Samples and populations Estimating with uncertainty

Review - order of operations

1.Parentheses 2.Exponents and roots 3.Multiply and divide 4.Add and subtract

Review - order of operations

Review - types of variables Categorical variables –For example, country of birth Numerical variables –For example, student height

Review - types of variables Categorical variables Numerical variables Discrete Continuous

Review - types of variables Categorical variables Numerical variables Discrete Continuous Nominal Ordinal

Review - types of variables Categorical variables –Nominal - no natural order –Ordinal - can be placed in an order

Review - types of variables Categorical variables –Nominal - no natural order Example - country of birth –Ordinal - can be placed in an order

Review - types of variables Categorical variables –Nominal - no natural order Example - country of birth –Ordinal - can be placed in an order Example - educational experience –Some high school, high school diploma, some college, college degree, masters degree, PhD

Sampling from a population We often sample from a population Consider random samples –Each individual has an equal and identical probability of being selected

Body mass of 400 humans

Random sample of 10 people

Population mean:  = 70.8 kg

Population mean:  = 70.8 kg Sample mean: x = 76.7 kg

Another sample…

Population mean:  = 70.8 kg Sample mean: x = 69.2 kg

What if we do this many times? Example: gene length

n = 20,290

 =  =

Sample histogram

n = 100 Y = s =

Y = s = Y = s = Y = s = Y = s =

Y = s = Y = s = Y = s = Y = s = Sampling distribution of the mean

1000 samples Sampling distribution of the mean

 = Mean of means: Sampling distribution of the mean

Y = s = Y = s = Y = s = Y = s =

s = s = s = s = Sampling distribution of the standard deviation

100 samples Population  = Mean sample s = Sampling distribution of the standard deviation

1000 samples Population  = Mean sample s = Sampling distribution of the standard deviation

Sampling distribution of the mean, n=10 Sampling distribution of the mean, n=100 Sampling distribution of the mean, n = 1000

Sampling distribution of the mean, n=10 Sampling distribution of the mean, n=100 Sampling distribution of the mean, n = 1000

Larger sample size

Group activity #2 Form groups of size 2-5 Get out a blank sheet of paper Write everyone’s full name on the paper

How many toes do aliens have?

Instructions You have measurements from a population of 400 aliens Use your random number table to select a sample of ten measurements Calculate your sample mean and, if you have a calculator or a large brain, your sample standard deviation On your paper, answer the following: 1.What was your sample mean and standard deviation? 2.How did you randomly choose your sample?

Distribution of the sample mean No matter what the frequency distribution of the population: The sample mean has an approximately bell-shaped (normal) distribution Especially for large n (large samples)

How precise is any one estimated sample mean?

The standard error of an estimate is the standard deviation of its sampling distribution. The standard error predicts the sampling error of the estimate.

Standard error of the mean

Estimate of the standard error of the mean

Confidence interval –a range of values surrounding the sample estimate that is likely to contain the population parameter 95% confidence interval –plausible range for a parameter based on the data

The 2SE rule-of-thumb

Confidence interval

Pseudoreplication The error that occurs when samples are not independent, but they are treated as though they are.

Example: “The transylvania effect” A study of 130,000 calls for police assistance in 1980 found that they were more likely than chance to occur during a full moon.

Example: “The transylvania effect” A study of 130,000 calls for police assistance in 1980 found that they were more likely than chance to occur during a full moon. Problem: There may have been 130,000 calls in the data set, but there were only 13 full moons in These data are not independent.