Author: Justin

Lesson 8: Inverse Normal Curve Problems

Review:

  • Normal Curve problems
  • 3 types: Less than, Greater than, Between
  • Schedule
    • Review for Exam 2 on Mon Sep 28
    • Exam 2 on Wed Sep 30

Presentation:

  • Inverse Normal Curve calculations
    • Z = (x – xbar)/s
    • x = xbar + Z * s
    • xbar = mean
    • s = standard deviation
  • Problem Example
    • 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:

  • For each problem, draw and shade the Normal curve first. Use the Z-table to find the Z-score corresponding to the stated proportion.
  • Assume SAT scores are approximately Normal with mean = 1026 and standard deviation = 209, i.e., N(1026, 209)
    • Standard Normal Curve
      What Z-score has 75% of observations below it?
    • Standard Normal Curve
      What Z-score has 20% of observations below it?
    • SAT — 80th Percentile
      What SAT score separates the top 20% of students from the remaining 80%?
    • SAT — Bottom 25%
      What SAT score separates the bottom 25% of students from the rest?
    • SAT — Top 5%
      What SAT score would a student need to be in approximately the top 5% of SAT scores?
    • SAT — Bottom 10%
      Below approximately what SAT score would the lowest 10% of students fall?
    • SAT — Middle 80%
      Find the two SAT scores that contain the middle 80% of students.
    • SAT — Middle 50%
      Find the two SAT scores that contain the middle 50% of students.
  • Exam Scores
    Scores on an exam are approximately Normal with a mean of 76 and a standard deviation of 9.
    What score marks the 85th percentile?
  • Commute Times
    Employee commute times are approximately Normal with a mean of 32 minutes and a standard deviation of 8 minutes.
    What commute time separates the shortest 30% of commutes from the rest?
  • Annual Salaries
    Starting salaries for a group of graduates are approximately Normal with a mean of $58,000 and a standard deviation of $7,500.
    What salary marks the beginning of the top 15%?
  • Package Weights
    Package weights are approximately Normal with a mean of 24 ounces and a standard deviation of 3.2 ounces.
    Below what weight would the lightest 10% of packages fall?
  • Customer Spending
    Customer purchases are approximately Normal with a mean of $84 and a standard deviation of $18.
    What purchase amount marks the 70th percentile?
  • Delivery Times
    Delivery times are approximately Normal with a mean of 42 minutes and a standard deviation of 6 minutes.
    What delivery time separates the fastest 5% of deliveries from the remaining 95%?
  • Employee Productivity
    Daily output is approximately Normal with a mean of 115 units and a standard deviation of 14 units.
    Find the two values that contain the middle 90% of employee output.
  • Monthly Electricity Use
    Household electricity use is approximately Normal with a mean of 920 kWh and a standard deviation of 180 kWh.
    Find the two values that contain the middle 60% of households.
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Lesson 7: Types of Normal Curve Problems

Review:

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

Presentation:

Solving Normal Curve Problems

  •  Z-Table
  • 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
  • Demonstration Problem 1
    • Suppose exam scores are normally distributed with a mean of 75 and a standard deviation of 10.
    • Find the proportion of students who score:
      • Less than 85
      • Greater than 85
      • Between 65 and 85
    • Z-score for 85:z=85−7510=1.00z=\frac{85-75}{10}=1.00
    • From the Z-table:P(Z<1.00)=0.8413P(Z<1.00)=0.8413
    • Less than 85: P(X<85)=0.8413P(X<85)=0.8413
    • Greater than 85: P(X>85)=1−0.8413=0.1587P(X>85)=1-0.8413=0.1587
    • For 65:z=65−7510=−1.00z=\frac{65-75}{10}=-1.00
    • From the Z-table:P(Z<−1.00)=0.1587P(Z<-1.00)=0.1587
    • Between 65 and 85: P(65<X<85)=0.8413−0.1587=0.6826P(65<X<85)=0.8413-0.1587=0.6826
  • Demonstration Problem #2
    • Suppose delivery times are normally distributed with a mean of 28 minutes and a standard deviation of 6 minutes.
    • Find the proportion of deliveries that take:
      • Less than 33 minutes
      • Greater than 33 minutes
      • Between 24 and 33 minutes
    • Z-score for 33 minutes:z=33−286=0.83z=\frac{33-28}{6}=0.83
    • From the Z-table:P(Z<0.83)=0.7967P(Z<0.83)=0.7967
    • Less than 33 minutes: P(X<33)=0.7967P(X<33)=0.7967
    • Greater than 33 minutes: P(X>33)=1−0.7967=0.2033P(X>33)=1-0.7967=0.2033
    • Z-score for 24 minutes:z=24−286=−0.67z=\frac{24-28}{6}=-0.67
    • From the Z-table:P(Z<−0.67)=0.2514P(Z<-0.67)=0.2514
    • Between 24 and 33 minutes: P(24<X<33)=0.7967−0.2514=0.5453P(24<X<33)=0.7967-0.2514=0.5453

