Glossary

7.2 Confidence Intervals for Means

3 min read•july 23, 2024

Confidence intervals help estimate population means using sample data. They provide a range of likely values for the true population average, accounting for sampling variability and uncertainty.

For large samples or known population standard deviations, we use z-distributions. With small samples or unknown standard deviations, t-distributions are applied. Understanding sample size requirements and interpreting results are crucial for accurate business insights.

Confidence Intervals for Population Means

Confidence intervals with z-distribution

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  • Used when sample size is large (n≥30n \geq 30) or population is normally distributed and population standard deviation (σ\sigma) is known
  • Confidence interval formula: xˉ±zα/2⋅σn\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}
    • xˉ\bar{x} represents
    • zα/2z_{\alpha/2} represents critical value from standard
    • α\alpha represents significance level and 1−α1 - \alpha represents (95%, 99%)
    • nn represents sample size
  • Example: Estimating average customer spending (\sigma = \20,, n = 100,, \bar{x} = $50$, 95% confidence level)

Confidence intervals with t-distribution

  • Used when sample size is small (n<30n < 30), population is normally distributed, and population standard deviation is unknown
  • Sample standard deviation (ss) used as estimate for population standard deviation
  • Confidence interval formula: xˉ±tα/2,n−1⋅sn\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}
    • tα/2,n−1t_{\alpha/2, n-1} represents critical value from with n−1n-1 degrees of freedom
  • t-distribution has heavier tails compared to standard normal distribution accounting for additional uncertainty when using sample standard deviation
  • Example: Estimating average employee satisfaction score (n=25n = 25, xˉ=3.8\bar{x} = 3.8, s=0.6s = 0.6, 90% confidence level)

Sample size for confidence intervals

  • (EE) represents maximum acceptable difference between sample mean and
  • Sample size formula for known population standard deviation: n=(zα/2⋅σE)2n = (\frac{z_{\alpha/2} \cdot \sigma}{E})^2
  • Sample size formula for unknown population standard deviation: n=(tα/2,n−1⋅sE)2n = (\frac{t_{\alpha/2, n-1} \cdot s}{E})^2
    • Iterative process or software often used to solve for nn since sample size appears on both sides of equation
  • Example: Determining sample size for estimating average customer wait time (E=2E = 2 minutes, 95% confidence level, σ=10\sigma = 10 minutes)

Interpretation of confidence intervals

  • Provides range of plausible values for population mean
  • Confidence level (95%, 99%) represents long-run probability that interval will contain true population mean
  • Business applications:
    • Estimating average sales, revenue, or customer satisfaction score for product or service
    • Comparing mean performance of different business units or strategies
    • Determining if process change has resulted in significant improvement in key metric
  • Consider width of confidence interval and practical significance of results when making decisions based on
  • Example: Comparing average sales between two store locations (\bar{x}_1 = \1000,, \bar{x}_2 = $1200,95, 95% CI for difference: $50toto$350$)

Additional Considerations

Assumptions and limitations

  • Assumes sample is randomly selected from population
  • Assumes population is normally distributed or sample size is large enough for to apply
  • Violations of assumptions may lead to inaccurate confidence intervals
  • Provides estimate of population mean but does not prove causality between variables
  • Example: Non-random sampling may result in biased estimate of population mean

Key Terms to Review (18)

