EXAMPLESDistributionsTopic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION 10-1 Topic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION As has been mentioned repeatedly, we use sample information to estimate the unknown information about the population we are interested in. EXAMPLES 1. Rats raised in an enriched environment appear to have larger cortexes than those raised in unenriched environments. What is the average cortex weight, µ, for the population of cortex weights for every rat which could be raised in an enriched environment? We’ll grow a sample of rats in the enriched environment and use our sample data to estimate µ. 2. Fiddler crabs are known as such because they have asymmetrically sized pincers. Like human handedness, it is estimated that approximately 10% of fiddler crabs are “left-pincer”, i.e. the left pincer is larger than the right. What is the true proportion, π, of left-pincered crabs on a remote island in the South Pacific? We’ll fly to the S. Pacific, take a random sample of crabs and use the sample proportion p to estimate π.Topic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION 10-2 Several ways we could report the results: We could report a POINT ESTIMATE, which is a single number from the sample. It has little context or meaning unless additional information is provided. For example, when reporting the sample mean it usually is better to also report something about its sampling variability, such as including the sample size n and the sample standard deviation s. So, other possibilities for reporting: 1. Report x and s. Not useful since s is not the standard deviation of x 2. Report x and nσσx=. Usually can’t because we don’t know the value of σ 3. Report x and nsSEM =. • is STANDARD ERROR OF THE MEAN SEM• is an unbiased estimate of SEMxσ (under random sampling)Topic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION 10-3 • one common report is to list SEMx± but it’s not the best choice since it isn’t directly interpretable as an interval estimate We would like to report a range or interval of plausible values for the population parameter we’re estimating. Such a report is called an INTERVAL ESTIMATE of the population parameter. When done right, the interval estimator includes our point estimate and an estimate of the accuracy of the point estimator and some measure of our “comfort” with the estimate we are providing. That is, we would like to be somewhat confident that the interval we report actually covers the true value of the parameter we are trying to estimate. Defn: A CONFIDENCE INTERVAL for a population characteristic is an interval of plausible values for that parameter. It is constructed so that the true value of the parameter is captured inside the interval with a chosen, specified level of confidence. EXAMPLE Political polls are almost always reported as follows: “The proportion of voters who would vote for Gore today is 48% ± 4% with 95% confidence”. What thatTopic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION 10-4 means is that based on their sample, the true proportion π of voters who would choose Gore is somewhere inside the interval 44% and 52%. The 95% confidence refers to the probability that the interval captures π. Defn: The CONFIDENCE LEVEL associated with a confidence interval is the probability that the interval estimate covers the true value of the parameter. One way of thinking of it is as the success rate of the method we are using to do the estimation! The method includes the choice of point estimator, the assumptions about the distribution of that estimator, and the sampling design. The confidence level is chosen by the researcher doing the reporting. Common levels in the life sciences are 90%, 95% and 99%. In social sciences you sometimes see 85% as well. When intervals are constructed in a way that uses knowledge of the sampling distributions of the point estimators, we can assign these probabilities and not just guess what they might be.Topic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION 10-5 π • • • • • e.g. 68% of all sample proportions are within the interval above. If we put a confidence interval of the same width around each estimate shown above, what percent of the new intervals would include π ? Important Point: the confidence level is not a probability about a particular interval including the true value. The interval either does or does not cover the true value (but you’ll never know that). The confidence level is the probability that the method you used will give an interval that includes the true value!!!Topic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION 10-6 1) INTERVAL ESTIMATION OF THE POPU-LATION PROPORTION π How does the Gallup polling company calculate the 95% confidence interval they report? They take a sample of size n and calculate the sample proportion of successes p (in this case people who indicate they would vote right now for Gore). Then they use: A LARGE SAMPLE 95% CONFIDENCE INTERVAL FOR THE POPULATION PROPORTION π is nppp)1(96.1−± i.e. ⎟⎟⎠⎞⎜⎜⎝⎛−+−−npppnppp)1(96.1,)1(96.1 • Large-sample refers to a sample size that is sufficiently large to invoke the Central Limit Theorem. • p is the point estimator of π so is at the center of the intervalTopic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION 10-7 • npp )1( −is the estimator of - we use this because it is the estimate of how much sampling variability there is in p pσ • 1.96 is the z-score, z*, that makes the following statement true: 0.95 = Pr(- z* < Z < + z*). We use this because we are using the CLT which states that sample proportions are normally distributed for large samples The formula is easily adapted for other confidence levels. Simply replace 1.96 with the appropriate number from the table below. The z critical values for common confidence levels are: Confidence Level Z critical values 80% 1.28 90% 1.645 95% 1.96 98% 2.33 99% 2.58 99.9% 3.29Topic (10) – ESTIMATION OF PARAMETERS OF A SINGLE POPULATION 10-8 EXAMPLE Our researcher did fly to the South Pacific and collected 150 crabs. For each crab she recorded whether the left or right pincer was dominant and observed that 20 crabs were
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