Confidence Intervals for the Mean (σ known)

Confidence Intervals for the Mean (σ known)

Introduction

• Scenario: A simple random sample of 100 graders are part of a new reading program. Their scores on a standardized test are collected to evaluate the program's effectiveness.

  • Population standard deviation: $\sigma = 15$
  • Sample mean score: $\bar{x} = 67.3$

• Goal: Estimate the population mean score ($\mu$) if all graders in the district participated.

Point Estimate

• The best estimate for the population mean ($\mu$) is the sample mean ($\bar{x}$).
• $\bar{x} = 67.3$ is the point estimate.
• A point estimate is a single number.
• It's improbable that $67.3$ exactly equals the population mean $\mu$. Therefore, it's essential to indicate how precise the estimate is likely to be.

Confidence Interval

• Using an Interval Estimate
  • For example, if the estimate could be off by 10 points, the interval for estimating $\mu$ would be $57.3$ to $77.3$, which can be expressed as $67.3 \pm 10$.
  • The $\pm$ number is the margin of error.

95% Confidence Interval

• Concept: A 95% confidence interval means you are 95% confident that the true population mean falls within the calculated interval.
• To build a 95% confidence interval, determine the margin of error so that the interval is likely to contain $\mu$.

Sampling Distribution of $\bar{x}$

• The sampling distribution of $\bar{x}$ is used to determine likely values of the sample mean.

• Central Limit Theorem: Because the sample size is large (n > 30), the sampling distribution of $\bar{x}$ is approximately normal.

  • Mean: $\mu$
  • Standard error: $\frac{\sigma}{\sqrt{n}}$

• Example: For the reading program:

  • Standard error = $\frac{15}{\sqrt{100}} = 1.5$

Constructing the 95% Confidence Interval

• Start with a normal curve and find the z-scores that bound the middle 95% of the area.
  • Z-scores: $1.96$ and $-1.96$
  • The value $1.96$ is the critical value.
• Margin of error = Critical value * Standard error
  • Margin of error = $1.96 * 1.5 = 2.94$
• The 95% confidence interval for $\mu$ is: $\bar{x} \pm 2.94$
  • $67.3 \pm 2.94$
  • Resulting confidence interval: $\mu$ is between $64.36$ and $70.24$
• We are 95% confident that the true population mean is between $64.36$ and $70.24$.

Confidence Level

• Definition: 95% is the confidence level for the confidence interval.
• Interpretation: The confidence level measures the success rate of the method used to construct the confidence interval.
• If we draw many samples and construct a confidence interval from each, then in the long run, 95% of the intervals would cover the true value of $\mu$.

Terminology Review

• Point Estimate: A single number to estimate an unknown parameter (e.g., population mean).
• Confidence Interval: An interval used to estimate the value of a parameter.
• Confidence Level: A percentage between 0% and 100% that measures the success rate of the method used to construct the confidence interval.
• Margin of Error: Calculated by multiplying the critical value by the standard error.

Assumptions

• The population standard deviation $\sigma$ is known.
• In practice, $\sigma$ is usually unknown, but assuming it is known allows us to use the normal distribution.
• Simple random sample.
• Sample size is large (n > 30) OR the population is approximately normal.

Finding Critical Values for Different Confidence Levels

• 95% is common, but other confidence levels can be used.
• Any confidence level between 0% and 100% can be used by finding the appropriate critical value.

Example: 90% Confidence Interval

• Sample mean: $\bar{x} = 7.1$
• Standard error: $\frac{\sigma}{\sqrt{n}} = 2.3$
• Construct a 90% confidence interval for $\mu$.
• Critical value for 90% confidence (from table A3): $1.645$
• Margin of error = $1.645 * 2.3 = 3.8$
• 90% confidence interval: $7.1 \pm 3.8$
• $\mu$ is between $3.3$ and $10.9$.

Z-score Notation

• $z_{\alpha}$: the z-score with an area of $\alpha$ to its right.

• $z_{\frac{\alpha}{2}}$: the z-score with an area of $\frac{\alpha}{2}$ to its right.

• If $1 - \alpha$ is the confidence level, then the critical value is $z<em>{\frac{\alpha}{2}}$. The area under the standard normal curve between $-z</em>{\frac{\alpha}{2}}$ and $z_{\frac{\alpha}{2}}$ is $1 - \alpha$.

• These z-scores can be found using table A2 or technology.

Example: Finding $z_{\frac{\alpha}{2}}$ for a 92% Confidence Interval

• Confidence level: 92%, so $1 - \alpha = 0.92$
• $\alpha = 0.08$
• $\frac{\alpha}{2} = 0.04$
• Critical value: $z_{0.04}$
• Area to the right of $z_{0.04}$ is $0.04$, so the area to the left is $1 - 0.04 = 0.96$.
• Using table A2 or technology, the critical value is approximately $1.75$.

Excel Example: Confidence Interval for Mean IQ Score

• Scenario: An IQ test given to a simple random sample of 75 students at a college.
  • Sample mean: $\bar{x} = 105.2$
  • Population standard deviation: $\sigma = 10$
• Goal: Construct a 90% confidence interval for the mean IQ score of students at this college.

Assumptions

• Simple random sample.
• Sample size is large (n > 30) OR the population is approximately normal.
• Assumptions are met since we have a simple random sample and n = 75.

Calculations

• Point estimate: $\bar{x} = 105.2$
• Using CONFIDENCE.NORM in Excel:
  • Takes three arguments: alpha, standard deviation, sample size.
  • Alpha is the complement of the confidence level (e.g., if confidence level is 0.90, alpha is 0.10).
  • Type =CONFIDENCE.NORM(0.1, 10, 75)
  • Margin of error: 1.8993
• Confidence interval:
  • Lower endpoint: $105.2 - 1.8993 = 103.3$
  • Upper endpoint: $105.2 + 1.8993 = 107.1$
• Confidence interval: $103.3$ to $107.1$

Interpretation

• Based on our sample, we are 90% confident that the population mean IQ score is between $103.3$ and $107.1$.

Example: Mathematics SAT Score

• Scenario: A simple random sample of 100 entering freshmen.
  • Sample mean mathematics SAT score: $\bar{x} = 458$
  • Population standard deviation: $\sigma = 116$
• Goal: Construct a 99% confidence interval for the mean mathematics SAT score for the entering freshman class.

Assumptions

• Simple random sample.
• Sample size is large (n = 100).
• Assumptions are met.

Calculations

• Critical value for 99% confidence: $2.576$
• Standard error: $\frac{\sigma}{\sqrt{n}} = \frac{116}{\sqrt{100}} = 11.6$
• Margin of error: $2.576 * 11.6 = 29.812$
• 99% confidence interval: $458 \pm 29.812$

Result

• We are 99% confident that the population mean is between $428.19$ and $487.81$.

Impact of Sample Size and Confidence Level on Margin of Error

• Margin of error = Critical value * $\frac{\sigma}{\sqrt{n}}$
• Sample Size:
  • If n decreases (e.g., from 100 to 75), the margin of error increases.
• Confidence Level:
  • A smaller confidence level leads to a smaller critical value.
  • A smaller confidence level results in a smaller margin of error.