Category: f19-busad265

Review for Exam 1

Review:

  • Normal Curve problems
  • Exam 1 on Wed Sep 18

Presentation:

  • Exam format
    • Open note, open book, open calculator
    • No phones, laptops, tablets, watches
    • No “sharing” with your neighbor
    • No leaving the classroom during the exam
  • Review Topics
    • Data distributions
      • Stemplots
      • Histograms
    • Measures of Central Tendency
      • Mean
      • Median
      • Percentiles, Quartiles
      • 5-number summary
    • 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:

Assignment:

  • Study for Exam 1
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Lesson 6: Solving Normal Curve Problems

Review:

  • Review Quiz 2 and Averages
  • Standard Normal Calculations with Z-Scores
  • Exam 1 on Wed, Sep 18

Presentation:

Solving Normal Curve Problems

  • Video (0:45-3:45, 6:10-11:45)
  • 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 (see pp. 63-67)
    • 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?

Study:

  • Read Text pp. 62-68
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Lesson 5: Normal Curve and Z-Scores

Review:

  • Quiz 2
  • Standard Deviation
  • Exam 1 on Wed Sep 18

Presentation:

  • 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 (until 4:30)
      • Demonstrate with student height data
  • Standard Normal Distribution
    • Mean = 0
    • Standard Deviation = 1
    • Calculating 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

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.
  • 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 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

Assignment:

  • Text Ch. 1 pp. 61-64
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Lesson 4: Measuring Variance and Standard Deviation

Review:

Presentation:

  • Demonstrate calculation
    • Find the variance and the standard deviation for the following (n=10) exam scores: 80 73 92 85 75 98 93 55 80 90
    • Demonstrate in Google Sheets
  • Video

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
  • Using the Student Height Data, calculate mean and standard deviation in Sheets
  • Text Ch. 1, p. 40-44
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Lesson 3: Measures of Center, Percentiles and Boxplots

Review:

  • Quiz 1
  • Data Distributions
    • Stemplots
    • Histograms
  • Attendance (use Quiz)

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 with height data
    • 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 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 Female height data
        • 152, 157, 160, 165(2), 168, 170(3), 173(2), 180
    • 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
        • 160, 168, 170(2), 173(3), 175(3), 177, 178(15), 180(4), 183(5), 185(4), 188(5), 191(4), 193, 196(2), 203
    • Video: Boxplots
    • Examples

Activity:

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 (#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 1.

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

Assignment:

  • Text: Ch. 1 p. 30-38
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Lesson 2: Data Distributions with Stemplots and Histograms

Review:

  • Attendance
  • Intro to Statistics
  • Syllabus
  • Textbook
  • Assessment Answers
    • a, d, e, d, b, e, d, e, b, e

Presentation:

  • Data Distributions
    • The “shape” of the data
    • Distribution of Exam Scores
    • Types of distributions
      • Skewed, bi-modal, u-shaped, uniform, etc
    • Theoretical Probability Distributions
    • Visualize the distribution
      • Picture is worth a thousand words
      • Not much training necessary to look at a data graphic
      • Improved approach for “patterns” and “participation“
      • Use stemplots for small datasets
      • Use histograms for large datasets
  • Unit 2: Stemplots
    • Video (0:45-2:00; 6:00-11:00)
    • Demonstrate stemplot construction
      • Female heights (cm)
      • 152, 157, 160, 165, 165, 168, 170, 170, 170, 173, 173, 180
      • Rules
        • Keep rows and columns neatly aligned
        • Don’t use commas
        • Display gaps in data with empty space
  • Unit 3: Histograms

Activity:

  • Using the vehicle mpg data (below), create three stemplots
    1. Top Selling Midsize Cars in the US
    2. Top Selling Midsize Cars in Europe
    3. Back to back comparison – US vs Europe
  • Using the same vehicle mpg data (below), create two histograms
    1. Top Selling Midsize Cars in the US
    2. Top Selling Midsize Cars in Europe

Study:

  • Text: Ch. 1, p. 9-15

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

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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