Alternative Hypothesis: The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect, a difference, or a relationship in the population. It serves as the focus of research, aiming to provide evidence that supports its claim over the null hypothesis through statistical testing and analysis.
Central Limit Theorem: The Central Limit Theorem states that, given a sufficiently large sample size, the sampling distribution of the sample mean (or sample proportion) will be normally distributed, regardless of the original population's distribution. This theorem is crucial because it allows for making inferences about population parameters using sample statistics, bridging the gap between descriptive statistics and inferential statistics.
Confidence Level: The confidence level is a statistical measure that reflects the degree of certainty in an estimate, typically expressed as a percentage. It indicates the proportion of times that a statistical procedure will produce an interval that contains the true parameter if the procedure were repeated numerous times. This concept is vital in constructing confidence intervals, conducting hypothesis tests, determining sample sizes, and understanding errors in statistical analysis.
Interval Estimate: An interval estimate is a range of values used to estimate a population parameter, providing more information than a single point estimate. It reflects the uncertainty associated with the estimation process by defining lower and upper bounds, allowing for a more reliable understanding of the parameter's true value. This approach is essential in assessing means and understanding the variability and confidence associated with statistical inferences.
Margin of error: The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It provides an estimate of the uncertainty around a sample statistic, helping to convey how much the results may differ from the true population value. This concept is crucial when interpreting data, as it indicates the range within which the true value is likely to fall and connects closely to confidence levels and sample size.
Normal distribution: Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This characteristic forms a bell-shaped curve, which is significant in various statistical methods and analyses.
Null hypothesis: The null hypothesis is a statement that assumes there is no effect or no difference in a given situation, serving as a default position that researchers aim to test against. It acts as a baseline to compare with the alternative hypothesis, which posits that there is an effect or a difference. This concept is foundational in statistical analysis and hypothesis testing, guiding researchers in determining whether observed data can be attributed to chance or if they suggest significant effects.
One-sample confidence interval: A one-sample confidence interval is a statistical range that estimates the true mean of a population based on a sample drawn from that population. This interval provides a margin of error around the sample mean, giving an idea of how much uncertainty is involved in estimating the population parameter. The width of the interval depends on factors like the sample size, variability in the data, and the desired confidence level.
Point Estimate: A point estimate is a single value derived from sample data that serves as a best guess or approximation of an unknown population parameter. It represents the most likely value for a characteristic of the population, such as the mean or proportion, based on observed data. Point estimates are essential for making inferences about populations, often being the starting point for constructing confidence intervals that provide a range of plausible values for the parameter.
Population Mean: The population mean is the average of a set of values within a defined population, calculated by summing all the values and dividing by the total number of observations. This term is fundamental in understanding the characteristics of a population and plays a crucial role in various statistical analyses, particularly in the assessment of sampling distributions and confidence intervals.
Power of a Test: The power of a test is the probability that it correctly rejects a null hypothesis when the alternative hypothesis is true. This concept is crucial because it reflects the test's ability to detect an effect or difference when one exists, and it is closely tied to the risks of Type I and Type II errors, as well as the design of studies involving confidence intervals and model assumptions.
Sample mean: The sample mean is the average value calculated from a set of data points in a sample. It serves as a point estimate of the population mean and is central to various statistical analyses, including understanding the sampling distribution, constructing confidence intervals, and conducting hypothesis tests. The sample mean helps summarize the data and provides insights into the overall characteristics of the population from which the sample was drawn.
Sample size determination: Sample size determination is the process of calculating the number of observations or replicates needed in a study to ensure that the results will be statistically significant and reflective of the population. This concept is crucial because an appropriately chosen sample size helps to achieve a desired level of confidence in estimates, whether for means or proportions, and aids in effective quality control. Proper sample size determination helps balance accuracy and resource constraints while minimizing errors.
T-distribution: The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. It is used primarily in statistics for estimating population parameters when the sample size is small and the population standard deviation is unknown. This distribution becomes more like the normal distribution as the sample size increases.
Two-sample confidence interval: A two-sample confidence interval is a statistical method used to estimate the difference between the means of two independent groups while accounting for uncertainty. It provides a range of values within which the true difference in means is likely to lie, based on sample data and a specified confidence level. This concept is crucial for comparing groups, evaluating effectiveness of treatments, or assessing variations in processes.
Type I Error: A Type I error occurs when a null hypothesis is incorrectly rejected when it is actually true, leading to a false positive conclusion. This concept is crucial in statistical hypothesis testing, as it relates to the risk of finding an effect or difference that does not exist. Understanding the implications of Type I errors helps in areas like confidence intervals, model assumptions, and the interpretation of various statistical tests.
Type II Error: A Type II Error occurs when a statistical test fails to reject a false null hypothesis. This means that the test concludes there is no effect or difference when, in reality, one exists. Understanding Type II Errors is crucial for interpreting results in hypothesis testing, as they relate to the power of a test and the implications of failing to detect a true effect.
Z-score: A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean, allowing for comparison between different datasets and understanding the relative position of a value within a distribution.
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