Confidence Intervals and Sample Sizes

Handout Overview

• The handout has three columns:

  • Section 7.2

  • Section 7.3 (today's topic)

  • Section 7.1

• Sections 7.2 and 7.3 concern means and numerical data, estimating the average or mean.

• The primary difference between sections 7.2 and 7.3 will be the availability of the population standard deviation.

Section 7.2: Minimum Sample Size

• We didn't complete Section 7.2, specifically how to calculate the minimum sample size.

• Formula is based on the margin of error:
  Margin \space of \space Error = z * (Standard \space Deviation / \sqrt{n})

• We solve for n in this equation.

• The formula includes a z-score, determined by the desired confidence level.

• For 90% confidence:

  • Sampling distribution is considered.

  • 90% lies in the middle, bounded by lower and upper limits.

  • In terms of x, these are lower and upper bounds; in terms of z, negative and positive z values for standard normal distribution.

Finding Z-Scores

• Use inverse norms to find Z-scores.

• For 90% confidence:

  • 90% inside, remainder outside.

  • 100 - 90 = 10

  • 10% is outside, split in half.

  • 10 / 2 = 5

  • 5% on each tail.

• Negative z-score calculation:

  • Inverse \space Norm(left \space tail, mean, standard \space deviation)

  • Inverse \space Norm(0.05, 0, 1)

• Result: z-score = -1.645.

• A table provides pre-calculated z-scores for common confidence levels.

• For 90% confidence, the z-score of 1.645 is readily available on the table.

• Using the table is generally easier than calculating inverse norms.

Minimum Sample Size Formula

• Formula:
  n = (z * Standard \space Deviation / Margin \space of \space Error)^2

• Aim is to determine the necessary amount of data to collect.

• Example: Estimating the amount of money people spend at a store.

• Margin of error:

  • Indicates desired accuracy.

  • Keyword: "within".

  • Example: "Accurate to within $1" implies margin of error is 1.

Example Problem (Number 8)

• Estimate the mean number of minutes (confidence interval for average).

• Determine the required sample size.

• Using the sample size formula

• Given:

  • 95% confidence

  • Margin of error: 7 minutes

  • Standard deviation: 47

• 95% confidence implies z-score = 1.96.

• Plugging in values:
  n = (1.96 * 47 / 7)^2

• Result: n = 173.18

• Round up to 174, as you cannot sample a fraction of a person.

Verifying Z-Score with Table

• For 95% confidence, the table confirms the z-score of 1.96.

• Alternatively, calculate using inverse norm with a left tail of 0.025.

Homework Questions: Problems 7.1 and 7.2

• Problem Seven

Problem Seven Set-Up

• Sample of 68 corridors, mean of 61.2 decibels, population standard deviation of 9.7.

• Construct a 95% confidence interval.

• This is a confidence interval for the mean.

• Differentiating Z and T Interval

  • Z-interval (7.2 question): population standard deviation is known.

  • T-interval (7.3 question): will be discussed later.

• Since this problem provides population standard deviation, use a z-interval.

Z-Interval Procedure

• Using calculator, go to Stat, then Tests, and choose option 7 (Z-interval).

• Input:

  • Summary Stats

  • Population \space Standard \space Deviation = 9.7

  • Mean = 61.2

  • Sample \space Size = 68

  • Confidence \space Level = 95

• Enter the values into calculator. And the Results are:

  • Lower bond: 58.9

  • Upper bond: 63.5

Using Raw Data: Problem Number 5

• Enter raw data into a list (Stat, Edit, option 1).

• Go to Stat, Tests, option 7 (Z-interval).

• Select "Data" instead of "Stats."

• Input population standard deviation (e.g., 2.1).

• Specify the list where data is entered (e.g., L2).

• Set frequency to 1.

• Choose confidence level (e.g., 99%).

• Calculate. You get the results.

• Example interpretation: First-time married couples stay together between roughly 5.4 and 7.4 years.

Section 7.3

• Focus: Constructing confidence intervals.

• No minimum sample size calculations.

• Two methods:

  • Formula

  • Calculator function

• Calculator function is similar to 7.2, but uses a different function

  • Rationale: Lack of population standard deviation.

  • Instead, use sample standard deviation.

  • Creates more uncertainty because sample standard deviation is a guess of population value.

T-Distribution

• Incorporates added uncertainty.

• Formula: x \space bar +- t * (s/ \sqrt{n}) * Instead of population standard deviation, use the sample standard deviation.

  • Instead of Z-score, use the t-score.

T-Distribution Table

• Provides t-scores.

• Example: 95% confidence interval, sample size of 20.

• Degrees of freedom = n - 1 (loss of one degree of freedom due to the guess involved).

  • 20 - 1 = 19

• Table usage:

  • Locate the 95% under confidence level column

  • Find the intersection with 19 degrees of freedom.

  • T-score: Approximately 2.093.

Using Calculator for T-Scores

• Function: inverse t.

  • Input the left tail, 0.025, and the degrees of freedom, 19.

T-Distribution characteristics

• Bell shaped

• Symmetric around the mean, median, mode: 0 at center

• Approaches the axis but doesn't touch it.

• Standard Deviation Variance is not 1

• Standard normal: mean is 0, standard deviation: 1.

Problem Solving Techniques

• Construct Confidence Intervals

• Calculator function is very easy; Problems are typically 15 seconds long

• Raw Data:

  • Put Raw data into the list

  • Go to Stat, Tests, Option 8

Confidence Level

• Level Column; sample size is 10

• N (Sample Size) = 10

• Degrees of freedom will be 10 - 1 = 9

• T Score: 2.262

Important Notes

• Encouraging the use of the t-interval

• T Distribution: Bell-Shaped, Symmetric, Centered at Zero

• Calculator More Accurate Than Table Because It Uses the Exact Value Of T Score Compared to Estimations With Table

• T Interval is Fast and Accurate, Whereas Formula Is Slower