Lesson 10: The t-distribution for Small Sample Inference

Review:

• Estimation with Confidence Intervals (n>=30)

Presentation:

• t-distribution
  • For larger samples, n ≥ 30 use Z-table
  • For small samples, n < 30 use t-distribution
  • Only impacts calculation of the test statistic
    • margin of error for confidence intervals
    • test statistic and p-value for significance testing (don’t worry, we’ll cover  these topics later)
  • t distribution critical values
    • print the t distribution table for the next exam
      • use the “0.025” column for estimation with 95% confidence intervals
    • select the row corresponding to df = n – 1
      • df = degrees of freedom
      • n = number of records in sample data
    • find the t* critical value
      • row = df = n – 1
      • column = .025 (top label) or 95% confidence level (bottom label)
      • e.g., if n = 13
        • df = n – 1 = 12
        • go to the row where df = 12 and move to the .025 column
        • you should find t* = 2.179
  • t-statistic formulas
    • for confidence intervals
      • x-bar ± t * s/√n
      • instead of the Z equivalent:  x-bar ± z * σ/√n
  • Video
  • Examples:
    • Identify appropriate t distribution critical values
      • n=10 with 95% confidence, t* = 2.262
      • n=20 with 95% confidence, t* = 2.093
    • A sample of size n = 25 produced the sample mean, x-bar = 36.0 and standard deviation, s = 9.0. Construct a 95% confidence interval to estimate the population mean.
      • n = 25, x-bar = 36.0, s = 9.0
      • df = 24, t* = 2.064
      • m = 2.064*(9/(√25)) = 3.7152
      • Estimate for μ = 36.0 ± 3.7152
      • 95% confidence interval: [32.2848, 39.7152]
        • previous lesson interval with Z=1.96 and n=400: [35.118, 36.882]
        • sample size can make a big difference

Activity:

• Problem 1. Use the t distribution table, assuming 95% confidence, to find the value of t* for each of the following sample sizes.
  • n = 12
  • n = 18
  • n = 24
  • n = 28
• Problem 2. The weights (in lbs) from a random sample of n=16 four-year-old children who took part in a study on childhood obesity are provided below. Estimate the mean weight of four-year-old children using a 95% confidence interval. The sample standard deviation equals 4.2.
  • {37.1, 26.7, 36.1, 36.2, 40.3, 43.9, 36.2, 40.7, 42.5, 34.8, 37.9, 34.5, 31.1, 36.4, 35.7, 33.4}
• Problem 3. After purchasing a new fuel-efficient vehicle, the owner calculates miles per gallon (mpg) the first n=12 times he fills the fuel tank. His calculations are provided below. Estimate vehicle fuel efficiency using a 95% confidence interval. You will need to calculate standard deviation.
  • {42, 43, 37, 36, 34, 45, 48, 43, 38, 42, 43, 46}

These examples and problems are from the Against All Odds video series and Introduction to the Practice of Statistics (6th Ed.) by Moore et al.

Lesson 9: Estimation with Confidence Intervals

Review:

• Quiz 3
• Sampling Distributions

Presentation:

• How to Estimate a Population Mean (μ)
  • Conduct a Simple Random Sample (SRS) of size n
  • Calculate the sample mean, x-bar = (∑x)/n
  • Calculate the sample standard deviation, s = √(∑(x – x-bar)²/(n-1)
  • If n is sufficiently large (≥30), we can
    • Assume x-bar is Normally distributed
    • Estimate the population mean, μ = x-bar (point estimate)
    • Calculate margin of error, m = z*σ
      • where z = 1.96 for 95% confidence
      • where σ = s/(√n)
  • Estimation with confidence interval for μ = x-bar ± m
• Watch video
• Example: A sample of size n = 400 produced the sample mean, x-bar = 36.0 and standard deviation, s = 9.0. Construct a 95% confidence interval to estimate the population mean.
  • n = 400
  • x-bar = 36.0
  • s = 9.0
  • m = 1.96*(9/(√400)) = 0.882
  • Estimate for μ = 36.0 ± 0.882
  • 95% confidence interval: [35.118, 36.882]

Activity:

• Problem 1. A sample of size n = 100 produced the sample mean, x-bar = 16.0 and standard deviation, s = 3.0. Construct a 95% confidence interval to estimate the population mean.
• Problem 2. An operation manager at a large plant observed 120 workers assembling an electronic component. The average time needed for assembly was 16.2 minutes with a standard deviation of 3.6 minutes. Construct a 95% confidence interval to estimate the mean assembly time.
• Problem 3. A computer technician installs new hard drives on 64 different computers. The average installation time is 42 minutes with a standard deviation of 5 minutes. Construct a 95% confidence interval for the mean installation time.
• Problem 4. A research firm conducted a survey of regular smokers to estimate the average amount spent per week on cigarettes. A sample of 49 regular smokers revealed average spending on cigarettes to be $21.55 with a standard deviation of $5.21. Construct a 95% confidence interval to estimate mean weekly cigarette spending.

Assignment:

• Read pp. 353-362, Introduction to Inference and Estimation with Confidence

Lesson 8: Sampling Distributions and the Central Limit Theorem

Review:

• Exam 1 Results
• Design of Experiments
• Census and Sampling
• Sampling and Surveys

Presentation:

• Sampling Distributions
  • Take multiple Simple Random Samples, sample size = n, and calculate each sample mean
  • “Sampling Distribution” is the distribution of sample means
  • Standard deviation of sample means = s/√n
  • Figure 5.8 on p. 336: 
    
  • Example 5.18 on p. 338

• Central Limit Theorem
  • Simple random samples of size n from any population with mean μ and finite standard distribution σ. When n is large (typically ≥ 30), the sampling distribution of the sample mean is approximately Normal.
  • Example 5.19 on p. 339
• Video

Activity:

• Test the Central Limit Theorem
• Calculate sample means (use Vehicle Year)
• Report results

Study:

• For practice complete Exercises 5.36, 5.37, 5.38, 5.39 on pp. 336-340
• Read pp. 335-344, Sampling Distribution

Mean	2007.65
Standard Error	0.63
Median	2006.50
Mode	2013.00
Standard Deviation	5.77
Range	23.00
Minimum	1996.00
Maximum	2019.00
Sum	168643.00
Count	84.00
Largest(1)	2019.00
Smallest(1)	1996.00
sampling distribution mean	2007.65
sampling distribution stdev	0.63

Lesson 7: Census, Sampling, Surveys and Inference

Review:

• Exam 1 Results
• Starfish reporting
• Exam solutions

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: Google Survey
      • 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
• Samples and Surveys
  • Toward Statistical Inference
    • Population parameters
    • Sample statistics
    • Sampling distribution
    • Bias and Variability
      • to reduce bias, use random sampling
      • to reduce variability, use larger sample sizes (p. 215)
    • Margin of Error Calculator
  • Video: Samples and Surveys

Activity:

• 2020 Iowa Caucus
• Choose a recent poll
• Find the sample size
• Estimate the population size
• Calculate margin of error using 95% confidence

Assignment:

• Watch Designing Experiments

Exam 1 Results

Mean	90.5
Median	95.0
Mode	94.6
Standard Deviation	15.8
Range	100.0
Minimum	0.0
Maximum	100.0
Sum	9047.7
Count	100.0