Activity:

  • For each problem, draw and shade the curve, calculate the z-score(s), use the Z-table, and find the requested proportion.
    • Exam scores are normally distributed with a mean of 72 and a standard deviation of 9. What proportion of students score less than 80?
    • Adult heights are normally distributed with a mean of 67.5 inches and a standard deviation of 2.8 inches. What proportion are taller than 70 inches?
    • Delivery times are normally distributed with a mean of 32 minutes and a standard deviation of 6 minutes. What proportion of deliveries take between 27 and 39 minutes?
    • Monthly sales are normally distributed with a mean of $48,000 and a standard deviation of $6,500. What proportion of months have sales below $55,000?
    • SAT Math scores are approximately normally distributed with a mean of 510 and a standard deviation of 95. What proportion of students score above 625?
    • Package weights are normally distributed with a mean of 19.5 pounds and a standard deviation of 2.2 pounds. What proportion weigh between 17 and 22 pounds?
    • Employee commute times are normally distributed with a mean of 26 minutes and a standard deviation of 7 minutes. What percentage of employees commute more than 34 minutes?
    • Restaurant bills are normally distributed with a mean of $42 and a standard deviation of $9. What percentage of bills are between $35 and $51?
    • Battery life is normally distributed with a mean of 10.5 hours and a standard deviation of 1.7 hours. What percentage of batteries last less than 12 hours?
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Lesson 4: Time Series Analysis

Review:

  • DataCamp scores
  • New Google Sheets Add-In
  • Exam 1 next week
    • Bring a laptop and be sure it’s charged

Presentation:

What Is Time Series Data?

  • Define a time series as observations recorded over time.
  • Introduce notation:Yt=observed value at time tY_t = \text{observed value at time } t
  • Emphasize that order matters in time series data.
  • Examples:
    • Monthly sales
    • Weekly customer traffic
    • Quarterly profits
    • Daily stock prices
    • Annual population

Plot the Time Series Values on a Chart

  • Put time on the horizontal axis.
  • Put the observed variable YtY_t on the vertical axis.
  • Look for important patterns:
    • Trend — long-term upward or downward movement
    • Seasonality — repeating patterns over regular intervals
    • Random variation — unpredictable fluctuations

Trend Models

  • Review the idea of using regression to model change over time.
  • Linear trend:Yt=a+btY_t=a+bt
  • Consider polynomial trend models when the relationship isn’t a straight line (curve fitting).
  • Use trend models to generate forecasts for future periods.

Naive Forecasting

  • Simplest forecasting approach:Ft+1=YtF_{t+1}=Y_t
  • The forecast for the next period is simply the most recent observation.
  • Use as a baseline for comparison with other forecasting methods.

Moving Average (MA) Forecasting

  • Use an average of the most recent observations.
  • Example: 3-period moving averageFt+1=Yt+Yt−1+Yt−23F_{t+1}=\frac{Y_t+Y_{t-1}+Y_{t-2}}{3}
  • Moving averages smooth random fluctuations.
  • More time periods averaged = more smoothing.
  • More smoothing means slower reaction to changes.

