Estimation and Confidence Intervals Study Guide

Chapter 3: Estimation - How Large is the Effect?

Section 3.3: Confidence Intervals for Single Means

• Purpose:

  • To find a likely range of values for a given parameter.

• General Form:

  • The formula for calculating the confidence interval for a single mean is given as:
    ext{Point Estimate} ext{ (Sample Mean)} ext{ } ar{x} ext{ } ext{± Margin of Error}

  • The margin of error is calculated using:
    ext{Margin of Error} = M imes S

• 95% Confidence Interval for Proportions:

  • The formula is specified as:
    p ext{ } ext{±} 2 imes ext{ } ext{sqrt}igg(p(1 - p)/nigg)
    where:

  • $p$ = sample proportion

  • $n$ = sample size

• Question posed:

  • What changes would occur in a 95% Confidence Interval for Means?

General Formula for Confidence Interval

• Formula:

  • The confidence interval for a single mean can be expressed as:
    ar{x} ext{ } ext{±} M imes S

• Validity Conditions:

  • The sample size must be greater than 20.

  • The distribution should not be strongly skewed.

• Standard Error (SE):

  • Standard Error is defined as:
    S = rac{s}{ ext{(sqrt)(n)}}
    where:

  • $s$ = sample standard deviation

  • $n$ = sample size

  • This definition is consistent with the one used in the standardized statistic from Chapter 2.

• 95% Confidence Interval Calculation:

  • The multiplier of 2 is applied for constructing the 95% confidence intervals:
    C.I. = ar{x} ext{ } ext{±} 2 imes S .

  • This results in limits defined as:
    (L{low}, L{up})
    where $L{low}$ and $L{up}$ represent the lower and upper limits, respectively.

Example and Interpretation

• Given:

  • A random sample of 30 textbooks from the Cal Poly campus bookstore shows an average price of $65.02, with a standard deviation of $51.42.

  • The distribution of textbook prices is not strongly skewed.

• **Approximate 95% Confidence Interval Calculation:

  • Point Estimate:**

  • Mean price (sample mean) = $65.02

  • Applying Formula:
    ar{x} ext{ } ext{±} M_{m} imes S

  • Which results in:
    ext{C.I.} ext{ } = ext{(lower limit, upper limit)}

• Interpretation of Confidence Interval:

  • We are ___ % confident that the long run average price of textbooks at Cal Poly falls between ____ and ____.

• Different Scenarios:

  • If the null hypothesis value falls inside the interval:

  • It is a likely value.

  • Fail to reject the null hypothesis.

  • If the null hypothesis value falls outside the interval:

  • It is NOT a likely value.

  • Reject the null hypothesis.

Section 3.4: Width of Confidence Interval

• Factors Influencing the Width of Confidence Intervals:

  • Confidence Level:

  • Significance Level + Confidence Level = 100%

  • An increase in the confidence level results in a decrease in the significance level.

  • More likely values occur when rejecting less often, leading to a wider interval.

  • Conversely, a decrease in the confidence level increases the significance level and narrows the interval.

• Sample Size:

  • Increasing the sample size leads to:

  • Less variability.

  • A narrower confidence interval.

  • A decrease in the sample size results in:

  • More variability.

  • A wider confidence interval.

  • "A larger sample size makes you more confident in your answer."

• General Formula for Width:

  • The formula remains consistent:
    ar{x} ext{ } ext{±} M imes S

• Effects of Standard Error:

  • If the standard error increases:

  • The margin of error increases.

  • Resulting in a wider interval.

  • If the standard error decreases:

  • The margin of error decreases.

  • Resulting in a narrower interval.

• Summary of Effects:

  • Confidence Level Increases → Interval Widens

  • Confidence Level Decreases → Interval Narrows

  • Sample Size Decreases → Interval Widens

  • Sample Size Increases → Interval Narrows

  • Standard Error Increases → Interval Widens

  • Standard Error Decreases → Interval Narrows