Introduction

In this document, we will solve the two confidence interval problems presented earlier—one using the Z-Interval and the other using the t-Interval—with step-by-step instructions for performing these calculations using a TI-84 calculator.

Problem 1: Solving a Z-Interval (Known Population Standard Deviation)

Problem Statement

We are given weekly study time data for 10 freshmen. We know the population standard deviation \(\sigma = 3\) hours, and the sample mean is \(\bar{x} = 11\) hours. We need to calculate a 95% confidence interval for the true population mean study time.

  • Data: 10, 8, 12, 14, 11, 9, 15, 13, 8, 10
  • Known Population Standard Deviation: \(\sigma = 3\) hours
  • Sample Mean: \(\bar{x} = 11\) hours
  • Confidence Level: 95%

Steps on the TI-84 Calculator

  1. Turn on the Calculator:
    • Press the ON button to power on your TI-84.
  2. Enter the Z-Interval Calculation:
    • Press STAT.
    • Scroll right to the TESTS menu using the arrow keys.
    • Scroll down to 7:ZInterval.
    • Press ENTER.
  3. Input the Z-Interval Parameters:
    • On the next screen, select Stats by pressing ENTER (since we are using summary statistics).
  4. Enter the Summary Data:
    • Enter \(\sigma = 3\) under σ, then press ENTER.
    • Enter the sample mean \(\bar{x} = 11\), then press ENTER.
    • Enter the sample size \(n = 10\), then press ENTER.
    • Enter the confidence level \(C = 0.95\), then press ENTER.
  5. Calculate the Interval:
    • Scroll down to Calculate and press ENTER.
    • The result will display the confidence interval: \([9.14, 12.86]\).

Thus, the 95% confidence interval is \([9.14, 12.86] \, \text{hours}\), indicating that we are 95% confident that the true population mean study time lies within this range.

Problem 2: Solving a t-Interval (Unknown Population Standard Deviation)

Problem Statement

We are given the heights of 12 randomly selected students. The sample standard deviation is \(s = 3.12\) inches, and the sample mean is \(\bar{x} = 69.25\) inches. We need to calculate a 95% confidence interval for the population mean height.

  • Data: 65, 67, 72, 70, 68, 65, 74, 69, 70, 68, 71, 73
  • Sample Standard Deviation: \(s = 3.12\) inches
  • Sample Mean: \(\bar{x} = 69.25\) inches
  • Confidence Level: 95%

Steps on the TI-84 Calculator

  1. Turn on the Calculator:
    • Press the ON button to power on your TI-84.
  2. Enter the t-Interval Calculation:
    • Press STAT.
    • Scroll right to the TESTS menu using the arrow keys.
    • Scroll down to 8:TInterval.
    • Press ENTER.
  3. Input the t-Interval Parameters:
    • On the next screen, select Stats by pressing ENTER (since we are using summary statistics).
  4. Enter the Summary Data:
    • Enter the sample mean \(\bar{x} = 69.25\), then press ENTER.
    • Enter the sample standard deviation \(s = 3.12\), then press ENTER.
    • Enter the sample size \(n = 12\), then press ENTER.
    • Enter the confidence level \(C = 0.95\), then press ENTER.
  5. Calculate the Interval:
    • Scroll down to Calculate and press ENTER.
    • The result will display the confidence interval: \([67.27, 71.23]\).

Thus, the 95% confidence interval is \([67.27, 71.23] \, \text{inches}\), indicating that we are 95% confident that the true population mean height lies within this range.