Chapter 8: Confidence Intervals – Business Statistics Study Notes

Confidence Intervals

Introduction

Confidence intervals are a fundamental concept in inferential statistics, providing a range of values within which a population parameter is likely to fall. In business statistics, confidence intervals are used to estimate means and proportions, helping decision-makers understand the reliability of sample-based estimates.

8.1 Point Estimates

Definition and Purpose

• Point Estimate: A single value that best describes the population of interest, such as the sample mean (\( \bar{x} \)) or sample proportion (\( \hat{p} \)).

• Point estimates are easy to calculate but do not provide information about their accuracy or variability.

• Interval Estimate: Provides a range of values that best describes the population, accounting for sampling variability.

8.2 Calculating Confidence Intervals for the Mean When the Standard Deviation (\( \sigma \)) of a Population Is Known

Key Concepts

• Confidence Interval for the Mean: An interval estimate around a sample mean that provides a range for the true population mean.

• Confidence Level: The probability that the interval estimate will include the population parameter (e.g., 90%, 95%, 99%).

Assumptions

• Sample size is at least 30 (n ≥ 30).

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

Standard Error of the Mean

• Formula: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ where \( \sigma \) is the population standard deviation and \( n \) is the sample size.

Confidence Interval Formulas

• Upper Confidence Limit (UCL): $UCL_{\bar{x}} = \bar{x} + z_{\alpha/2} \sigma_{\bar{x}}$

• Lower Confidence Limit (LCL): $LCL_{\bar{x}} = \bar{x} - z_{\alpha/2} \sigma_{\bar{x}}$

• Critical z-score (\( z_{\alpha/2} \)): Value from the standard normal distribution corresponding to the desired confidence level.

Example

• Sample mean: $129.20$, $n = 32$, $\sigma = 40.60$

• Standard error: $\sigma_{\bar{x}} = \frac{40.60}{\sqrt{32}} = 7.173$

• For 90% confidence, $z_{\alpha/2} = 1.645$

• Interval endpoints: $UCL_{\bar{x}} = 129.20 + (1.645 \times 7.173) = 141.00$ $LCL_{\bar{x}} = 129.20 - (1.645 \times 7.173) = 117.40$ Interpretation: We are 90% confident that the average online order size for Gap customers is between $117.40 and $141.00.

Margin of Error

• Margin of Error (ME): The width of the confidence interval between the sample mean and its upper or lower limit. $ME = z_{\alpha/2} \sigma_{\bar{x}}$

• Example: $ME = 1.645 \times 7.173 = 11.80$

Interpreting Confidence Intervals

• It is incorrect to say there is a 90% probability that the population mean is within the interval; rather, 90% of such intervals from repeated samples will contain the true mean.

• Each sample produces its own confidence interval, but the margin of error remains the same if sample size and standard deviation are unchanged.

Sample	Mean	Margin of Error	Lower Limit	Upper Limit
1	129.20	11.80	117.40	141.00
2	130.00	11.80	118.20	141.80
...	...	...	...	...

Changing Confidence Levels

Effect on Interval Width

• The confidence level is the complement of the significance level: $1 - \alpha$.

• Higher confidence levels (e.g., 99%) require wider intervals to ensure the true parameter is captured.

Confidence Level (%)	Significance Level (%)	Critical z-score (\( z_{\alpha/2} \))
80	20	1.28
90	10	1.645
95	5	1.96
98	2	2.33
99	1	2.575

Using Excel for Confidence Intervals (Sigma Known)

Excel Function

• CONFIDENCE.NORM: Calculates the margin of error for confidence intervals. =CONFIDENCE.NORM(alpha, standard_dev, size)

• alpha: Significance level

• standard_dev: Population standard deviation

• size: Sample size

Confidence Intervals for the Mean with Small Samples (Sigma Known)

Key Points

• When n < 30 and \( \sigma \) is known, the population must be normally distributed.

• The Central Limit Theorem does not apply for small samples.

Example

• Sample mean: $8,150$, $n = 15$, $\sigma = 2,970$

• Standard error: $\sigma_{\bar{x}} = \frac{2,970}{\sqrt{15}} = 767.44$

• For 99% confidence, $z_{\alpha/2} = 2.575$

• Interval endpoints: $UCL_{\bar{x}} = 8,150 + (2.575 \times 767.44) = 10,126.16$ $LCL_{\bar{x}} = 8,150 - (2.575 \times 767.44) = 6,173.84$

8.3 Calculating Confidence Intervals for the Mean When the Standard Deviation (\( \sigma \)) of a Population Is Unknown

Key Concepts

• When \( \sigma \) is unknown, use the sample standard deviation (s).

• Formula for sample standard deviation: $s = \sqrt{ \frac{ \sum_{i=1}^{n} (x_i - \bar{x})^2 }{ n - 1 } }$

Student's t-distribution

• Used when s replaces \( \sigma \).

• Properties:

  • Bell-shaped and symmetrical

  • Shape depends on degrees of freedom (df = n - 1)

  • Flatter and wider than the normal distribution

  • As n increases, t-distribution approaches normal

Confidence Interval Formulas (\( \sigma \) Unknown)

• Upper Confidence Limit: $UCL_{\bar{x}} = \bar{x} + t_{\alpha/2} \frac{s}{\sqrt{n}}$

• Lower Confidence Limit: $LCL_{\bar{x}} = \bar{x} - t_{\alpha/2} \frac{s}{\sqrt{n}}$

• t-score (\( t_{\alpha/2} \)): Value from the t-distribution table for the desired confidence level and degrees of freedom.

