7.3 Justifying a Claim Based on a Confidence Interval for a Population Mean

• What is a confidence interval for a population mean?

  Range of confident values
• The margin of error is affected by both the standard deviation and the sample size.


  True
• A 95% confidence interval for the mean weight of apples (145g, 155g) means we are 95% confident the true mean weight lies within this range.


  True
• Increasing the sample size reduces the margin of error.


  True
• A confidence interval helps quantify the uncertainty around a sample estimate.


  True
• What effect does a larger standard deviation have on the margin of error in a confidence interval?

  Increases the margin of error
• What does the critical value in a confidence interval determine?

  Width of the interval
• What does the confidence level in a confidence interval represent?

  Probability of true mean
• What does a 90% confidence interval of (10, 12) indicate?

  True mean is in (10, 12)
• Which distribution is used when the population standard deviation (σ) is known?

  z-distribution
• What is the formula for calculating the margin of error when σ is unknown?

  $E =$$t *$$\frac{s}{\sqrt{n}}$
• The coverage of a confidence interval is the probability that the interval contains the true population mean
• Match the confidence level with its approximate critical value:

  90% ↔️ 1.645

  95% ↔️ 1.96

  99% ↔️ 2.576
• What does the margin of error measure?

  Uncertainty around sample mean
• The critical value in a confidence interval determines the width of the interval
• If the population standard deviation is known, the margin of error is calculated using the z-distribution
• A larger sample size reduces the margin of error in a confidence interval.


  True
• The sample mean is the average value from the sample data
• Common confidence levels include 90%, 95%, and 99%
• A 99% confidence interval is wider than a 90% confidence interval for the same data.


  True
• The t-distribution is used when the population standard deviation (σ) is unknown
• The confidence interval is calculated as the sample mean plus or minus the margin of error
• One-tailed intervals are appropriate when testing a specific directional hypothesis.
• If a 95% confidence interval for the mean weight of apples is (145g, 155g), we are 95% confident the true mean lies between 145g and 155g.
• Match the confidence level with its impact on interval width:

  90% ↔️ Narrower

  95% ↔️ Medium

  99% ↔️ Wider
• Match the confidence level with its impact on the interval width:

  90% ↔️ Narrower

  95% ↔️ Medium

  99% ↔️ Wider
• Steps for calculating confidence intervals:

  1️⃣ Choose the appropriate distribution

  2️⃣ Calculate the margin of error

  3️⃣ Construct the confidence interval
• Two-tailed confidence intervals are more commonly used because they make fewer assumptions about the direction of the difference between means.

  True
• If the sample mean is within the confidence interval, the claim that the population mean is equal to the sample mean is justified
• What is the primary purpose of using a confidence interval to justify a claim about a population mean?

  Compare claimed value to interval
• Order the following confidence levels from smallest to largest critical value:

  1️⃣ 90%

  2️⃣ 95%

  3️⃣ 99%
• The margin of error is calculated using the z-value when the population standard deviation is known.
• Match the component of a confidence interval with its definition:

  Margin of Error ↔️ Range added to sample mean

  Confidence Level ↔️ Probability interval contains true mean

  Critical Value ↔️ Determines interval width
• What happens to the critical value as the confidence level increases?

  Increases
• When should the standard normal (z-distribution) be used in calculating confidence intervals?

  When σ is known
• Why are two-tailed intervals more commonly used than one-tailed intervals?

  They make fewer assumptions
• What happens to the margin of error when the sample size decreases?

  Increases
• What are common confidence levels used in statistics?

  90%, 95%, and 99%
• As the confidence level increases, the margin of error increases
• Arrange the factors affecting the margin of error in order of their impact:

  1️⃣ Standard Deviation (Increases)

  2️⃣ Sample Size (Decreases)