# Statistical inference | Be US Census Bureau’s Chief Analyst (Lesson 4 of 5) | 6-8 ### Student Objective

Students will be able to:
1. Make statistical inferences from sample means and proportions, using US American Community Survey data for one state

### Instructions

Materials Needed

Vocabulary Introduced:

• statistically significant
• statistical inference
• t-tests
• means
• proportions

### Step 1: Compare two averages: “Is the difference between these two averages significant?”

• Think-Pair-Share:
• Show students the following table of information and ask them, “How has each state changed in terms of the percentage of residents that own their home from 2010 to 2019?  Where do you see large changes?  Modest changes?  And little or no change?”
 Percentage of Homes Owned by Year and State State Name 2010 2019 Montana 50.00% 100.00% Nebraska 83.33% 61.54% New York 41.67% 46.15% Wisconsin 41.67% 69.23%

• Exemplar Response:
• There is a clear increase in home ownership for Montana
• Modest increase in Wisconsin
• There is a clear decrease in home ownership for Nebraska
• There is not much of a difference in home ownership for New York
• Push Thinking:
• It’s clear for places like Montana, Nebraska, and Wisconsin that the percentage of home ownership has clearly changed.
• But, what about New York?
• How do we know if that small percentage difference is actually that different than 10 years ago?

### Step 2: Enduring Understanding

• Today we ask ourselves, “How do we know if the difference between statistical values from two different samples are statistically significant?”
• Define key terms:
• Statistical significance = the calculation of a statistic is extremely unlikely to be common or expected, as decided upon a statistical significance test
• Statistical inference = a conclusion based off the analysis of statistical calculations (that are either statistically significant or not)
• T-Tests = a process in the form of a test used to determine whether a statistic is probably significant or not
• means = statistically fancy word for averages

### Step 3: Calculate Sample Statistics According to Their Decided Criteria for Evaluation of One State over 2 Time Periods

• Using the data from “FiveStatesOfData_Students” (each student should have their copy) and using pivot Tables in Google Sheets, have students:
• calculate the averages for their selected variables for analysis of the ONE state they chose, for each year of analysis (2010 and 2019)
• They should fill out the following table on a google slide assigned to them (Teacher makes copy and pre-assigns)

### Step 4: Conceptual Understanding of Distributions and “Extreme Values”

• [Insert mini-lesson on conceptual understanding of “extreme values” in a distribution]
• Bottom line:
• if a calculated statistic (like a mean difference = average1 – average 2) lands on the extreme right or left tail of a distribution, then that statistic may actually be a part of an entirely different distribution.
• Therefore, a t-test measures whether a difference in two means from two different samples are extremely far away from 0 (which would imply that there is a statistically significant difference between two means)

### Step 5: Calculate Mean Differences and other Important Values to Prepare for a t-test

• Students fill out values in an assigned google slide titled, “Stat Significance” (teacher makes multiple copies, assigns, and shares)
• Students fill out the following columns for their respectively chosen variables of analysis:
• Variable Name
• Number of Houses 2010
• Number of Houses 2019
• Average of Variable 2010
• Average of Variable 2019
• Difference in Average
• Teacher provides students the following Standard Errors for students to fill out their table:
• Standard Deviation for each variable (in slides 3-7)
• NOTE: For the variables that are proportions:
• the “standard deviation” is actually the calculation of: P1 x (1 – P1) and P2 x (1 – P2)

### Step 6: Students Run T-Tests to compare mean differences and Make Statistical Inferences

• Have students independently walk through T-Test procedures following this scaffolded document for each of their variables for evaluation (make a copy and have students start from Step 4.0 on page 6 and down, after deleting the previous steps)
• NOTE:
• for the variables that are proportions, look on page 13 to give students the modified STEP 4.0 for running a T-Test using proportions

### Step 7: Students Fill Out Table Summarizing Their Results of T-Tests

• Now that students have performed their T-Tests for each of their variables, have them fill out their “Stat Significance” slide for the following columns:
• T-Stat Value
• P-value
• Stat Sig?

### Step 8: Stamp End of Lesson and Preview Next Lesson

• Today we asked ourselves, “How do we know if the difference between statistical values from two different samples are statistically significant?”
• We ran a statistical inference test, the T-Test, to answer our question.
• Next lesson, we will synthesize our results and conclusions in a report to the Governor of the state you chose!

### Justification

In this activity, students use real census data from the American Community Survey (ACS) to run 2-sample, 2-tailed T-tests on sample averages and proportions.