Confidence Intervals and Estimation of Population Parameters

Section 8.1 Point and Interval Estimates of Population Parameters

Statistical Inference Methods

Statistical inference methods are essential in statistics for drawing conclusions about population parameters based on sample data. These methods rely on probability calculations and sampling distributions.

• Randomization Assumption: Probability calculations assume data are gathered from a random sample or randomized experiment.

• Sampling Distribution: The probability calculations refer to the sampling distribution of a statistic, which is often approximately normal.

Types of Statistical Inference

• Estimation: Determining population parameter values.

• Hypothesis Testing: Testing hypotheses about parameter values.

• The most informative estimation method constructs a confidence interval, an interval believed to contain the unknown parameter value.

Point Estimate and Interval Estimate

Estimation can be done using a single value or an interval.

• Point Estimate: A single number that is the best guess for the parameter.

• Interval Estimate: An interval of numbers believed to contain the actual value of the parameter.

• A point estimate alone does not indicate how close it is to the parameter; an interval estimate incorporates a margin of error to gauge accuracy.

Properties of Point Estimators

• Unbiasedness: An estimator is unbiased if its sampling distribution is centered at the parameter it estimates.

• The sample mean ($\bar{x}$) is an unbiased estimator of the population mean ($\mu$).

• The sample proportion ($\hat{p}$) is an unbiased estimator of the population proportion ($p$).

• Small Standard Deviation: A good estimator has a small standard deviation, meaning it tends to fall closer to the parameter than other estimators.

• For a normal distribution, the sample mean has a smaller standard deviation than the sample median.

Confidence Interval

A confidence interval is an interval containing the most believable values for a parameter. The probability that this method produces an interval containing the parameter is called the confidence level, commonly set at 0.95 (95%).

Section 8.2 Constructing a Confidence Interval to Estimate a Population Proportion

Logic Behind Constructing a Confidence Interval

Confidence intervals are constructed using properties of sampling distributions, especially for proportions.

• The sampling distribution of the sample proportion ($\hat{p}$) is approximately normal for large samples, with $np \geq 15$ and $n(1-p) \geq 15$.

• Mean of sampling distribution: $p$ (population proportion).

• Standard deviation: $\sqrt{\frac{p(1-p)}{n}}$

Margin of Error

• Approximately 95% of a normal distribution falls within 1.96 standard deviations of the mean.

• Margin of error: $1.96 \times \text{standard deviation}$

• 95% confidence interval for a proportion: $\hat{p} \pm 1.96 \times \text{standard deviation}$

Example 1: Confidence Interval for a Proportion

• Sample: 315 of 1285 respondents agreed with a statement.

• Estimated standard deviation: 0.01

• Margin of error: $1.96 \times 0.01 = 0.02$

• 95% CI: $0.31 \pm 0.02 = (0.29, 0.33)$

• Interpretation: The population proportion is predicted to be between 0.29 and 0.33.

Finding the 95% Confidence Interval for a Population Proportion

• Population proportion symbol: $p$

• Sample proportion symbol: $\hat{p}$ ("p-hat")

• For large samples, $\hat{p}$ is approximately normal.

• Margin of error: $1.96 \times \text{standard deviation}$

• Standard deviation: $\sqrt{\frac{p(1-p)}{n}}$ (unknown $p$ is estimated by $\hat{p}$)

• Standard error: $se = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• 95% CI: $\hat{p} \pm 1.96(se)$

Example 2: Paying Higher Prices to Protect the Environment

• Sample: $n = 1361$, $637$willing ($\hat{p} = 0.468$)

• Standard error: $se = \sqrt{\frac{0.468 \times 0.532}{1361}} = 0.0135$

• 95% CI: $0.468 \pm 1.96 \times 0.0135 = 0.468 \pm 0.026 = (0.442, 0.494)$

• Interpretation: Estimated population proportion is between 44% and 49%.

Sample Size Needed for Validity

• For validity of the 95% CI: $n\hat{p} \geq 15$ and $n(1-\hat{p}) \geq 15$

• If not met, use an adjusted formula (see Section 8.4).

Using Confidence Levels Other Than 95%

• Higher confidence level (e.g., 99%) increases the chance of correct inference but widens the interval.

• Lower confidence level narrows the interval but increases the risk of incorrect inference.

