Statistics – 7.1 Notes Name: _____________________________ What is a Sampling Distribution?

Sampling Distributions x Sampling distributions are the key to inference when data are produced by random

sampling.

x Because the results of random samples include an element of chance, we can’t guarantee that our inferences are correct. What we can guarantee is that our methods usually give correct answers.

x The reasoning of statistical inference rests on:

“How often would this method give a correct answer if I used it very many times?” OR

“What would happen if we did this many times?”

We write 𝜇 (𝑡ℎ𝑒 𝐺𝑟𝑒𝑒𝑘 𝑙𝑒𝑡𝑡𝑒𝑟 𝑚𝑢) for the population mean and �̅� (x-bar) for the sample mean. We use P to represent a population proportion. The sample proportion �̂� (p-hat) is used to estimate the unknown parameter P. Practice problem: Identify the population, the parameter, the sample, and the statistic in each of the following settings.

1. The Gallup Poll asked a random sample of 515 U.S. adults whether or not they believe in ghosts. Of the respondents, 160 said “yes”.

2. During the winter month, the temperatures outside the Starnes’s’ cabin in Colorado can stay well below freezing (32 Celsius or 0 Fahrenheit) for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at 50 Fahrenheit. She wants to know how low the temperature actually gets in the cabin. A digital thermometer records the indoor temperature at 20 randomly chosen times during a given day. The minimum reading is 38 Fahrenheit.

A parameter is a number that describes some characteristic of the population.

A statistic is a number that describes some characteristic of a sample.

Key

stakstiosyParameters

I Mf p

population is all vSadultsandtheparameter of interest is P proportion

ofadultswhobelieveinghostsThesampleis 515peopleinterviewedandthestatistic is p 6,03 0.31theproportionwhosaytheybelieve inghosts

population alltimesduringthedayinquestionParameterofinterest Thetrueminimumtemp inthecabinthatdaySample 20tempreadings at randomly selectedtimesstatistic minimum of38 F

Statistics – 7.1 Notes Name: _____________________________ What is a Sampling Distribution?

How can �̅� be an accurate estimate of 𝜇? After all, different random samples produced different values of �̅�.

x This basic fact is called sampling variability: the value of a statistic varies in repeated random sampling.

x To make sense of sampling variability, we ask, “What would happen if we took many

samples?” Sampling Distribution If we took every one of the possible samples of size n from a population, calculated the sample proportion for each, and graphed all of those values, we’d have a sampling distribution.

Describing Sampling Distributions Center: Biased and unbiased estimators

Spread: Low variability is better!

Larger ______________________ have a clear advantage over smaller samples. They are much

more likely to produce an estimate close to the true value of the ________________________.

n= n=

The ______________________ distribution of a statistic is the distribution of values taken by the

statistic in all-possible samples of the same size from the same population.

A statistic used to estimate a parameter is an unbiased _________________________ if the mean of its

sampling distribution is equal to the true value of the parameter being estimated.

Sampling

estimator

samplesparameter

Statistics – 7.1 Notes Name: _____________________________ What is a Sampling Distribution?

Practice Problem

The histogram above left shows the intervals (in minutes) between eruptions of Old Faithful geyser for all 222-recorded eruptions during a particular month. For this population, the median is 75 minutes. We used Fathom software to simulate taking 500 SRS’s of size 10 from the population. The 500 values of the sample median are displayed in the histogram above right. The mean of these 500 values is 73.5.

1. Is the sample an unbiased estimator of the population mean? Justify your answer.

2. Suppose we had taken samples of size 20 instead of size 10. Would the spread of the sampling distribution be larger, smaller, or about the same? Justify your answer.

3. Describe the shape of the sampling distribution Bias, Variability, and Shape We can think of the true value of the population parameter as the bull’s- eye on a target and of the sample statistic as an arrow fired at the target.

x Bias means that our aim is off and we consistently miss the bull’s-eye in the same direction. Our sample values do not center on the population value.

x High variability means that repeated shots are widely scattered on the target. Repeated samples do not give very similar results.

Nobecausethe near ofthe approximate sampling distribution ofthesamplemedian 73.5 isnotequaltothemedian ofthepopulation 75

Increasingthesample size willdecrease thespreadof thesamplingdistributionLargersamplesprovidemorepreciseestimatesbecausetheyincludemore informationaboutthepopulation

skewed left andunimodal

Largersamples meansmoreprecisebut notnecessarilymoreaccurate