Lesson 8: Sampling and Inference

Review:

• Exam 1 Results

• 

Presentation:

• Census and Sampling
  • Census = attempt to count entire population
    • US Census every 10 years
    • Most costly non-military federal government operation (except bailing out Wall Street)
  • Sample = gather info from a portion of the population
    • Sample Types
      • Voluntary response sample
        • Complete an optional survey
        • Inherent bias
        • Example: Survey Monkey or Google Forms
      • Simple random sample
        • select from a population
        • each individual has equal chance of being selected
        • Example: random selection of subset of enrolled students
      • Stratified random sample
        • divide population into groups or strata
        • random sampling, equal chance of selection within each group
        • Example: random selection within major (30% CIS, 35% Mgmt, 35% Econ)
  • Video: Census and Sampling

Activity:

• Design a 3-6 question survey (on paper)
• Select a topic of interest to you and your classmates (e.g., favorite food, study habits, music preferences, coffee vs. tea, campus activities).
• Survey must include at least one categorical, one ordinal (ranking), and one numerical question (see examples below).
• Keep it short and simple.

Example Survey:

Example Survey Questions: Student Study & Lifestyle Habits

Q1. Categorical
What is your preferred study environment?

• ☐ Library
• ☐ Coffee shop
• ☐ Dorm/apartment
• ☐ Outdoors
• ☐ Other

Q2. Ordinal
How often do you review class notes after lecture?

• ☐ Never
• ☐ Rarely
• ☐ Sometimes
• ☐ Often
• ☐ Always

Q3. Numeric
On average, how many hours do you study in a typical week? ____ hours

How many credit hours are you registered for this term? ____ credits

Lesson 7: Review for Exam 1

Review:

• Normal Curve problems
• Exam 1 on Thu

Presentation:

• Exam format
  • No phones, laptops, tablets, watches
  • No leaving the classroom during the exam
  • Bring a Z-table
  • You may bring a 1-page “cheat sheet”
    • Side 1 – any helpful content you’d like
    • Side 2 – leave blank
  • No “sharing” with your neighbor
• Review Topics
  • Data distributions
    • Stemplots
    • Histograms
  • Measures of Central Tendency
    • Mean
    • Median
    • Percentiles, Quartiles
    • 5-number summary
    • Boxplots
  • Measure of spread/variability
    • Variance
    • Standard Deviation
  • Normal Distribution
    • Standard Normal
    • Z-Scores
    • Z-table
  • Normal Curve problems
    • “Less than”
    • “Greater than”
    • “Between”
    • Inverse or Reverse look-up (solve for X instead of Z)

Activity:

• Practice exam

Assignment:

• Study for Exam 1

Lesson 6: Solving Normal Curve Problems

Review:

• Standard Normal Calculations with Z-Scores
• Review Lesson 5 problems

Presentation:

Solving Normal Curve Problems

• Against All Odds – Video Unit 7
• Find probabilities for Z-Scores with the Z-Table (use this link or back page of textbook)
• Normal Distribution Problems: 3 Types
  • p(X<a) = “Less Than”
  • p(X>b) = “Greater Than”
  • p(a<X<b) = “Between” (between 2 values)
• Steps to finding Standard Normal Proportions
  • Draw a picture of the distribution
  • Convert given values (a, b) to Z-Scores and locate on the horizontal axis
  • Look up corresponding proportion in the Z-Table
  • Decide if it’s a “Less Than”, “Greater Than” or “Between” problem
    • If “Less Than”, shade under the curve to the left of Z-Score; the Z-Table proportion you found is the answer.
    • If “Greater Than”, shade under the curve to the right of the Z-Score; subtract the Z-Table probability from 1 to find the answer.
    • If “Between”, you will have 2 proportions from the Z-Table
      • Shade the area under the curve between the two Z-Scores
      • Find the proportion for the larger value (further to the right)
      • Find the proportion for the smaller value (further to the left)
      • Subtract the smaller from the larger to find the “Between” proportion
• Inverse Normal Curve calculations
  • Z = (x – xbar)/s
  • x = xbar + Z * s
  • xbar = mean
  • s = standard deviation
• Problem Examples
  • What proportion of observations on a standard Normal curve are less than Z=1.47?
  • Using N(1026,209) …
    • What proportion of students who take the SAT have scores of at least 820?
    • What proportion of students who take the SAT would be NCAA “partial qualifiers”, i.e., between 720 and 820.
    • What score is necessary to place in the top 10% of all students taking the SAT?

Activity:

• Problem 1
  • In a recent year, 10th grade students took a standardized English language exam. The mean score was 572 and the standard deviation was 51, i.e., N(572,51).
    1. What proportion of students scored less than 600?
    2. What proportion of students scored greater than 600?
    3. What proportion of students scored between 600 and 650?
    4. What score is necessary to be in the top 5% of student test takers?
    5. 60% of students will score above x on the exam. What is x?
• Problem 2
  • Repeat each question in Problem 1 using a different normal distribution of scores: N(505,110).
    1. What proportion of students scored less than 600?
    2. What proportion of students scored greater than 600?
    3. What proportion of students scored between 600 and 650?
    4. What score is necessary to be in the top 5% of student test takers?
    5. 60% of students will score above x on the exam. What is x?