Example

• Sample mean: $103$,$n = 18$,$s = 10.6$

• Standard error: $\frac{10.6}{\sqrt{18}} = 2.50$

• For 95% confidence, $t_{\alpha/2} = 2.110$

• Interval endpoints: $UCL_{\bar{x}} = 103 + (2.110 \times 2.50) = 108.28$ $LCL_{\bar{x}} = 103 - (2.110 \times 2.50) = 97.72$

Excel Functions for t-distribution

• TINV.2T: Finds the critical t-score. =TINV.2T(alpha, degrees_of_freedom)

• CONFIDENCE.T: Calculates the margin of error for confidence intervals when \( \sigma \) is unknown. =CONFIDENCE.T(alpha, standard_dev, size)

8.4 Calculating Confidence Intervals for Proportions

Key Concepts

• Proportion data follow the binomial distribution, which can be approximated by the normal distribution if $np \geq 5$ and $n(1-p) \geq 5$.

• Sample Proportion: $\hat{p} = \frac{x}{n}$ where $x$ is the number of successes and $n$ is the sample size.

• Standard Error of Proportion: $SE_{\hat{p}} = \sqrt{ \frac{ \hat{p}(1-\hat{p}) }{ n } }$

Confidence Interval Formulas for Proportion

• Upper Confidence Limit: $UCL_{\hat{p}} = \hat{p} + z_{\alpha/2} SE_{\hat{p}}$

• Lower Confidence Limit: $LCL_{\hat{p}} = \hat{p} - z_{\alpha/2} SE_{\hat{p}}$

Margin of Error for Proportion

• $ME_{\hat{p}} = z_{\alpha/2} SE_{\hat{p}}$

8.5 Determining the Sample Size

Key Concepts

• Increasing sample size reduces the margin of error, resulting in a more precise confidence interval.

• Required sample size for a desired margin of error can be calculated.

Sample Size Formula for Mean

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

Sample Size Formula for Proportion

• $n = \frac{ z_{\alpha/2}^2 \hat{p}(1-\hat{p}) }{ ME^2 }$

• If no pilot sample is available, use $\hat{p} = 0.5$ for the most conservative estimate.

8.6 Calculating Confidence Intervals for Finite Populations

Key Concepts

• For finite populations, the standard error is adjusted using the finite population correction factor.

• Finite Population Correction Factor: $\sqrt{ \frac{ N - n }{ N - 1 } }$ where $N$ is population size and $n$ is sample size.

Confidence Interval Formulas (Finite Population)

• For mean (\( \sigma \) known): $UCL_{\bar{x}} = \bar{x} + z_{\alpha/2} \sigma_{\bar{x}} \sqrt{ \frac{ N - n }{ N - 1 } }$ $LCL_{\bar{x}} = \bar{x} - z_{\alpha/2} \sigma_{\bar{x}} \sqrt{ \frac{ N - n }{ N - 1 } }$

• For mean (\( \sigma \) unknown): $UCL_{\bar{x}} = \bar{x} + t_{\alpha/2} \frac{s}{\sqrt{n}} \sqrt{ \frac{ N - n }{ N - 1 } }$ $LCL_{\bar{x}} = \bar{x} - t_{\alpha/2} \frac{s}{\sqrt{n}} \sqrt{ \frac{ N - n }{ N - 1 } }$

• For proportion: $UCL_{\hat{p}} = \hat{p} + z_{\alpha/2} SE_{\hat{p}} \sqrt{ \frac{ N - n }{ N - 1 } }$ $LCL_{\hat{p}} = \hat{p} - z_{\alpha/2} SE_{\hat{p}} \sqrt{ \frac{ N - n }{ N - 1 } }$

Example

• Population size: $N = 1,600$, sample size: $n = 100$, sample mean: $7,277$, sample standard deviation: $1,480$

• Finite population correction factor: $\sqrt{ \frac{ 1,600 - 100 }{ 1,600 - 1 } }$

• For 95% confidence, $t_{\alpha/2} = 1.984$

• Interval endpoints: $UCL_{\bar{x}} = 7,277 + (1.984 \times 1,480 \times \text{correction factor})$ $LCL_{\bar{x}} = 7,277 - (1.984 \times 1,480 \times \text{correction factor})$

Focus on Analytics: Confidence Intervals and Surveys

Application in Business

• Confidence intervals are widely used in surveys to report the uncertainty of estimates.

• Companies like Google and Pew Research Center use modeled margin of error (MOE) to describe how much survey estimates may vary if repeated.

• Error bars in survey results visually represent confidence intervals.

Summary Table: Confidence Interval Formulas

Parameter	Standard Error	Critical Value	Interval Formula
Mean (\( \sigma \) known)	$\frac{\sigma}{\sqrt{n}}$	$z_{\alpha/2}$	$\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
Mean (\( \sigma \) unknown)	$\frac{s}{\sqrt{n}}$	$t_{\alpha/2}$	$\bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}$
Proportion	$\sqrt{ \frac{ \hat{p}(1-\hat{p}) }{ n } }$	$z_{\alpha/2}$	$\hat{p} \pm z_{\alpha/2} \sqrt{ \frac{ \hat{p}(1-\hat{p}) }{ n } }$

Additional info: These notes expand on the textbook slides by providing full formulas, step-by-step examples, and context for business applications, ensuring students can apply confidence intervals in real-world scenarios.