STAT 252 - WEEK 1: Construction and Interpretation of Confidence Intervals part 5

Construction of Confidence Intervals for a Population Proportion

Two-Tailed Confidence Interval for a Population Proportion

• Formula for Calculation:
  • Estimate: $\hat{p} \pm z^*\sqrt{\frac{\hat{p}\hat{q}}{n}}$
  • Where:
    • $\hat{p} = \text{sample proportion}$
    • $\hat{q} = 1 - \hat{p}$
    • $n = \text{sample size}$
    • $p = \text{population proportion}$ (the parameter we are estimating)
    • $\text{z}^* = \text{critical value}$, a widening factor determining the width of the confidence interval to achieve a specific confidence level in capturing $p$.
• Confidence Interval Structure:
  • $\hat{p} - z^<em>\sqrt{\frac{\hat{p}\hat{q}}{n}} < p < \hat{p} + z^</em>\sqrt{\frac{\hat{p}\hat{q}}{n}}$
• Margin of Error (E):
  • Equivalent to half the interval width: $E = z^*\sqrt{\frac{\hat{p}\hat{q}}{n}}$

Required Statistical Conditions for Proportions

• To construct a two-tailed confidence interval for a population proportion, the raw number of successes AND failures observed in the sample must BOTH be at least 10.
• If this condition is not met, the confidence interval cannot be constructed.

Example: 95% Confidence Interval for a Population Proportion (p. 319)

• Scenario: A random sample of $516$ voters from a county's $10,000$ voters found $289$ would vote for a political measure. The rest ($516 - 289 = 227$) would not.
• Conditions Met? Yes, $289$ (successes) $\ge 10$ AND $227$ (failures) $\ge 10$.
• Calculations:
  • Implied $\hat{p} = 289/516 \approx 0.5601$
  • Using the formula above, the calculated 95% Confidence Interval is: $0.5172 < p < 0.6029$
• Interpretation:
  • "We are $95\%$ confident that the true population proportion of all voters in the county who say they would vote for the certain political measure is between $0.5172$ and $0.6029$.
  • This is the standard, default interpretation for confidence intervals.

Common Confidence Levels and Associated Critical Values (z*)

• For two-tailed intervals, $z^*$ is always a critical value from the z-curve.
• Confidence Level (CI) | z*
  • $90\%$ | $1.645$
  • $95\%$ | $1.960$
  • $99\%$ | $2.576$

Interpreting "Confidence" (e.g., 95% Confidence)

• Note: Interpretations for $90\%$, $95\%$, and $99\%$ confidence levels are fundamentally the same, only differing in the percentage.

Correct Interpretations of a Confidence Interval (e.g., $0.5172 < p < 0.6029$ for the voter example)

• Standard Interpretation: "We are $95\%$ confident that the true population proportion of all voters who would vote for the certain political measure is between $0.5172$ and $0.6029$.
• Repeat Sampling Interpretation: "If $100$ people made a confidence interval, $95$ of them would, on average, capture the true population proportion, $p$.
  • This is the definition: we are not $95\%$ confident that $p$ is precisely the calculated $\hat{p}$, but rather that this specific $95\%$ confidence interval has captured the true value of $p$.
• Probabilistic Interpretation (Chance): "Another confidence interval made using the same method has a $95\%$ chance of capturing $p$.
  • A $95$ out of $100$ chance is equivalent to $0.95$ chance.
• Probabilistic Interpretation (Probability): "Another confidence interval made using the same method has a $.95$ probability of capturing $p$.

Incorrect Interpretations of a Confidence Interval

• Claiming identical intervals: "If $100$ people made a confidence interval, $95$ of them will get an interval of $0.5172 < p < 0.6029$"
  • Incorrect because each person's sample will yield a unique $\hat{p}$ (e.g., $0.5601$ in the example) and thus a unique confidence interval. It is highly improbable that multiple samples would yield the exact same confidence interval.
• Claiming fixed interval for all samples: "$95\%$ of all samples will yield confidence intervals of $0.5172 < p < 0.6029$"
  • Incorrect for the same reason as above; different samples yield different $\hat{p}$ values and consequently different intervals.
• Misinterpreting the interval limits as probability: "There is between a $0.5172$ and $0.6029$ chance that our $95\%$ confidence interval captured $p$.
  • Incorrect. Our confidence interval had a $95\%$ chance (or $.95$ probability) of capturing $p$ at the time of its construction. The limits only define the range of values we are $95\%$ confident $p$ is within.
• Claiming point estimate for the true proportion: "We are $95\%$ confident that the true population proportion of voters who support the certain political measure is $0.5601$.
  • Incorrect. A confidence interval provides a range of values for the population parameter, not a single point estimate ($\hat{p} = 0.5601$ is a sample statistic, a point estimate).

