Confidence Intervals & Margin of Error — Step-by-Step Guidance

Q1. Given a 95% confidence interval for the average time students spend studying per week as (12.4, 18.6): i. Find the sample mean. ii. Find the margin of error.

Background

Topic: Confidence Intervals for Means

This question tests your understanding of how to interpret a confidence interval and extract the sample mean and margin of error from the interval endpoints.

Key Terms and Formulas

• Confidence Interval: An interval estimate for a population parameter, usually written as (lower bound, upper bound).

• Sample Mean ($\bar{x}$): The midpoint of the confidence interval.

• Margin of Error (ME): The distance from the sample mean to either endpoint of the interval.

Key formulas:

• $\text{Sample Mean} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$

• $\text{Margin of Error} = \text{Upper Bound} - \text{Sample Mean}$

Step-by-Step Guidance

1. Identify the lower and upper bounds of the confidence interval: 12.4 and 18.6.

2. Calculate the sample mean by finding the midpoint between the two bounds: $\bar{x} = \frac{12.4 + 18.6}{2}$

3. Calculate the margin of error by subtracting the sample mean from the upper bound: $\text{ME} = 18.6 - \bar{x}$

4. Alternatively, you can also subtract the lower bound from the sample mean to check your calculation.

Try solving on your own before revealing the answer!

Q2. Given a confidence interval for the average daily screen time (in hours) as (5.2, 7.8): i. Find the sample mean. ii. Find the margin of error.

Background

Topic: Confidence Intervals for Means

This question is similar to Q1 and tests your ability to interpret a confidence interval and calculate the sample mean and margin of error.

Key Terms and Formulas

• Confidence Interval: (5.2, 7.8)

• Sample Mean ($\bar{x}$): The midpoint of the interval.

• Margin of Error (ME): The distance from the sample mean to either endpoint.

Key formulas:

• $\text{Sample Mean} = \frac{5.2 + 7.8}{2}$

• $\text{Margin of Error} = 7.8 - \bar{x}$

Step-by-Step Guidance

1. Identify the lower and upper bounds: 5.2 and 7.8.

2. Calculate the sample mean: $\bar{x} = \frac{5.2 + 7.8}{2}$

3. Calculate the margin of error: $\text{ME} = 7.8 - \bar{x}$

4. Check your calculation by also subtracting the lower bound from the sample mean.

Try solving on your own before revealing the answer!

Q3. Explain how increasing the sample size (n) affects the margin of error. - Does the margin of error increase or decrease? - Why?

Background

Topic: Margin of Error and Sample Size

This question tests your understanding of the relationship between sample size and margin of error in confidence intervals.

Key Terms and Formulas

• Margin of Error (ME): The amount added/subtracted from the sample mean to create the confidence interval.

• Sample Size ($n$): The number of observations in the sample.

• Standard Error (SE): $\text{SE} = \frac{\sigma}{\sqrt{n}}$ (or $\frac{s}{\sqrt{n}}$ for sample standard deviation)

• Margin of Error formula: $\text{ME} = z^* \times \text{SE}$

Step-by-Step Guidance

1. Recall that the margin of error depends on the standard error, which is inversely related to the square root of the sample size.

2. As $n$ increases, $\sqrt{n}$ increases, so $\frac{1}{\sqrt{n}}$ decreases.

3. This means the standard error gets smaller as sample size increases.

4. Since margin of error is calculated as $z^* \times \text{SE}$, a smaller SE leads to a smaller margin of error.

Try explaining this relationship in your own words before checking the answer!

Q4. Explain how changing the confidence level (e.g., from 90% to 99%) affects the margin of error. - Does the margin of error increase or decrease?

Background

Topic: Margin of Error and Confidence Level

This question tests your understanding of how the confidence level impacts the margin of error in a confidence interval.

Key Terms and Formulas

• Confidence Level: The probability that the interval contains the true population parameter.

• Critical Value ($z^*$): Increases as confidence level increases.

• Margin of Error formula: $\text{ME} = z^* \times \text{SE}$

Step-by-Step Guidance

1. Recall that a higher confidence level requires a larger critical value ($z^*$).

2. As $z^*$ increases, the margin of error increases because $\text{ME} = z^* \times \text{SE}$.

3. This means the interval becomes wider to ensure greater confidence that it contains the true mean.

Try explaining this effect in your own words before checking the answer!

Q5. Two studies estimate the same population mean: a. Study A: Narrow confidence interval b. Study B: Wide confidence interval Which study provides a more precise estimate, and why?

Background

Topic: Precision of Confidence Intervals

This question tests your understanding of how the width of a confidence interval relates to the precision of an estimate.

Key Terms and Formulas

• Precision: How close the estimate is likely to be to the true value.

• Confidence Interval Width: The difference between upper and lower bounds.

Step-by-Step Guidance

1. Compare the width of the confidence intervals for both studies.

2. A narrower interval means less uncertainty about the estimate.

3. A wider interval means more uncertainty about the estimate.

4. Think about which interval gives you a more precise estimate of the population mean.

Try to reason through which study is more precise before checking the answer!