Course Information

  • Course Code: STT 200

  • Course Name: Statistical Methods

Learning Goals

  • Parametric Confidence Intervals for a Mean

    • Determining sample size

  • Parametric Hypothesis Test for a Mean

Announcements

  • Exam Date: April 1, 2026 (Wednesday)

    • Time: 7:45 PM to 9:00 PM

    • Arrival: 7:30 PM at the testing venue

  • Venues for Exam:

    • Sections 49-58: Business N130

    • Sections 59-60: Wells Hall B117

  • What to Bring to the Exam:

    • MSU ID (needed for building access)

    • A #2 pencil

    • Graphing calculator

Exam 2 Topics

  • Topics Covered: Same as HWs 5, 6, and 7:

    • 4.1 The Central Limit Theorem

    • 4.2 The normal distribution

    • 4.3 Mathematical model for the sampling distribution of a single proportion

    • 4.4 Parametric hypothesis test for a single proportion

    • 4.5 Bootstrap confidence interval for a single proportion

    • 4.6 Parametric confidence interval for a single proportion

    • 4.7 Chi-squared goodness of fit (GOF) test

    • 4.8 Chi-squared independence test

    • 5.1 Bootstrap confidence interval for a single mean

    • 5.2 Mathematical model for the sampling distribution of a single mean and t-distributions

    • 5.3 Parametric confidence interval for a single mean

5.3 Parametric Confidence Intervals for a Single Mean

Example: The Campus That Never Sleeps

  • A random sample of 50 STT 200 students at MSU was surveyed about their nightly sleep.

  • Statistical Summary of Responses:

    • Sample mean, 𝑥̄: 7.19

    • Sample standard deviation, s: 1.19

    • Sample size, n: 50

    • Minimum sleep reported: 5 hours

    • Maximum sleep reported: 10 hours

Constructing a 95% Confidence Interval

  • Method: Use TInterval function from the calculator's STATS>TESTS menu

Accessing More Decimal Places with TInterval

  • To view more decimal places:

    • Press VARS key, select Statistics (option 5)

    • Scroll to TEST menu, select H (lower) for lower bound, I (upper) for upper bound

True or False Questions for Confidence Intervals

  • The 95% confidence interval for sleep is approximately (6.85, 7.53).

Assessing Statements:

  1. False: "95% of STT 200 students get between 6.85 and 7.53 hours of sleep."

  2. True: "We are 95% sure the average number of hours STT 200 students sleep is between 6.85 and 7.53."

Additional True or False:

  1. True: "If we take 1000 random samples, about 950 confidence intervals will contain the true average sleep hours."

  2. False: "About 950 of the sample means will fall between 6.85 and 7.53."

Deconstructing a Confidence Interval

  • Form: point estimate ± margin of error (ME)

  • Margin of Error Equation:
    ME = t^* imes rac{s}{
    }

  • To Reduce Margin of Error:

    1. Lower the confidence level (making t* smaller)

    2. Increase sample size

Determining Sample Size Given Margin of Error (ME)

  • Margin of Error Equation:
    ME = t^* imes rac{s}{n}

  • Solve for n:
    n = rac{(t^*s)}{ME^2}

  • Estimation: Use previous sample to estimate s and the degrees of freedom for t^*

  • Rounding: Always round n up to the next integer value

Example: The City That Never Sleeps

  • Revisit the previous sample of 50 STT 200 students to determine necessary sample size for estimating average sleep within 0.25 hours at 95% confidence:

    • Data:

    • Sample mean, 𝑥̄: 7.19

    • Sample standard deviation, s: 1.19

    • Sample size, n: 50

5.4 Parametric Hypothesis Tests for a Population Mean

General Steps in Hypothesis Testing

  1. State Hypotheses:

    • Null Hypothesis (H0): 𝜇 = 𝜇0

    • Alternative Hypothesis (HA): 𝜇 ≠ 𝜇0 or < or >

  2. Check Conditions:

    • Observations must be independent.

    • Sample size sufficient: n ≥ 30 (can relax for normal populations)

  3. Compute Test Statistic and p-value:

    • Test Statistic Formula:
      t = rac{ar{x} - ext{μ}_0}{ rac{s}{
      }}

    • p-value: Area beyond t in t-distribution with n - 1 degrees of freedom

    • Find area using tcdf() function

  4. Evaluate Results:

    • Interpret p-value:

      • p > 0.10: Little evidence

      • 0.05 < p ≤ 0.10: Some evidence

      • 0.01 < p ≤ 0.05: Strong evidence

      • 0.001 < p ≤ 0.01: Very strong evidence

      • p ≤ 0.001: Extremely strong evidence

Example: Classical Music

  • Research Objective: Test if high school students complete a maze faster while listening to classical music.

  • Assumed mean completion time for the general population: 40 seconds.

  • Sample Size: 100 HS students; Mean: 39.1 seconds; Standard Deviation: 4 seconds.

Step 1: Stating Hypotheses

  • Parameter of Interest: Let 𝜇 represent the average time to complete the maze while listening to classical music.

  • Hypotheses:

    • H0: \ 𝜇 = 40

    • HA: \ 𝜇 < 40

Step 2: Checking Conditions

  1. Observations: Independent? Yes

  2. Sample size: n > 30? Yes, n = 100

Step 3: Computing Test Statistic and Finding p-value

  • Test Statistic Calculation:
    t = rac{39.1 - 40}{ rac{4}{ ext{√}100}} = -2.25

  • Finding p-value:

    • Use t-distribution with 99 degrees of freedom:
      p ext{-value} = ext{tdf}(-10^10, -2.25, 99) = 0.0133

Step 4: Evaluating Results

  • Interpret p-value: Strong evidence for HA.

Using the Calculator

  • T-Test Input:

    • X̄: 39.1

    • Sx: 4

    • n: 100

    • t-statistic: -2.25

    • p-value: 0.0133