Measuring Forecast Error

  • Compare forecasts with what actually occurred.
  • Forecast error:et=Yt−Fte_t=Y_t-F_t
  • Absolute error:∣Yt−Ft∣|Y_t-F_t|
  • Measure overall forecasting accuracy:MAE=∑∣Yt−Ft∣nMAE=\frac{\sum |Y_t-F_t|}{n}
  • Lower MAE indicates a more accurate forecasting method.
  • Note: MAE is also commonly called MAD in forecasting.

Compare Forecasting Methods

  • Naive forecast
  • Moving average
  • Trend model
  • Which method has the lowest MAE?

Time Series Forecasting Process

  • Time series forecasting uses past observations to estimate future values.
  • The basic forecasting process is:
  • Plot the data → identify patterns → choose a forecasting method → calculate forecasts → measure forecast error → compare models.

Demonstrate with fictional oil price data in Sheets

Activity:

  • Recreate the forecasting and error measurement approach.
  • Use the “Activity” data tab.
  • Look in the same fictional oil price data in Sheets file.
  • This is your practice exam.
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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 – Normal Calculations
  • Calculate Z-Scores
    • Z = (x – mean)/stdev
  • Use Z-Scores
    • Find probabilities with the Z-Table (download and print)
    • Z-Table tells you the proportion below or “less than” any specific Z-Score
    • The look-up table requires Z-scores with two decimal precision
  • Typical Steps in a simple Normal Curve problem
    • Draw a picture of the bell curve and label the axes
    • For a specific value of x, calculate Z = (x – mean)/stdev
    • Use the resulting Z-score to look up the corresponding “less than” proportion
    • For example, if we want to estimate the proportion of females below a certain height
      • In the video
        • mean female height = 63.8 in.
        • standard deviation = 4.2 in.
        • x = 70 in.
      • To calculate the Z score associated with 70 in.
        • Z = (70 – 63.8)/4.2 = 1.48
      • To calculate the proportion below Z = 1.48
        • Use the Z-Table (check both pages to be sure you find positive 1.48)
        • Look-up 1.48 by finding the row Z = 1.4 and then the column Z = .08
        • You should find the value 0.9306
        • This tells us that 93.06% of female heights are below 70 in.
        • It also tells us that 6.94% of female heights are above 70 in.
      • So, to calculate the proportion above Z = 1.48
        • Use the Z-Table
        • Look-up 1.48 by finding the row Z = 1.4 and then the column Z = .08
        • You should find the value 0.9306
        • Subtract from 1.0000
        • 1.0000 – 0.9306 = 0.0694 (or 6.94%)

Activity:

1. Test scores have a mean of 70 and a standard deviation of 10.

  • Find the z-score for a score of 90.
  • Assuming 90 and above is an A, what proportion of students got an A on the test?
  • Find the z-score for a score of 60.
  • Assuming below 60 is an F, what proportion of students got an F on the test?

2. Suppose commute times are approximately normal with a mean = 30 minutes and stdev = 5 minutes.

  • What proportion of commutes are less than 35 minutes?
  • What proportion of commutes are more than 45 minutes?
  • What proportion of commutes are less than 10 minutes?

3. SAT Math scores are approximately normal with mean = 500 and stdev = 100. ACT Math scores are approximately normal with mean = 18 and stdev = 6.

Julie earned a 630 on the SAT. John earned a 22 on the ACT.

  • Calculate Julie’s Z-score.
  • Calculate John’s Z-score.
  • Who performed better?
  • What proportion of students score above 650?
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Lesson 5: Normal Curve and Z-Scores

Review:

  • Exam 1 Results

Presentation:

  • Normal Curves
    • Normal Curve explained in 1 min
    • 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
      • Video – Normal Curves
      • 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 = 1.40
        • Z-Score for woman 60 inches tall
          • Z = (60 – 64.5)/2.5 = -1.80
    • 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 2.00 standard deviations above the mean
      • And, 64.5 – 5 = 59.5 is 2.00 standard deviations below the mean
      • Using the Empirical Rule, 95% of women are between 59.5 and 69.5 inches tall.

Activity:

Problem 1. Normal Curves and the 68–95–99.7 Rule

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. Draw a Normal curve and label the mean.
  2. Label the values one, two, and three standard deviations above and below the mean.
  3. Give the interval containing approximately 68% of scores.
  4. Give the interval containing approximately 95% of scores.
  5. Give the interval containing approximately 99.7% of scores.
  6. Approximately what percentage of students scored above 674?
  7. Approximately what percentage scored below 521?

