Exam 1 Results

Lesson 1-5 Scoring

Review for Exam 1

Review:

• Normal Curve Problems
• Exam 1 on Wed

Presentation:

Review topics for Exam 1

• Distributions
  • Stemplots
  • Histograms
• Measures of Center
  • Mean
  • Median
  • Mode
  • Percentiles
• Boxplots
  • Quartiles
  • 5-number summary
  • IQR
• Measuring the “spread” of the data
  • Variance
  • Standard Deviation
• Normal Curves
  • Standard Normal
  • Z-Scores
  • Z-table
  • Solving normal curve problems
    • less than
    • greater than
    • between
    • reverse look up (solve for x rather than z)

Lesson 5: Solving Normal Curve Problems

Review:

• Standard Normal Calculations with Z-Scores
• Assignment Grading
• Exam 1 on Wed, Sep 13

Presentation:

Solving Normal Curve Problems

• Find probabilities for Z-Scores with the Z-Table (use this link or back page of textbook)
• Normal Distribution Probability 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 Probabilities
  • Draw a picture of the distribution
  • Convert given values (a, b) to Z-Scores and locate on the horizontal axis
  • Look up corresponding probabilities 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 probability 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 probabilities from the Z-Table
      • Shade the area under the curve between the two Z-Scores
      • Find the probability for the larger value (further to the right)
      • Find the probability for the smaller value (further to the left)
      • Subtract the smaller from the larger to find the “Between” probability
• Inverse Normal Curve calculations
  • Z = (x – xbar)/s
  • x = xbar + Z * s
  • xbar = mean
  • s = standard deviation
  • Examples
    • Step through examples 1.29, 1.30 and 1.31 on pp. 65-66
    • What proportion of observations on a standard Normal curve are less than Z=1.47?
    • Use 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?

Assignment:

• 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).
    • What proportion of students scored less than 600?
    • What proportion of students scored greater than 600?
    • What proportion of students scored between 600 and 650?
    • What score is necessary to be in the top 5% of student test takers?
    • 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).
    • What proportion of students scored less than 600?
    • What proportion of students scored greater than 600?
    • What proportion of students scored between 600 and 650?
    • What score is necessary to be in the top 5% of student test takers?
    • 60% of students will score above x on the exam. What is x?

Study:

• Read pp. 64-68

Lesson 4: Standard Normal Calculations with Z-Scores

Review:

• Standard Deviation
• Normal Curves
• Exam 1 on Wed, Sep 13

Presentation:

• Standard Normal Distribution
  • Mean = 0
  • Standard Deviation = 1
  • 68-95-99.7 rule
• Normal Calculations video
• Z-Scores
  • (x – mean)/std dev
  • Example (p. 61)
    • 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
  • Calculating proportions/probabilities with Z-scores
    • Z-table
    • less than
    • greater than
    • between

Assignment:

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-year-old woman who is 6 feet tall? Label point.
• Between what two values do the heights of the central 95% of young women lie? Shade this area.
• 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 American College Testing program (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.
• Assuming that both tests measure the same kind of ability, who has the higher 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
• −1 < z < 2
• -2 < z < 0

Study:

• Review AgainstAllOdds_StudentGuide_Unit08
• Read Ch. 1 pp. 61-64

Lesson 3: Standard Deviation and Normal Curves

Review:

• Measures of Center
• Boxplots
• Slack
• Attendance

Presentation:

• Standard Deviation
  • How to calculate
    • See formulas on pp. 40-41
    • Example 1.19 on p. 41
    • Video
    • Demonstrate with student height data

• Normal Curves
  • 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 Rule
    • Video
    • Demonstrate with student height data

Assignment:

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.
54 59 35 41 46 25 47 60 54 46 49 46 41 34 22 

Calculate the mean and standard deviation

Problem 2. Six ninth-grade students and six 12th-grade students were asked: How many movies have you seen this month? Here are their responses.
Ninth-grade students: 5, 1, 2, 5, 3, 8
12th-grade students: 4, 2, 0, 2, 3, 1

Calculate the mean and standard deviation for each of these data sets. Which is more spread out, the ninth-grade or 12th-grade data set?

Problem 3. Height of 4-year-old boys is approximately normally distributed with mean μ = 40 inches and standard deviation σ = 1.5 inches. Sketch a copy of this curve. On the horizontal axis, mark the location of the mean. Then mark points on the horizontal axis that are one standard deviation on either side of the mean.

Problem 4. IQ scores are normally distributed with mean 100 and standard deviation 15.

Sketch a copy of this curve. Then on the horizontal axis mark the mean, 100, and one standard deviation on either side of the mean. Label the horizontal axis as IQ Test Scores. Scores above 120 represent superior intelligence. Shade the area under your normal curve that represents the proportion of people with superior intelligence.

