Estimating a Population Mean: Confidence Intervals

Sampling Distributions & Confidence Intervals: Mean

Estimating a Population Mean Using Confidence Intervals

Confidence intervals are used in statistics to estimate the range within which a population parameter, such as the mean, is likely to fall. When the population standard deviation is unknown, the sample mean and sample standard deviation are used to construct the interval, typically employing the t-distribution.

• Confidence Interval: An interval estimate that gives a range of plausible values for a population parameter, based on sample data.

• Population Mean (μ): The average value of a variable in the entire population.

• Sample Mean (x̅): The average value calculated from a sample drawn from the population.

• Sample Standard Deviation (s): A measure of variability within the sample.

• t-Distribution: Used when the population standard deviation is unknown and the sample size is not very large.

Formula for Confidence Interval for the Mean (Unknown Population Standard Deviation)

The confidence interval for the population mean μ is calculated as:

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

• x̅: Sample mean

• s: Sample standard deviation

• n: Sample size

• tα/2: t-score corresponding to the desired confidence level and degrees of freedom (df = n - 1)

Example 1: Estimating Mean Body Temperature

A random sample of 106 body temperatures has a mean of 98.20°F and a standard deviation of 0.62°F. Construct a 99% confidence interval estimate of the mean body temperature of all healthy humans.

• Sample mean (x̅): 98.20°F

• Sample standard deviation (s): 0.62°F

• Sample size (n): 106

• Confidence level: 99%

Using a calculator or statistical software (e.g., TInterval function), the confidence interval is:

Result: (98.04, 98.36) or $98.04 < \mu < 98.36$

Interpretation: We are 99% confident that the mean body temperature of all healthy humans lies between 98.04°F and 98.36°F.

Example 2: Estimating Mean Weight of Bumble Bats

The world's smallest mammal, the bumble bat, has weights (in grams) from a sample: 1.7, 1.6, 1.5, 2.0, 2.3, 1.6, 1.6, 1.8, 1.5, 1.7, 2.2, 1.4, 1.6, 1.6, 1.6. Construct a 95% confidence interval estimate of their mean weight.

• Raw data: Entered into statistical software or calculator.

• Confidence level: 95%

Using the TInterval function, the confidence interval is:

Result: (1.57, 1.86) or $1.57 < \mu < 1.86$

Interpretation: We are 95% confident that the mean weight of bumble bats lies between 1.57 g and 1.86 g.

Steps for Constructing a Confidence Interval for the Mean

1. Collect a random sample and calculate the sample mean (x̅) and sample standard deviation (s).

2. Determine the desired confidence level (e.g., 95%, 99%).

3. Find the appropriate t-score for the confidence level and degrees of freedom (df = n - 1).

4. Calculate the margin of error: $t\_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)$

5. Construct the interval: $x\_\bar \pm$ margin of error.

Comparison Table: Confidence Interval Parameters

Parameter	Symbol	Role in CI Calculation
Sample Mean	x̅	Center of the interval
Sample Standard Deviation	s	Measures variability
Sample Size	n	Affects margin of error
t-score	tα/2	Determines width of interval
Confidence Level	--	Probability that interval contains μ

Additional info: The t-distribution is used instead of the normal distribution when the population standard deviation is unknown and the sample size is small (typically n < 30). For larger samples, the t-distribution approaches the normal distribution.