Finding Confidence Intervals for Proportions with Different Levels of Confidence

• Revisit Chamber of Commerce Scenario: 99% Confidence Interval
  • $289$ successes out of $516$ voters.
  • Assumptions/Conditions: MET ($289 \ge 10$ successes and $227 \ge 10$ failures).
  • Calculations: Using $z^* = 2.576$ for a $99\%$ CI:
    • The $99\%$ Confidence Interval is: $0.5038 < p < 0.6164$
  • Interpretation: "We are $99\%$ confident that the true population proportion of all voters in the county who say they would vote for the certain political measure is between $0.5038$ and $0.6164$.
• Note: All else being equal, a higher confidence level results in a wider confidence interval.

Interval Width & Margin of Error

• Interval Width: Upper Limit (UL) - Lower Limit (LL)
• Margin of Error (E): Interval Width / $2$

Example 1: Reuters/Ipsos Poll (Oct 2024 - Biden Approval)

• 95% Confidence Interval: $0.3220 < p < 0.3771$
• Interval Width: $0.3771 - 0.3220 = 0.0551$
• Margin of Error (E): $0.0551 / 2 = 0.0276$
  • Interpretation: The poll had a margin of error of $2.76\%$ percentage points.

Example 2: Reuters/Ipsos Poll (Oct 2020 - Trump Approval)

• 95% Confidence Interval: $0.3890 < p < 0.4510$
• Interval Width: $0.4510 - 0.3890 = 0.0620$
• Margin of Error (E): $0.0620 / 2 = 0.0310$
  • Interpretation: The poll had a margin of error of $3.10\%$ percentage points.

Construction of Confidence Intervals for a Population Mean (One Sample t-Interval)

Two-Tailed Confidence Interval for a Population Mean, $\mu$

• Formula: $\bar{x} \pm (t\text{ critical value}) \frac{s}{\sqrt{n}}$
  • Or, expressed as an interval: $LL < \mu < UL$
• Where:
  • $\bar{x} = \text{sample mean}$
  • $s = \text{sample standard deviation}$
  • $n = \text{sample size}$
  • $\mu = \text{population mean}$ (the parameter we are estimating)
  • $\text{t critical value}$ (often written as $t^*$) is found using the t-distribution.
  • $LL = \text{lower limit}$, $UL = \text{upper limit}$.

Required Statistical Conditions for Means

• To construct a two-tailed confidence interval for a population mean, one of the following must be true:
  1. $n > 50$
  2. The population from which the sample is drawn must be KNOWN (i.e., judged) TO BE NORMALLY DISTRIBUTED.
• If neither of these conditions is met, the confidence interval cannot be constructed.

Guided Example: Insurance Profits (p. 359-60)

• Scenario: A random sample of $30$ insurance plans showed a sample mean profit ($\bar{x}$) of $\$1438.90$ and a sample standard deviation ($s$) of $\$1329.60$. The data is determined to be from a normally distributed population.
• Conditions Met? Yes, the population is judged to be normally distributed.
• Calculations for a 95% Confidence Interval:
  • Using the formula $\bar{x} \pm (t\text{ critical value}) \frac{s}{\sqrt{n}}$, the calculated interval is: \942.47 < \mu < \$1935.33$
• Interpretation:
  • "We are $95\%$ confident that the true population mean of all insurance plans sold by the company’s sales representatives is between $\$942.47$ and $\$1935.33$.

Critical Values for a Population Mean (t*)

• To find the critical value (t*) from the t-critical value table, three pieces of information are needed:
  1. Number of tails: Confidence intervals are typically two-tailed by default in this course.
  2. Significance Level ($\alpha$): Calculated as $100\% - \text{Confidence Level Percentage}$.
  3. Degrees of Freedom (df): Calculated as $df = n - 1$.