Lesson 5: Normal Curve and Z-Scores

Review:

• 8675309
• Standard Deviation
• Exam 1 on Mon Feb 16

Presentation:

• Normal Curves
  • Normal Curve explained in 1 min
  • Normal Curves explained on 1 page
  • Density Curves
    • Height of curve indicates proportion of values
    • Area under the curve = 1.0
    • Any sub-area under the curve is then a proportion (% of values)
  • Normal Curves
    • A special case of density curves
    • Bell shaped and symmetrical
    • Mean and Standard deviation
    • 68 – 95 – 99.7 Empirical Rule
    • Against All Odds – Video Unit 7
    • Demonstrate with student height data
• Standard Normal Distribution
  • Mean = 0
  • Standard Deviation = 1
  • Calculating Z-Scores
    • (x – mean)/std dev
    • Example
      • Young women heights normally distributed
      • Mean = 64.5 in
      • Standard Deviation = 2.5 in
      • Z-Score for woman 68 inches tall
        • Z = (68 – 64.5)/2.5 = 1.4
      • Z-Score for woman 60 inches tall
        • Z = (60 – 64.5)/2.5 = -1.8
  • 68 – 95 – 99.7 Empirical Rule
    • Add or subtract multiples of Standard Deviation from the mean to find specific heights
    • For example, 64.5 + 5 = 69.5 is 2 standard deviations above the mean
    • And, 64.5 – 5 = 59.5 is 2 standard deviations below the mean
    • Using the Empirical Rule, 95% of women are between 59.5 and 69.5 inches tall.

Activity:

Problem 1. The distribution of heights of young women (18–24 years old) is approximately normal with mean = 64.5 inches and standard deviation = 2.5 inches.

• Draw a normal curve with a horizontal axis. Label mean.
• What is the z-score for a 20-yr-old woman who is 6 ft tall? Label point.
• What % of young women are more than two standard deviations taller than the mean?

Problem 2. There are two national college-entrance examinations, the SAT and the ACT. Scores on individual SAT exams are approximately normal with mean = 500 and standard deviation=100. Scores on the ACT exams are approximately normal with mean = 18 and standard deviation = 6.

• Julie’s SAT Math score is 630. John’s ACT Math score is 22.
• Calculate the standardized Z-scores for both scores.
• Assuming both tests measure similar ability, who has the better score?
• What percent of all SAT scores are above 600?

Problem 3. Find the percentage of observations from a standard normal distribution that satisfy the following. Also, draw a normal curve and shade the corresponding area under the curve.

• z < −1
• z < 1
• z > -2
• z > 2

Problem 4. The Indiana Statewide Testing for Educational Progress (ISTEP) is a program for assessing the skills of students in various grades. In a recent year, 76,531 tenth grade Indiana students took the English language arts exam. The mean score was 572 and the standard deviation was 51. Assume the ISTEP scores are approximately Normally distributed to answer the following questions.

1. Use the 68-95-99.7 rule to give a range of scores that includes 95% of students.
2. Use the 68-95-99.7 rule to give a range of scores that includes 99.7% of students.
3. Calculate the Z-score for a student who scored 600.
4. Calculate the Z-score for a student who scored 500.

Lesson 4: Measuring Variance and Standard Deviation

Review:

• Measures of Center and Boxplots
  • Mean, Median, Percentiles, Quartiles
  • Five-number summary
  • Min, Q1, Median, Q3, Max
  • Quartiles and Boxplots (U. Illinois)
  • Boxplot Grapher: http://www.imathas.com/stattools/boxplot.html

Presentation:

• Demonstrate calculation
  • Calculate the standard deviation for a simple data set: 2, 4, 6, 8, 10
  • Against All Odds – Video Unit 6
  • Calculate the standard deviation for the following (n=10) exam scores:
    • 80 73 92 85 75 98 93 55 80 90
    • Demonstrate by hand
    • Demonstrate in Google Sheets

Activity:

Problem 1. Below are the number of home runs that Babe Ruth hit in each of his 15 years with the New York Yankees, 1920 – 1934. Calculate the mean and standard deviation.
54 59 35 41 46 25 47 60 54 46 49 46 41 34 22 

Problem 2. Six 9th-grade students and six 12th-grade students were asked: how many movies have you seen this month? See their responses below. Calculate the mean and standard deviation for each data set. Which is more spread out, the 9th-grade or 12th-grade data?
9th-grade: 5, 1, 2, 5, 3, 8
12th-grade: 4, 2, 0, 2, 3, 1

Assignment:

Repeat Problems 1 and 2 (above) using Sheets