Study:

• Textbook
  • Ch. 1, p. 40-44
• Preview
  • Normal Calculations: AgainstAllOdds_StudentGuide_Unit08
  • Checking Assumptions of Normality: AgainstAllOdds_StudentGuide_Unit09

Lesson 2: Measures of Center and Boxplots

Review:

• Stemplots
• Histograms
• Assignment for Lesson 1
• Slack
  • student name with Slack username
• Attendance
• Tutoring Available
  • David Mould: dj.mould@pack.csupueblo.edu
  • Fridays 9-11 am in HSB 108 or by appointment

Presentation:

• Measures of Center
  • Mean
    • Same as Average
    • Sum of values divided by count, or ∑x/n
    • Example with height data
  • Median
    • Center value within ordered sequence of values
    • Same as 50th Percentile
    • Position = (n + 1)/2
      • n = total number of observations
      • Another approach with larger data sets, position = 0.50*n
      • If the position is between two numbers then take the average of those two values.
    • Example with height data
  • Video: Measures of Center

• Percentiles and Boxplots
  • Percentiles
  • To find the xth percentile
    • calculate x/100 * n (rounded to the nearest integer or take average of two values on either side)
    • result is the position of the value in an ordered (smallest to largest) data set
    • 25th percentile = 25/100 * n
    • 50th percentile = 50/100 * n = Median
    • 75th percentile = 75/100 * n
    • Example with height data
  • Boxplots
  • Five number graphic summary
    • 25th Percentile
    • 50th Percentile = Median
    • 75th Percentile
    • Minimum value
    • Maximum value
    • Example with height data
  • Video: Boxplots
  • Examples
    • Stacking Bar Charts, Histograms and Boxplots
    • Boxplot paired with Histogram

Assignment:

Problem 1.

Here are the starting salaries, in thousands of dollars, offered to 20 students who earned bachelor’s degrees in computer science in 2011.
63 56 66 77 50 53 78 55 90 65 64 69 59 76 48 54 49 68 51 50
a. Make a stemplot.
b. Find the median, mean and mode.
c. Find the five-number summary.
d. Make a boxplot.
e. Compute the range and interquartile range (IQR).

Problem 2.

A consumer testing lab measured calories per hot dog in 20 brands of beef hot dogs. Here are the results:
186 181 176 149 184 190 158 139 175 148 152 111 141 153 190 157 131 149 135 132
a. Make a stemplot.
b. Find the median, mean and mode.
c. Find the five-number summary.
d. Make a boxplot.
e. Compute the range and interquartile range (IQR).

These problem descriptions are from “Against All Odds”, modified slightly and copied here for convenience.

Please upload to Slack channel #lesson02

Study:

• Textbook Read Ch. 1 p. 30-38
• Preview:
  • Standard Deviation: AgainstAllOdds_StudentGuide_Unit06
  • Normal Curves: AgainstAllOdds_StudentGuide_Unit07

Lesson 1: Stemplots and Histograms

Review:

• Intro to Statistics
• Syllabus
• Slack
• Textbook
• Attendance
  • Student height

Presentation:

• Data Distributions
  • The “shape” of the data
  • Distribution of Exam Scores
  • Types of distributions
    • Skewed, bi-modal, u-shaped, uniform, etc
  • Make a picture: Data Visualization is the future
    • “patterns” and “participation”
• Unit 2: Stemplots
  • Video
  • Student height data on the board
  • Demonstrate stem and leaf plot construction
• Unit 3: Histograms
  • Video
  • Student height data on the board
  • Demonstrate histogram construction

Assignment:

• Complete Exercises #2 on p. 13-14 in AgainstAllOdds_StudentGuide_Unit02
• Complete Exercises #3 on p. 18-19 in AgainstAllOdds_StudentGuide_Unit03
• Instructions for submitting assignment
  • Compile and organize solutions into one, easy to read document
  • Convert to PDF
  • Upload a single PDF file to Slack

Study:

• Preview
  • AgainstAllOdds_StudentGuide_Unit04
  • AgainstAllOdds_StudentGuide_Unit05
• Textbook
  • Ch. 1, p. 9-15

Intro to Statistics

Presentation:

• Attendance
• Course Overview
• Syllabus (PDF): BUSAD 265 FA17 Holman
• Video: What is Statistics?

Assignment:

• Join my “Slack” team
  • Pueblo Stats
  • This is where you’ll upload completed exercises (PDF only)
  • This is a forum where you can ask questions, publicly or privately, anytime
  • This is an experiment, please help make it a useful learning environment

Study:

• Preview materials
  • AgainstAllOdds_StudentGuide_Unit02 (Stemplots)
  • AgainstAllOdds_StudentGuide_Unit03 (Histograms)
• Link to Companion Textbook
• Textbook Reading
  • Ch. 1, p. 1-9