not biased and that the sample is representative of the population.)

10. Sampling Distributions of Proportions Conditions: 2) 10% Rule. In order to insure independence, we can not take a sample that is too large without replacement. As long as our sample is no more than 10% of our population size, we protect independence.

11. Sampling Distributions of Proportions Conditions: 3) Success/Failure. To insure that the sample size is large enough to approximate normal, we must expect at least 10 successes and at least 10 failures. np  10 and n(1 – p)  10

12. Sampling Distributions of Proportions

13. Sampling Distributions of Sample Means Sampling Distribution of a Sample Mean – Quantitative Data Sample means are when a distribution is created from the means of many samples when the data is quantitiative. We do this because: *Averages are less variable than individual observations *Averages are more normal than individual observations

14. Sampling Distributions of Sample Means It makes sense that the shape of the distribution x depends on the shape of the population distribution. ** If the population distribution is normal, then so is the distribution of the sample mean regardless of sample size.

15. Sampling Distributions of Sample Means It makes sense that the shape of the distribution x depends on the shape of the population distribution. **Even for skewed or odd shaped distributions, if the sample size is large enough, the sampling distribution will still be approximately normal. This idea leads us to…

16. Sampling Distributions of Sample Means The Central Limit Theorem (CLT) CLT addresses two things in a distribution, shape and spread. As the sample size increases: The shape of the sampling distribution becomes more normal The variability of the sampling distribution decreases

17. Sampling Distributions of Sample Means The Law of Large Numbers Draw observations at random from any population with finite mean . As the number of observations drawn increases, the mean of the observed values (x-bar) gets closer and closer to the true mean, .

18. Sampling Distributions of Sample Means We are allowed to use normal probability calculations to answer questions about sample means as long as we meet the following conditions.

19. Sampling Distributions of Sample Means Conditions: 1) Randomization. The sample should be a simple random sample (SRS) of the population. (This is often difficult to achieve in reality. We at least need to be very confident that the sampling method was not biased and that the sample is representative of the population.)

20. Sampling Distributions of Sample Means Conditions: 2) 10% Rule. In order to insure independence, we can not take a sample that is too large without replacement. As long as our sample is no more than 10% of our population size, we protect independence.

21. Sampling Distributions of Sample Means Conditions: 3) Large Enough Sample. The truth is, it depends. There is no “for sure” way to tell. It is common practice to say any sample where n ≥ 30, you are safe to assume normality for the sampling distribution. (Remember, if the distribution is given normal, then any sample size is OK)

22. Sampling Distributions of Sample Means

23. Sampling Distributions of Sample Means

24. Sampling Distributions We said at the beginning that in most real life cases, we will not know the population parameters (µ, σ, p or π) so we will have to use the sample statistics as estimates of those. Our terminology changes just a little…

25. Sampling Distributions

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27. Sampling Distributions