Problem 2. Heights and Z-Scores

The heights of young women ages 18–24 are approximately Normally distributed with mean 64.5 inches and standard deviation 2.5 inches.

  1. Draw a Normal curve and label the mean.
  2. Label the values one, two, and three standard deviations above and below the mean.
  3. A 20-year-old woman is 6 feet tall. Calculate her z-score.
  4. Interpret the z-score in words.
  5. Approximately what percentage of young women are taller than 69.5 inches?
  6. Approximately what percentage are between 59.5 and 69.5 inches?

Problem 3. Standard Normal Areas

For a standard Normal distribution, find the percentage of observations that satisfy each condition. For each one, draw a Normal curve and shade the appropriate area.

  1. z<−1z < -1
  2. z<1z < 1
  3. z>1z > 1
  4. −1<z<1-1 < z < 1
  5. z>2z > 2
  6. −2<z<2-2 < z < 2

Problem 4. Comparing Scores with Z-Scores

Scores on the SAT Math exam are approximately Normally distributed with mean 500 and standard deviation 100. ACT Math scores are approximately Normally distributed with mean 18 and standard deviation 6.

Julie earned a 630 on SAT Math. John earned a 22 on ACT Math.

  1. Calculate Julie’s z-score.
  2. Calculate John’s z-score.
  3. Interpret each z-score in words.
  4. Assuming the two exams measure similar mathematical ability, who performed better relative to other test takers?
  5. What percentage of SAT Math scores are above 600?
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Lesson 3: Linear Regression and Curve Fitting

Review:

  • Histograms
  • Scatterplots
  • DataCamp assignment

Presentation:

  • Linear Regression
    • Scatter plots
    • Line plots
    • Trend lines
  • Curve Fitting
    • Polynomial fit
    • Forecasting
  • Sample Data

Activity:

  • Download recent Oil Prices (same as Oil Prices link above) from the St. Louis Fed
  • Fit a 2nd and 3rd order Polynomial curve.
    • Record the polynomial Equations (e.g., y = 0.04*x^2 + 0.22*x + .45)     
    • Record the R-Squared value (e.g., R^2 = 0.84)
  • Use your preferred polynomial equation to generate forecasts for the next 5 time periods (n=5 forecasts)

Assignment:

  • Complete DataCamp assignment
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Lesson 2: Data Aggregation and Visualization

Review:

  • Attendance
  • Course Intro
  • Syllabus revision (Data camp=10%, 3 Exams @ 30% each = 90%)
  • DataCamp

Presentation:

Activity:

  • Download and Pivot
    • Download US County Dataset
    • Create a Pivot Table
      • 49 rows and 4 columns
      • Each row corresponds to 48 States and Washington DC (Alaska and Hawaii are excluded)
      • Each column corresponds to four variables: Population, GDP, Median HH Income, and Poverty %
      • Calculate the aggregate “Sum” for Population and GDP by State
      • Calculate the aggregate “Average” for Median HH Income and Poverty % by State
  • Visualize State Data
    • Create graphs (use Insert > Chart)
      • Use the resulting State aggregation (from the Pivot table)
      • Make four (4) histograms
        • (1) Population
        • (2) GDP
        • (3) Median HH Income
        • (4) Poverty %
      • Make four (4) scatter plots:
        • (1) Population (x) vs GDP (y)
        • (2) GDP (x) vs Poverty % (y)
        • (3) GDP (x) vs Median HH Income (y)
        • (4) Median HH Income (x) vs Poverty % (y)
      • All graphs must have axis labels
      • All scatter plots must include a “Trendline” along with labels for the “Equation” and “R^2 value”

Assignment:

  • Continue working on your DataCamp assignment
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Lesson 4: Standard Deviation

Review:

  • Measures of Center and Boxplots
    • Mean, Median
    • Five-number summary: Min, Q1, Median, Q3, Max
    • Boxplots for comparison

Presentation:

  • Demonstrate calculation

Activity:

Problem 1. During the 2026 FIFA World Cup, the U.S. Men’s National Team scored 11 goals in 5 matches and reached the Round of 16. The team scored 4 goals vs Paraguay, 2 goals vs Australia, 2 goals vs Türkiye, 2 goals vs Bosnia & Herzegovina, and 1 goal vs Belgium.

a. Calculate the mean and standard deviation of goals scored per match.

