Jill VanderStoep�Hope College – Holland, MI�

EAPOST Workshop:

Test for a Single Mean

Overview

• Information on our curriculum
• How I run my class
• Test for a Single Mean:
• Exploring Quantitative Data
• Simulation-Based Approach Exploration
• Questions

Information on our Curriculum

• Created to follow GAISE
• Begins with 4 pillars of inference:
• Significance
• Estimation
• Generalization
• Causation
• Features a spiral approach using the 6 steps of statistical investigation
• Utilizes simulation to actively understand the investigation process

Information on our Curriculum

• The 6 steps of statistical investigation
1. Ask a research question
2. Design a study and collect data
3. Explore the data
4. Draw inferences beyond the data (Logic of inference: Significance and estimation)
5. Formulate conclusions (Scope of inference: Generalization and causation)
6. Look back and ahead

How I Run my Class

• 15 weeks instruction/1 week final exams
• First 5 weeks, cover the 4 pillars: significance, estimation, generalization, causation.
• Second 5 weeks, explore inference comparing two groups (including matched pairs)
• Third 5 weeks, explore inference comparing multiple groups as well as regression

How I Run my Class

• Students read or watch videos on the upcoming class content through an example.
• They take a reading/video quiz before class on that content
• Ideally they start the HW
• In class we recap the content from the example
• In class we go through an exploration of a different content
• After class they complete HW

How I Run my Class

• Test for a single mean is typically week 3

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Test for a Single Mean: Exploring Quantitative Data

Beth Chance workshop on sampling distribution of a single mean linked here

Test for a Single Mean: Simulation-Based Approach Exploration

Backpack Weights

Backpack Weights

• Carrying a heavy backpack can be a source of chronic, low-level trauma and can cause long-lasting shoulder, neck, and back pain.
• The American Academy of Orthopedic Surgeons recommends that a backpack not weigh more than 10% to 15% of the wearer’s body weight.
• College students at a public university in California (about 20,000 enrollment) wondered if college students at their university were following these backpack weight guidelines.

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Backpack Weights

• STEP 1: Ask a research question
• Do university students carry backpacks that weigh on average something different than the recommended 10% of their body weight?
• Let’s take a closer look at this study

Backpack Weights

• STEP 2: Design a study and collect data
• Each of 4 consecutive days, student researchers set up a scale at a different location on their campus. At each location, data was gathered on students who agreed to participate until they had data on 25 students at that location (100 total students).
• Each student had their backpack weight recorded to the nearest pound.
• Each student filled out a short survey that asked their weight in pounds, biological sex, what college they were in, their major, their year in school, whether they were a graduate or undergraduate student, their total course units, and whether they experienced any back problems.

Backpack Weights

• What are the observational units?
• The observational units are the 100 students.
• What measured variables are categorical?
• Biological sex, college, major, year in school, back problems.
• What measured variables are quantitative?
• The backpack weight (lbs), the student weight (lbs), the number of course units.

Backpack Weights

• Other variables calculated
• Percentage of student’s body weight the backpack is calculated as a ratio
• Spread sheet/data sheet

Hypotheses

• State the null and alternative hypotheses
• Null hypothesis: The unknown long run mean ratio of backpack weight to body weight is 0.10
• Alternative hypothesis: The unknown long run mean ratio of backpack weight to body weight is different from 0.10

Parameters

• We can also use a symbol for the unknown parameter we are trying to estimate.
• The parameter of interest is: 
• µ = the unknown long-run mean ratio of backpack weight to body weight

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Hypotheses

• Mu (µ) is the parameter for population mean ratio.
• Using this symbol, we can rewrite the hypotheses as follows: 

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H₀: µ = 0.10

H_a:µ ≠ 0.10

Hypotheses

Remember

• The hypotheses are about the parameter value in the population, not just for these 100 students.
• Hypotheses are always about populations or processes, not the sample data.
• Step 3: explore the data comes next & we will use the descriptive statistics applet

Step 3: Explore the data

Results

• The sample mean was different than 0.10
• What are two possible explanations for why we observed a sample mean that was different from 0.10?
• #1 The population mean ratio really is 0.10 and our sample mean of 0.078 from 100 students happened by random chance.
• #2 The population mean ratio is different from 0.10 and that is why our sample mean ratio was different from 0.10.

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Step 4: Draw inferences beyond the data

• Is it possible to get a mean ratio of 0.078 even if in the population the ratio is 0.10?
• Yes, it’s possible, how likely though?
• A p-value will tell how likely it is to get a mean ratio that different from 0.10.
• Let’s use our 3S strategy

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Statistic

Simulation

• We can simulate samples of size 100 from a population where the mean ratio of backpack weight to body weight is 0.10
• Create a hypothetical population of ratios
• Assume the variability seen in our sample is the variability in the population
• Make many, many copies of our sample
• Slide this distribution to center on 0.10.

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Simulation

• Take repeated samples of size 100 from this hypothetical population each time calculating the mean ratio
• Do this repeated sampling many times to develop a null distribution of mean ratios
• Let’s see what this looks like

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Simulation

Simulation

Simulation

Simulation

One mean applet

• Let’s see how this process is done with the One Mean applet.
• We will use the backpack/bodyweight ratio as the variable we calculated.

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P-value

• We should have obtained a null distribution like the one shown.
• We can see that our observed statistic of 0.078 (or smaller) or the comparable value (0.122 or larger)didn’t even occur once in 1,000 samples.
• Therefore, our p-value is less than 1/1,000 or approximately 0.

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P-value

• What does this p-value mean?
• Assuming that the population mean ratio of backpack weight to body weight truly was 0.10, if we were to repeatedly take samples of size 100 and in each sample average the ratios, we would find the study sample mean ratio of 0.078 or one more extreme in about 0 of the repeated samples.
• Therefore, we have very strong evidence that the backpack weight to body weight ratio is significantly different from 0.10

Standardized statistic

Theory-based analysis

Confidence Interval

Step 5: Formulate �Conclusions

Was the sample randomly selected from a larger population?

• No, the sample was not randomly sampled from a larger population of university students. However, if you believe that university students in California are not different in their body composition or the amount of weight they carry in their backpacks compared to other university/college students across the US, then we can generalize to all college/university students in the US.

Were the observational units randomly assigned to treatments?

• No, there was no random assignment in this study, so we cannot make any causal conclusions.

Conclusions

• While we have strong evidence that the mean ratio is different from 0.10 (10%), is the difference impressive?
• Recall that for children in elementary school and secondary school, pediatric surgeons recommended that the ratio of backpack weight to body weight not exceed 10% to 15%.
• The average ratio of 0.08 (7.8%) is well below this recommendation.
• It does seem that college/university students are carrying more reasonable weights in their backpacks.

Step 6: Look back and �ahead

• Looking back: Did anything about the design and conclusions of this study concern you? In particular, are there things that could have been done to give a better chance finding strong evidence of a true difference between the two groups?
• Always nice to have a larger sample size. A random sample. Researchers should try to replicate their results. These replicated studies could be at different colleges/universities across the U.S.
• Looking ahead: What should the researchers’ next steps be to fix the limitations or build on this knowledge?
• See if these results hold when other variables are looked at. For example do these ratios hold within subpopulations like males and females, or between different majors. Could backpack weights be decreasing due to e-books (all books on one laptop).

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Questions

• Any questions about the analysis?
• THANK YOU!