Category: s26-busad265

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
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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?
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Lesson 5: Normal Curve and Z-Scores

Review:

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

Presentation:

  • Normal Curves
  • 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.
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Lesson 4: Measuring Variance and Standard Deviation

Review:

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

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Lesson 3: Measures of Center, Percentiles and Boxplots

Review:

  • Visualizing the shape of the distribution
  • Stemplots
  • Histograms
    • Against All Odds Video – Unit 3
    • Demonstrate spreadsheet histogram construction
    • Histogram ≠ Bar Chart
    • UN health and economy (Sheets)

Presentation:

  • Measures of Center
    • Mean
      • Same as Average
      • Sum of values divided by count, or ∑x/n
      • n = total number of observations/measurements/values
      • Example
    • Median
      • Center value within ordered sequence of values
      • Same as 50th Percentile
      • Position = 0.50*n
        • n = total number of observations
        • round to the nearest whole number
        • alternative for small (n<30) data sets = (n+1)/2
      • Example
    • 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 Female height data
    • Boxplots
    • Five number summary
      • 25th Percentile = Q1
      • 50th Percentile = Q2 = Median
      • 75th Percentile = Q3
      • Minimum value
      • Maximum value
      • Range = Max – Min
      • Interquartile Range (IQR) = Q3 – Q1
      • Example with Male height data
    • Video: Boxplots

Activity:

Problem 1.

Here are the starting salaries, in thousands of dollars, offered to 21 students who earned bachelor’s degrees in computer science in 2021.
63 56 66 77 50 53 78 55 90 65 64 69 59 76 48 54 49 68 51 50 70
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 19 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
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 (#1 and #2) are from “Against All Odds”, modified slightly and copied here for convenience.

Problem 3.

Use the MPG data for top selling midsize cars in the US and European markets from Lesson 2.

a. Find the five-number summaries.
b. Produce 2 boxplots, one for each market, and put them both on the same chart to facilitate comparison.

Assignment:

  • For class on Monday
    • Use the UN health and economy data (filter to display North and South America only).
    • Use Google Sheets (or Excel) to create a histogram for Life Expectancy, Infant Mortality and GDP Per Capita (3 separate histograms)
    • Compute the 5-number summary for Life Expectancy, Infant Mortality and GDP Per Capita (1 5-number summary for each variable).
    • Be prepared to share your histograms and 5-number summaries.
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Lesson 2: Data Distributions with Stemplots and Histograms

Review:

  • Attendance
  • Intro to Statistics
  • Syllabus
  • Textbook

Presentation:

  • Data Distributions
    • The “shape” of the data
    • Distribution of Exam Scores
    • Types of common distributions
      • Normal Distribution: A symmetric bell-shaped curve, where most data points cluster around the mean.
      • Uniform Distribution: A flat distribution, where all outcomes are equally likely.
      • Skewed Distribution (Right Skewed): A distribution with a longer tail on the right, indicating a few high values.
      • Bimodal Distribution: A distribution with two distinct peaks, showing two different groups within the data.
      • Exponential Distribution: A distribution with a rapid decline, often used to model time between events.
      • Poisson Distribution: A distribution that models the number of events occurring within a fixed interval of time.
      • Binomial Distribution: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It’s widely used in scenarios where there are only two possible outcomes, often labeled as “success” and “failure.”
    • Visualize the distribution of data you want to understand
      • Picture is worth a thousand words
      • Not much training necessary to look at a data graphic
      • Improved approach for detecting “patterns” and inviting “participation“
      • Use stemplots for small datasets
      • Use histograms for large datasets
  • Stemplots (aka Stem and Leaf plots)
    • Against All Odds Video – Unit 2
    • Demonstrate stemplot construction
      • Female heights (cm): 152, 157, 160, 165, 165, 168, 170, 170, 170, 173, 173, 180
      • Stemplot Rules
        • Keep rows and columns neatly aligned
        • Don’t use commas, decimals or other visible separators
        • Display gaps in data with empty space
    • Activity: Using the vehicle mpg data (see below), create three stemplots
      • Top Selling Midsize Cars in the US
      • Top Selling Midsize Cars in Europe
      • Back to back comparison – US vs Europe
  • Histograms

Activity Data:

Top Selling Midsize Cars in the US – MPG

  • Toyota Camry – 32
  • Honda Accord – 33
  • Nissan Altima – 30
  • Subaru Outback – 29
  • Ford Fusion – 27
  • Chevy Malibu – 26
  • Kia Optima – 27
  • Volkswagen Passat – 29
  • Subaru Legacy – 29
  • Hyundai Sonata – 30

Top Selling Midsize Cars in the Europe – MPG

  • Volkswagen Passat – 35
  • Skoda Superb – 66
  • Opex Insignia – 53
  • Ford Mondeo – 48
  • Renault Talisman – 35
  • Toyota Avensis – 63
  • Mazda 6 – 30
  • Peugeot 508 – 62
  • Kia Optima – 58
  • Hyundai i40 – 67
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Lesson 1: Introduction to Statistics

Presentation:

  • Attendance
  • Why Statistics?
  • Statistics is different
    • This won’t be the same old math class.
    • Give me and statistics a chance.
  • Course Overview
    • Syllabus (pdf)
    • Bookmark this course page
    • Classroom activities
    • Exams
      • Reviews
      • Format of Midterms vs the Final
      • Make-up policy
    • Grading
      • Assignments/activities
      • Exams
      • Scores posted on Bb
    • Textbook (optional): Link to Used Textbook on Amazon
    • Mostly pen/paper, some spreadsheet work
  • Tips for success in this class
    • Attend class!
    • Engage with in-class activities and assignments
    • Attend the review session prior to each exam
    • Study by practicing the problems we work on during class

Activity:

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