Confidence Intervals and Estimation (Statistics for Business)

Confidence Intervals and Estimation

Introduction

In business statistics, estimation is a fundamental process used to infer population parameters based on sample data. This chapter focuses on point estimates and confidence intervals, which are essential tools for quantifying uncertainty in statistical inference.

Point and Interval Estimates

Definitions and Concepts

• Point Estimate: A single value calculated from sample data that serves as the best guess for an unknown population parameter.

• Confidence Interval: A range of values, derived from sample statistics, that is likely to contain the true value of the population parameter with a specified level of confidence.

The width of the confidence interval reflects the precision of the estimate: narrower intervals indicate more precise estimates.

Key Table: Estimating Population Parameters

We can estimate a Population Parameter …	with a Sample Statistic (a Point Estimate)
Mean (μ)	Sample Mean ($\overline{X}$)
Proportion (π)	Sample Proportion (p)

Confidence Intervals

Purpose and Interpretation

• Quantifies the uncertainty associated with a point estimate.

• Provides more information about a population characteristic than a point estimate alone.

• Such interval estimates are called confidence intervals.

Properties of Confidence Intervals

• Gives a range of plausible values for the population parameter.

• Accounts for sample-to-sample variability.

• Based on data from a single sample.

• Expressed in terms of a confidence level (e.g., 95%, 99%).

• Can never be 100% confident due to inherent sampling variability.

Constructing Confidence Intervals

General Formula

• The general form for a confidence interval is:

$\text{Point Estimate} \pm \text{Critical Value} \times \text{Standard Error}$

• Point Estimate: The sample statistic estimating the population parameter.

• Critical Value: A value from the sampling distribution (Z or t) corresponding to the desired confidence level.

• Standard Error: The standard deviation of the point estimate.

Confidence Level

• The probability that the interval will contain the unknown population parameter in repeated sampling.

• Common confidence levels: 90%, 95%, 99%.

• For a 95% confidence level, $\alpha = 0.05$ (so $1 - \alpha = 0.95$).

• Interpretation: 95% of all intervals constructed in this way will contain the true parameter.

• For any specific interval, it either contains the parameter or it does not—probability is not assigned to a specific interval after the data are observed.

Confidence Interval for the Mean (σ Known)

Assumptions

• Population standard deviation ($\sigma$) is known.

• Population is normally distributed, or sample size is large ($n > 30$).

Formula

$\overline{X} \pm Z_{\alpha/2} \left( \dfrac{\sigma}{\sqrt{n}} \right)$

• $\overline{X}$: Sample mean

• $Z_{\alpha/2}$: Z-value for the desired confidence level

• $\sigma$: Population standard deviation

• $n$: Sample size

Finding the Critical Value ($Z_{\alpha/2}$)

• For a 95% confidence interval, $Z_{\alpha/2} = 1.96$

• For a 99% confidence interval, $Z_{\alpha/2} = 2.576$

• Critical values correspond to the area in the tails of the standard normal distribution.

Common Confidence Levels and Z-values

Confidence Level	1 - α	Zα/2
80%	0.80	1.282
90%	0.90	1.645
95%	0.95	1.960
98%	0.98	2.326
99%	0.99	2.576
99.8%	0.998	3.090
99.9%	0.999	3.291

Example: Confidence Interval for the Mean (σ Known)

• Population mean $\mu = 368$, $\sigma = 15$

• Sample size $n = 25$

• 95% confidence interval:

$368 \pm 1.96 \times 15/\sqrt{25} = 368 \pm 5.88 = (362.12, 373.88)$

• Interpretation: 95% of intervals constructed in this way will contain the true mean.

Table: Multiple Sample Confidence Intervals

Sample #	$\overline{X}$	Lower Limit	Upper Limit	Contains $\mu$?
1	362.30	356.42	368.18	Yes
2	369.50	363.62	375.38	Yes
3	360.00	354.12	365.88	No
4	362.12	356.24	368.00	Yes
5	373.88	368.00	379.76	Yes

Confidence Interval for the Mean (σ Unknown)

Assumptions

• Population standard deviation ($\sigma$) is unknown.

• Population is normally distributed.

• Use the Student's t distribution instead of the normal distribution.

Formula

$\overline{X} \pm t_{\alpha/2, n-1} \left( \dfrac{S}{\sqrt{n}} \right)$

• $\overline{X}$: Sample mean

• $t_{\alpha/2, n-1}$: t-value for the desired confidence level and degrees of freedom ($n-1$)

• $S$: Sample standard deviation

• $n$: Sample size

Student's t Distribution

• A family of distributions that varies with degrees of freedom (d.f. = n - 1).

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

• Critical values are larger for smaller samples, reflecting greater uncertainty.

Example: Confidence Interval for the Mean (σ Unknown)

• Sample of 11 circuits: $\overline{X} = 2.20$ ohms, $S = 0.35$ ohms

• 95% confidence interval:

$2.20 \pm 1.96 \times \dfrac{0.35}{\sqrt{11}} = 2.20 \pm 0.2068 = (1.9932, 2.4068)$

• Interpretation: We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms.

Summary Table: Z and t Critical Values

Confidence Level	t (20 d.f.)	t (30 d.f.)	Z
90%	1.725	1.697	1.645
95%	2.086	2.042	1.960
99%	2.845	2.750	2.576

Additional info: As sample size increases, t-values approach Z-values, reflecting the Central Limit Theorem.

Key Takeaways

• Point estimates provide single-value approximations of population parameters, but do not reflect uncertainty.

• Confidence intervals offer a range of plausible values and quantify the precision of estimates.

• The choice between Z and t distributions depends on whether the population standard deviation is known and the sample size.

• Interpret confidence intervals in the context of repeated sampling, not as a probability statement about a specific interval.