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AP Stats.
Chapter 7-1 Notes
1] What is a parameter? What is a statistic? How is one related to the other?
A parameter is a numerical characteristic of a population (e.g. mean height, proportion of adults that have smartphones, etc.)
A statistic is a numerical characteristic of a sample that is used to estimate a parameter. A statistic (from a sample)
estimates a parameter (from the population).
2] Identify the population, the parameter, the sample, and the statistic in each of the following:
(a) A pediatrician wants to know the 75th percentile for the distribution of heights of 10-year-old boys, so she takes a
sample of 50 patients and calculates Q3 = 56 inches.
A pediatrician wants to know the 75th percentile for the distribution of heights of 10 year-old boys, so she takes a sample
of 50 patients and calculates Q3 = 56 inches.
Population = all 10 year-old boys
Parameter = 75th percentile of heights of 10 year-old boys
Sample = 50 ten year-old boys
Statistic = Q3 (75th) = 56 in.
(b) A Pew Research Center Poll asked 1,102 12-to-17-year-olds in the United States if they have a cell phone. Of the
respondents, 71% said “Yes.”
Population = all 12-17 year olds in the US
Parameter = proportion of 12-17 year olds that have a cell phone
Sample = 1,102 12-17 year olds
Statistic = 𝑝̂ = 0.71
3] What is sampling variability?
When repeated simple random samples are taken from a population, the proportions produced by the samples will vary.
This is known as sampling variability.
4] What is a sampling distribution?
Suppose we take an SRS of size 100 from a large population and plot the sample proportion of interest. Then take a
second SRS of size 100 and compute and plot the sample proportion. Repeat this for ALL possible samples of size 100 –
𝑛
the number (
) gives the number of all SRSs of size 100. This distribution is the sampling distribution of size 100.
100
5] What is the difference between the distribution of the population, the distribution of the sample, and the sampling
distribution of a sample statistic?
The distribution of the population is the distribution of some numerical characteristic of the population for EVERY
member of the population.
The distribution of a sample is the distribution of that numerical characteristic but only for the members of the sample.
The distribution of a sample statistic is the sampling distribution (see #6) for the statistic in question. For example, say
you are interested in knowing the proportion of ECRCHS students that are in favor of banning handguns. Take an SRS of
size 100 and ask “Are you in favor of banning handguns?” and plot the sample proportion of “Yes.” Next repeat this
process FOR ALL POSSIBLE SAMPLES OF SIZE 100. The distribution produced is the distribution of the sample statistic,
which in this case is the sample proportion 𝑝̂ .
6] What is an unbiased estimator? What is a biased estimator? Provide some examples.
A statistic is an unbiased estimator if its sampling distribution is centered at the true population parameter. (The mean
of the statistic equals the population parameter that it is estimating.)
Examples:
𝑝̂ is an unbiased estimator of p, the population proportion.
𝑥̅ + 3𝑠𝑥 is a biased estimator of µ because it will consistently overestimate the mean of the population.
7] How can you reduce the variability of a statistic?
Increase the sample size. Specifically, to cut the standard deviation of a sample statistic in half, increase the sample size
by a factor of 4.
8] What effect does the size of the population have on the variability of a statistic?
The size of the population has NO effect on the variability of a sample statistic, provided that the sample size is no
greater than 10% of the population size.