Download Chapter 7

Chapter 7
1.18.2017
Activity
• You will receive 6 random cards
– Randomly choose 5 of the six
– Calculate the median of your 5 cards
– Put it on the dotplot
• Then trade 2 (randomly chosen) of your cards to another
person for 2 of theirs
– Then trade 2 (randomly chosen) again with a different person
• Now choose 5 cards again, and calculate the median again
• Trade again
– So you will put 3 total dots on the board
Parameters vs Statistics
• If we want to estimate something about a population, we
often take a sample
• Think of political polls—we take a sample of 500 or 1000
people to try to estimate how the entire country is feeling
• The percentage of people in the sample who support a
certain candidate is a statistic
• The percentage of people in the population that support a
certain candidate is a parameter
An Example
• I take a random sample of 10 AP Statistics students
• I measure their final exam scores. Their average was a
58.1%
– I conclude that the average final exam score for all AP
Statistics students was probably around a 58%
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The 58.1% for the 10 students is _________
The 58% estimated score is ____________
What is the population?
What is the sample?
Sampling Variability
• I take a different random sample of 10
students
– The mean is 60.05
• If I took another different sample, it would
probably be different from both of the first 2
– This is the idea of sampling variability
– 2 different random samples are likely to produce
somewhat different statistics
Sampling Variability
• If we take a bunch of different random
samples, the distribution of the estimates (or
“statistics”) will have a center very close to
the true value
• Look at the cards: True median is 6
Describing a Sampling Distribution
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Shape: roughly symmetric with a peak at 6
Center: mean/median of approximately 6
Spread: Standard deviation of roughly 1.8
Outliers: No outliers
Population Distribution
• This would be the distribution if we had the
entire population
– We never do
• But with enough samples, the sampling
distribution closely approximates the
population distribution
Unbiased Estimator
• A statistic used to estimate a parameter is
unbiased if the mean of the sampling
distribution is equal to (or very close to) the
true value of the parameter being estimated
• If they are different, then there is bias
– Why could there be bias in a statistic?
Unbiased Estimator
• A statistic used to estimate a parameter is
unbiased if the mean of the sampling
distribution is equal to (or very close to) the
true value of the parameter being estimated
• If they are different, then there is bias
– Why could there be bias in a statistic?
• Bad sampling is the most common reason—not
representative of the population
Variability
• Which gives us more confidence in our
conclusions?
– The mean of your statistic is 54.3 with a standard
deviation of 29.5
– The mean of your statistic is 54.3 with a standard
deviation of 8.4
Variability
• Which gives us more confidence in our
conclusions?
– The mean of your statistic is 54.3 with a standard
deviation of 29.5
– The mean of your statistic is 54.3 with a standard
deviation of 8.4
• A “tighter” distribution gives us more confidence that the
mean truly is near 54.3
• We would say that there is less variability (which is good)
Bias & Variability
• Which one has low bias with high variability?
• High bias, low variability?
• Low bias, low variability?
• High bias, high variability?
Bias & Variability
• Which one is best?
• Which one is worst?
Bias & Variability
• Which one is best?
– Low bias, low variability
• Which one is worst?
– High bias, high variability
• Debates about 2nd best
– I lean towards low bias, high variability
Question #1 on your Homework
• A random sample of 1000 people who signed
a card saying that they intended to quit
smoking were contacted 9 months later. 210
of the 1000 (21%) of the sampled individuals
had not smoked over the past six months
– Population:
– Parameter:
– Sample:
– Statistic
Question #1 on your Homework
• A random sample of 1000 people who signed a
card saying that they intended to quit smoking
were contacted 9 months later. 210 of the 1000
(21%) of the sampled individuals had not smoked
over the past six months
– Population: People who signed a card saying that they
would quit smoking
– Parameter: Proportion who actually quit (or
proportion who didn’t actually quit)
– Sample: Random sample of 1000 people who signed
the card
– Statistic: 0.21