Business Statistics Chapter 8

Chapter 8

• This chapter focuses on business statistics and confidence intervals.
• The authors are Robert A. Donnelly, Jr., Serina Al Haddad, and Stefan Ruediger.
• The content covers calculating confidence intervals for means (known and unknown standard deviations), proportions, sample sizes, and finite populations.
• The chapter highlights a practical application using data from the Bureau of Labor Statistics (BLS) regarding unemployment rates.
• The text discusses the uncertainty inherent in statistical sampling, explaining that a sample estimate of a population statistic provides a range of possible values; not a precise measure.
• Confidence intervals provide a range of values within which a population parameter is likely to fall, and the confidence level indicates the probability that the interval will contain the true population parameter.

8.1 Point Estimates

• A point estimate is a single value that best describes a population of interest.
• The sample mean estimates the unknown population mean.
• The sample proportion estimates the unknown population proportion.
• Point estimates are quick to calculate but don't reflect the level of accuracy of the estimate.

8.2 Calculating Confidence Intervals for the Mean (σ Known)

• Confidence intervals are range estimates around a sample mean.
• Confidence levels represent the probability that the confidence interval contains the true population mean value.
• This section establishes the formulas for calculating a confidence interval for a population mean when the population standard deviation is known.
• The data must adhere to certain conditions– the minimum sample size is 30 or more.
• The population standard deviation, 𝜎, must be known.

8.3 Calculating Confidence Intervals for the Mean (σ Unknown)

• Addresses the situation where the population standard deviation is unknown.
• The sample standard deviation, 𝑠, is used in place of the unknown population standard deviation,𝜎.
• The Student's t-distribution is utilized due to the use of 𝑠. The t distribution accounts for the smaller sample sizes.

8.4 Calculating Confidence Intervals for Proportions

• Proportion data follow the binomial distribution, and the normal distribution can be used for approximation.
• Conditions include: 𝑛𝑝 ≥ 5 and 𝑛(1−𝑝) ≥ 5.
• Formulas are given for sample proportion (𝑝) and the standard error of the proportion (𝜎).
• The population proportion (𝑝) is unknown, so it's estimated by the sample proportion (𝑝̄).

8.5 Determining the Sample Size

• Increasing sample size lowers margin of error and yields a narrower confidence interval.
• Formulas are given to calculate sample sizes needed to achieve a specified margin of error for the mean and proportion, given information about confidence level and a potential population standard deviation's value.

8.6 Calculating Confidence Intervals for Finite Populations

• Finite populations present a different situation than infinite populations when sampling as the formulas for standard error need a finite population correction factor.
• When calculating confidence intervals for a population mean and population proportion from finite populations, accounting for the sample's size in relation to the total population size is necessary.
• The correction factor is used when 𝑛/𝑁 > 0.05, where 𝑛 is the sample size, and 𝑁 is the population size.