Introductory Statistics

MA207

Day 5 - Using sampling distributions as evidence

Understanding Sampling Distributions (#1)

A few Carroll students were discussing the prevalence of underage drinking on campus. Based on what the students see from their peers leads them to believe that roughly 50% of under age Carroll students drink. They later find out that a survey was given to 76 randomly selected under age Carroll students. Only 32 of the 76 reported having participated in under age drinking.

• Identify the research question.
• Null Hypothesis:
• Alternative Hypothesis:

Understanding Sampling Distributions (#1)

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A few Carroll students were discussing the prevalence of underage drinking on campus. Based on what the students see from their peers leads them to believe that roughly 50% of underage Carroll students drink. They later find out that a survey was given to 76 randomly selected under age Carroll students. Only 32 of the 76 reported having participated in underage drinking.

• Let’s build a simulation so we have a better sense of how samples of size 76 behave.
• Assumptions:
• Each student surveyed has a 50% chance of saying “yes,” (this assumption is based on the belief above that 50% of student drink)
• 76 randomly selected students are surveyed. If a different set of 76 random students were surveyed, we would not expect identical results

If each person in the population has a 50% chance of saying Yes to the drinking question, then this is modeled by spinning our 50/50 spinner 76 times.

Build this spinner and try it several times, each time the “repeat” is set at 76 people.

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Be sure to plot your results.

Understanding Sampling Distributions

• A sampling distribution shows us how spread out we expect randomly selected samples to be.
• A sampling distribution gives us the power to recognize whether one sample (of size 76 in this case) is weird or typical.
• If a sample is particularly weird, we should suspect that
• a) It didn’t come from the population we thought it did ... [interesting]
• b) Our null hypothesis might be wrong … [interesting]
• c) Our participants are lying … [interesting]
• d) The sample was unusual because variability is part of statistics … [This is possible, but it has a low probability of being the explanation]
• If our sample is not very weird, its variability is probably just from random chance
• Our job is to quantify “weird” in a mathematical / statistical way

Back to the question

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A few Carroll students were discussing the prevalence of underage drinking on campus. Based on what the students see from their peers leads them to believe that roughly 50% of underage Carroll students drink. They later find out that a survey was given to 76 randomly selected under age Carroll students. Only 32 of the 76 reported having participated in underage drinking.

• How unusual would it be to get a sample that is as low as 32 out of 76, if the drinking rate is 50%?
• If you quantify your answer to (A) as a probability, you get the p-value for the sample. Describe what the p-value means in your own words and what conclusions you could draw from the study. (talk with a neighbor)

A modified scenario

At a larger state school, a study from 2005 indicated that 64% of underage students drank. The school engaged is a sustained campaign to lower the rates of drinking across campus. This year, a survey was given to 150 randomly selected underage students at the school. Only 83 of the students reported drinking (55.3%).

• What is the null hypothesis and how do you use it in your simulator?
• What is the alternative hypothesis and why does it not appear in your simulator?
• If the drinking rates haven’t changed since 2005, what’s the chance of getting a sample as low (or lower) than the one in this study? i.e. What portion of your randomly generate samples were this low?

When you have a working simulation, show me. Yes, you can just tweak the other simulation for the new scenario.

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A modified scenario

At a larger state school, a study from 2005 indicated that 64% of underage students drank. The school engaged is a sustained campaign to lower the rates of drinking across campus. This year, a survey was given to 150 randomly selected underage students at the school. Only 83 of the students reported drinking (55.3%).

• What is the p-value?

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• Can we draw a cause / effect relationship out of this study?

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A modified scenario

At a larger state school, a study from 2005 indicated that 65% of underage students drank. The school engaged is a sustained campaign to lower the rates of drinking across campus. This year, a survey was given to 150 randomly selected underage students at the school. Only 83 of the students reported drinking (55.3%).

• What is the p-value?

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• Can we draw a cause / effect relationship out of this study?
• No, because it is not a randomized controlled study.
• If our p-value is below 0.05, we can say there is evidence that the drinking rate has lowered since 2005. But we cannot say it is because of the school campaign.

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A new example - Accessing Health Services

At a typical 4 year college, about 25% of students are in each class. Build a sampler that has 25% Freshmen, 25% Sophomores, 25% Juniors, and 25% Seniors.

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A new example - Accessing Health Services

At a typical 4 year college, about 25% of students are in each class. Build a sampler that has 25% Freshmen, 25% Sophomores, 25% Juniors, and 25% Seniors.

The number of students who access health services on campus should be the same across the 4 classes, but there is a concern that freshmen are less likely to seek help. In one week, out of 120 students who visited health services, only 20% were freshmen. Is this 20% in the range of normal variability, or does it represent a statistically extreme situation?

• Build a simulation to find out how random samples of size 120 behave.
• Create a sampling distribution and determine where the middle 95% of samples are.
• How unusual is 20%?

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A delicious example

The M&Ms Mars company has a machine that fills the fun sized bags of M&Ms, but due to natural variation in the machines not every bag gets exactly the same weight in M&Ms. The data in the table shows the number of bags in each weight category on a given day.

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�The advertised weight of the fun size M&Ms is 13.5 grams

• A friend buys a bag and finds that it only weighs 11.9 grams?
• Null Hypothesis:
• Alternative Hypothesis:
• What is the p-value for finding a bag like your friends or more extreme? What conclusion can we make?