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. 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

a. Calculate the mean and standard deviation for both student groups.
b. Which is more spread out, the 9th-grade or 12th-grade data? Explain using the standard deviations.

Problem 3. Two coffee shops recorded the number of customers who entered during the same six one-hour periods.

Coffee Shop A: 42, 45, 47, 44, 46, 46
Coffee Shop B: 31, 40, 44, 48, 51, 56

a. Calculate the mean and standard deviation for each coffee shop.
b. Which coffee shop has the more consistent number of customers? Explain using the standard deviations.

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Lesson 3: Measures of Center and Boxplots

Review:

  • Visualizing the shape of the distribution
  • Dotplots
  • Stemplots
  • Histograms

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 using ∑x/n
    • Median
      • Center value within ordered sequence of values
      • Use (n+1)/2 to find the position
      • If nn is even: average the two middle values
    • Demonstrate Mean and Median using sleep, work and/or height data
    • Video: Measures of Center (00:50-08:00)
    • Activity:
      • Use the MPG data for Ford vehicles
        • Calculate the mean
        • Find the median
      • Use the MPG data for Honda vehicles
        • Calculate the mean
        • Find the median
  • Boxplots
    • Five-number summary
      • Minimum value
      • Quartile 1 (Q1)
      • Quartile 2 (Q2) = Median
      • Quartile 3 (Q3)
      • Maximum value
    • Demonstrate Boxplot construction with sleep, work and height data
    • Video: Boxplots (00:50-08:15)
    • Activity:
      • Use the MPG data for Ford vehicles
        • Find the 5-number summary
        • Make a boxplot
      • Use the MPG data for Honda vehicles
        • Find the 5-number summary
        • Make a boxplot
      • Combine on one axis for comparison
        • Make a single horizontal axis to be shared by both Ford and Honda
        • Add both boxplots to the shared axis to facilitate comparison
        • What do the boxplots suggest about the typical MPG and variability of Ford versus Honda vehicles?

Exercises:

Problem 1.

Here are the starting salaries, in thousands of dollars, offered to 23 students who earned bachelor’s degrees in computer science.
63 56 66 77 50 53 78 55 90 65 64 69 59 76 48 54 49 68 51 50 70 69 71
a. Make a stemplot.
b. Calculate the mean.
c. Find the five-number summary.
d. Make a boxplot.

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. Calculate the mean.
c. Find the five-number summary.
d. Make a boxplot.

Problem 3.

A group of 21 recent college graduates reported the number of minutes they commute to work each morning. Here are the results: 18 24 31 27 22 35 16 29 41 25 20 33 28 19 37 23 26 30 21 34 25

a. Make a stemplot.
b. Calculate the mean.
c. Find the five-number summary.
d. Make a boxplot.

Problem 4.

A sample of 20 grocery stores recorded the price of a particular brand of coffee. Here are the results: 4.90 5.40 6.10 6.80 5.60 7.30 6.50 5.90 5.10 6.90 7.50 5.70 6.30 4.60 7.00 5.80 6.60 5.20 7.40 6.10. Use the Key: 5 | 4 = $5.40.

a. Make a stemplot.
b. Calculate the mean.
c. Find the five-number summary.
d. Make a boxplot.

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Lesson 1: Introduction to Advanced Statistics

Presentation:

Activity:

  • Login to DataCamp
  • Begin the Introduction to Excel course
  • Confirm successful login and classroom membership (so you aren’t asked to pay for DataCamp) before leaving class today
  • Data Camp Tips
    • Refresh frequently during exercises.
    • If using a personal email, send the email address to me so I can send an official invite.
    • Be sure to complete all sections within each Assignment for full credit (otherwise, it will be incomplete)
    • Use hints but not too many. XP will be part of your grade.
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