• Common z-scores: 1.645 (90%), 1.96 (95%), 2.58 (99%)

Confidence Level	Error Probability	z-Score	Confidence Interval
0.90	0.10	1.645	$\hat{p} \pm 1.64(se)$
0.95	0.05	1.96	$\hat{p} \pm 1.96(se)$
0.99	0.01	2.58	$\hat{p} \pm 2.58(se)$

Example 3: Influenza Vaccine

• Sample: $n = 3900$, $26$developed influenza ($\hat{p} = 0.0067$)

• Standard error: $se = \sqrt{\frac{0.0067 \times 0.9933}{3900}} = 0.0013$

• 99% CI: $0.0067 \pm 2.58 \times 0.0013 = 0.0067 \pm 0.0034 = (0.003, 0.010)$

Summary: Confidence Interval for a Population Proportion

• Assumptions: Data obtained by randomization; sample size large enough ($n\hat{p} \geq 15$, $n(1-\hat{p}) \geq 15$)

Effects of Confidence Level and Sample Size on Margin of Error

• Margin of error increases as confidence level increases.

• Margin of error decreases as sample size increases.

Section 8.3 Constructing a Confidence Interval to Estimate a Population Mean

Confidence Interval for a Population Mean

To estimate a population mean, use the sample mean as the point estimate and construct a confidence interval using the standard error.

• Standard error: $se = \frac{s}{\sqrt{n}}$ where $s$ is the sample standard deviation.

• Confidence interval: $\bar{x} \pm t^* \times se$

The t Distribution and Its Properties

• When population standard deviation is unknown, use the t-distribution instead of the normal distribution.

• The t-distribution is bell-shaped, symmetric about 0, and has thicker tails than the normal distribution.

• Degrees of freedom: $df = n - 1$

• Margin of error: $t^* \times se$

Example 4: Buying on eBay

• Sample: $n = 11$, sample mean $\bar{x} = 583.64$, sample standard deviation $s = 22.15$

• Standard error: $se = \frac{22.15}{\sqrt{11}} = 6.68$

• Degrees of freedom: $df = 10$

• For 95% CI, $t^* = 2.228$

• 95% CI: $583.64 \pm 2.228 \times 6.68 = 583.64 \pm 14.88 = (568.8, 598.5)$

• Interpretation: With 95% confidence, the mean closing price is between $569 and $599.

Robustness of the t Method

• t methods are robust to most violations of normality, but outliers can affect validity.

• Randomization is required for valid inference.

Standard Normal Distribution as t Distribution with Infinite df

• As $df \to \infty$, the t-distribution approaches the standard normal distribution.

• For large $df$, $t^*$ values approach $z$ values (e.g., $t^* = 1.96$ for 95% CI).

Section 8.4 Choosing the Sample Size for a Study

Sample Size for Estimating a Population Proportion

• Decide desired margin of error ($m$) and confidence level.

• Sample size formula: $n = \frac{p(1-p)z^2}{m^2}$

• Use $p = 0.5$ if no prior estimate is available.

Example 5: Sample Size for Exit Poll

• Desired margin of error: $m = 0.02$

• Estimated $p = 0.5$, $z = 1.96$

• Required sample size: $n = \frac{0.5 \times 0.5 \times 1.96^2}{0.02^2}$

Sample Size for Estimating a Population Mean

• Sample size formula: $n = \frac{z^2 \sigma^2}{m^2}$

• If $\sigma$ unknown, estimate from similar studies or use range/6 for bell-shaped distributions.

Example 6: Estimating Mean Education in South Africa

• Range: 0 to 18 years, $m = 1$ year

• Estimate $\sigma = \frac{18}{6} = 3$

• Sample size: $n = \frac{1.96^2 \times 3^2}{1^2}$

Other Factors Affecting Sample Size

• Desired precision (margin of error)

• Confidence level

• Variability in the data

• Cost

Using Small Sample Sizes

• t methods for means are valid for any $n$, but caution is needed for outliers and non-normality.

• For proportions, need at least 15 successes and 15 failures for normal approximation.

Confidence Interval for a Proportion with Small Samples

• If sample does not meet the 15 successes/failures rule, adjust by adding 2 to both successes and failures (add 4 to $n$).

• Formula: $\hat{p} = \frac{x+2}{n+4}$