Chapter 7: Estimating Population Parameters and Sample Sizes

Estimating a Population Proportion

Point Estimate and Sample Proportion

In statistics, a point estimate is a single value used to approximate a population parameter. The sample mean ($\bar{x}$) is the best point estimate of the population mean ($\mu$), and the sample proportion ($\hat{p}$) is the best point estimate of the population proportion ($p$).

• Sample Proportion ($\hat{p}$): Calculated as the number of successes ($x$) divided by the sample size ($n$): $\hat{p} = \frac{x}{n}$

• Example: If 85% of 1007 adults know what Twitter is, then $\hat{p} = 0.85$ is the best estimate for $p$.

Confidence Interval for a Proportion

A confidence interval is a range of values used to estimate the true value of a population parameter. The margin of error is the maximum likely difference between the sample statistic and the population parameter, with a specified probability (confidence level).

• Confidence Level: The probability (often expressed as a percentage) that the confidence interval contains the population parameter. Common levels are 90%, 95%, and 99%.

• Level of Significance ($\alpha$): The probability that the confidence interval does not contain the parameter. For example, $\alpha = 0.05$ corresponds to a 95% confidence level.

• Critical Value ($z_{\alpha/2}$): The value separating the tail area $\alpha/2$ in the standard normal distribution.

To construct a confidence interval for a proportion, use the formula:

• $\hat{p}$: sample proportion

• $\hat{q} = 1 - \hat{p}$: sample proportion of failures

Formula for Confidence Interval:

$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}\hat{q}}{n}}$

Example: For a 95% confidence level, $z_{\alpha/2} = 1.96$.

Sample Size for Estimating Proportion

To determine the sample size needed for estimating a proportion:

• If an estimate $\hat{p}$ is known:

$n = \left( \frac{z_{\alpha/2}^2 \hat{p} \hat{q}}{E^2} \right)$

• If no estimate is known, use $\hat{p} = 0.5$:

$n = \left( \frac{z_{\alpha/2}^2 \cdot 0.25}{E^2} \right)$

• Round-Off Rule: Always round up to the next whole number if the computed sample size is not an integer.

Estimating a Population Mean

Confidence Interval for the Mean

To estimate the population mean ($\mu$), use a confidence interval. If the population standard deviation ($\sigma$) is unknown, use the sample standard deviation ($s$) and the Student t distribution.

• Formula for Confidence Interval:

$\bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)$

• Calculator Tip: Use TInterval on a TI-83/84 calculator for confidence intervals when $\sigma$ is unknown.

Sample Size for Estimating the Mean

To determine the sample size needed for estimating the mean:

$n = \left( \frac{z_{\alpha/2}^2 \sigma^2}{E^2} \right)$

• If $\sigma$ is unknown, use $s$.

• Round-Off Rule: Always round up to the next whole number.

Common Critical Values

Confidence Level	$\alpha$	Critical Value ($z_{\alpha/2}$)
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.576

Student t Distribution

Properties and Comparison to Normal Distribution

The Student t distribution is used when estimating the mean with small sample sizes and unknown population standard deviation. It has several important properties:

• The t distribution varies with sample size (degrees of freedom).

• It has a symmetric bell shape, similar to the normal distribution, but is wider for small samples.

• The mean of the t distribution is 0.

• The standard deviation is greater than 1 and decreases as sample size increases.

• As sample size increases, the t distribution approaches the normal distribution.

Chapter 7 Summary Table

Parameter	Confidence Interval Method	Sample Size Formula
Proportion ($p$)	1–PropZInt	$n = \left( \frac{z_{\alpha/2}^2 \hat{p} \hat{q}}{E^2} \right)$
Mean ($\mu$)	TInterval	$n = \left( \frac{z_{\alpha/2}^2 \sigma^2}{E^2} \right)$

Additional info: The notes include calculator instructions for constructing confidence intervals and emphasize rounding up sample sizes to ensure adequate precision.