The t-Distribution

• Origin: The t-distribution is used when we employ the sample standard deviation ($s$) instead of the population standard deviation ($\sigma$) to construct a confidence interval.
• Purpose: It provides an extra widening factor to ensure that the stated confidence level (e.g., $95\%$, is an accurate reflection of the probability of capturing the population parameter ($\mu$ or $p$).
• Shape: It is a bell-shaped curve, approximately (but not exactly) normally distributed. Only the z-curve is truly normally distributed.
• Application: Thanks to research, t-tables are used to extract t-critical values (widening factors) for statistical calculations involving confidence intervals.
• Relationship to Sample Size: Each unique sample size ($n$) has its own unique t-distribution, defined by its degrees of freedom ($df = n-1$).
• Key Rules:
  • If you have the sample standard deviation ($s$), you use the t-distribution.
  • If you have the population standard deviation ($\sigma$), you use the z-distribution.
• Terminology: The t-distribution, t-curve, and studentized-t distribution are synonymous terms.
• Area Under the Curve: Any t-curve always has a total area of $1.00$ under it.
• Approximation to z-curve: A t-curve is approximately normally shaped. As the degrees of freedom ($df$) increase, the t-curve becomes a closer approximation to the z-curve.
  • For t-curves with $df \ge 600$, the t-critical value becomes practically equivalent to a z-critical value (rounded to two decimal places).

Interpretations of Confidence Intervals (Re-visited for Means)

• The fundamental interpretations of confidence intervals for the population proportion ($p$) are identical to those used for the population mean ($\mu$).

Correct Interpretations (e.g., for $95\%$ CI: \942.47 < \mu < \$1935.33$)

• Standard Interpretation: "We are $95\%$ confident that the true population mean profit of all of the policies sold by the company's sales representatives is between $\$942.47$ and $\$1935.33$.
• Repeat Sampling Interpretation: "If $100$ people made a confidence interval, $95$ of them would, on average, capture the true population mean, $\mu$.
  • Again, this means we are not $95\%$ confident that $\mu$ is precisely the calculated $\bar{x}$, but that this specific interval has captured the true value of $\mu$.
• Probabilistic Interpretation (Chance): "Another confidence interval made using the same method has a $95\%$ chance of capturing $\mu$.
• Probabilistic Interpretation (Probability): "Another confidence interval made using the same method has a $.95$ probability of capturing $\mu$.

Incorrect Interpretations (e.g., for $95\%$ CI: \942.47 < \mu < \$1935.33$)

• Claiming identical intervals: "If $100$ people made a confidence interval $95$ of them will get an interval of \942.47 < \mu < \$1935.33$"
  • Incorrect because each sample will have a unique $\bar{x}$ (e.g., $\$1438.90$ in the example) and thus a unique confidence interval. It is highly improbable for multiple samples to yield the exact same confidence interval.
• Claiming fixed interval for all samples: "$95\%$ of all samples will yield confidence intervals of \942.47 < \mu < \$1935.33$"
  • Incorrect for the same reason: different samples produce different $\bar{x}$ values and consequently different intervals.
• Predicting individual values rather than population parameters: "There is a $.95$ probability that if an additional sales representative is sampled they would report a yearly profit between $\$942.47$ and $\$1935.33$"
  • Incorrect. This statement refers to a prediction interval for an individual future value ($x$), not a confidence interval which is constructed to estimate a population parameter ($\mu$). Confidence intervals predict population parameters, while prediction intervals predict specific values of $x$ within the population.
• Claiming point estimate for the true mean: "We are $95\%$ confident that the true population mean profit of all of the policies sold by the company sales representative is $\$1438.90$.
  • Incorrect. This is merely the sample statistic $\bar{x}$, a point estimate. Confidence intervals apply to a range of values.

Guided Example: Insurance Profits (Re-visited) - 99% Confidence Interval (p. 359-60)

• Scenario: Using the same insurance profits data ($n = 30$, $\bar{x} = \$1438.90$, $s = \$1329.60$) from a normally distributed population, calculate a $99\%$ confidence interval.
• Assumptions/Conditions: MET (population is judged to be normally distributed).
• Calculations for a 99% Confidence Interval:
  • Using the formula $\bar{x} \pm (t\text{ critical value}) \frac{s}{\sqrt{n}}$ with the appropriate $t^*$ for $99\%$ confidence and $df = 29$:
    • The $99\%$ Confidence Interval is: \769.88 < \mu < \$2107.92$
• Interpretation: "We are $99\%$ confident that the true population mean profit of all insurance plans sold by the company’s sales representatives is between $\$769.88$ and $\$2107.92$.