Business Statistics Chapter 8
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Questions and Answers

What does a point estimate represent?

A single value that best describes the population of interest

A [BLANK] estimate provides additional information about variability.

interval

What are the two types of confidence intervals discussed in this chapter?

  • Confidence intervals for the variance and confidence intervals for the standard deviation
  • Confidence intervals for the mean and confidence intervals for the proportion (correct)
  • Confidence intervals for the median and confidence intervals for the mode

    • For a confidence interval for the mean, σ known, the sample size must be greater than 30.

    <p>True (A)</p> Signup and view all the answers

    The [BLANK] level is the probability that the interval estimate will include the population parameter of interest.

    <p>confidence</p> Signup and view all the answers

    What is the formula for the standard error of the mean when σ is known?

    <p>σ / √n</p> Signup and view all the answers

    What is the purpose of generating a confidence interval?

    <p>To provide an estimate of the value of the population parameter (C)</p> Signup and view all the answers

    The limits of a confidence interval describe the range in which we have zero confidence that the actual population mean lies

    <p>False (B)</p> Signup and view all the answers

    The [BLANK] of error is the width of the confidence interval between a sample mean and its upper limit or between a sample mean and its lower limit.

    <p>margin</p> Signup and view all the answers

    How can we decrease the margin of error for a confidence interval?

    <p>Increasing the sample size while keeping the confidence level constant.</p> Signup and view all the answers

    The Student's t-distribution is used when σ is known.

    <p>False (B)</p> Signup and view all the answers

    What is the approximate standard error of the mean when σ is unknown?

    <p>s / √n (A)</p> Signup and view all the answers

    The t-distribution is always flatter and wider than the normal distribution.

    <p>False (B)</p> Signup and view all the answers

    What is the formula for the confidence interval for the mean when σ is unknown?

    <p>x ± ta/2 * s / √n</p> Signup and view all the answers

    The critical t-score for a given confidence level is always greater than the critical z-score for the same confidence level.

    <p>False (B)</p> Signup and view all the answers

    The [BLANK] distribution is used in place of the normal probability distribution when the sample standard deviation (s) is used in place of the population standard deviation (σ).

    <p>Student's t</p> Signup and view all the answers

    Flashcards

    Point Estimate

    A single value estimate of a population parameter.

    Confidence Interval

    An interval estimate that provides a range where the true population parameter lies.

    Confidence Level

    Probability that the confidence interval contains the true population parameter.

    Standard Error

    An estimate of the variability of a sample mean from the population mean.

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    Margin of Error

    The range within which the true population parameter is expected to lie, given a sample.

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    Z-Score

    A value that indicates how many standard deviations a data point is from the mean.

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    Critical Z-Score

    The z-score that corresponds to a specified confidence level.

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    t-Distribution

    A probability distribution used when sample size is small and population standard deviation is unknown.

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    Degrees of Freedom (df)

    The number of independent values that can vary in a statistical calculation.

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    Confidence Interval for Proportions

    An interval estimate for the true population proportion based on a sample proportion.

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    Sample Size Calculation

    Determining how many observations are needed to estimate a population parameter accurately.

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    Finite Population Correction Factor

    Adjustment to standard error formula for finite populations sampling without replacement.

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    Excel T.INV.2T Function

    Calculates the critical t-score for a given significance level and degrees of freedom in Excel.

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    Excel CONFIDENCE.NORM Function

    Calculates the margin of error for confidence intervals with known population standard deviation.

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    Excel CONFIDENCE.T Function

    Calculates the margin of error for confidence intervals with unknown population standard deviation.

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    Sample Proportion

    The ratio of the number of successes to the total number of observations in the sample.

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    Normal Distribution

    A symmetric probability distribution where most observations cluster around the central peak.

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    Sample Mean

    The average value in a given sample, used as a point estimate for the population mean.

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    Population Standard Deviation (σ)

    The measure of variability of all values in a population.

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    Sample Standard Deviation (s)

    Estimates the population standard deviation based on sample data.

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    Binomial Distribution

    A probability distribution that summarizes the likelihood of a value taking on two possible outcomes.

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    Sampling Error

    The difference between a sample statistic and the actual population parameter.

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    Critical T-Score

    The t-score corresponding to the desired confidence level and degrees of freedom.

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    Pilot Sample

    A small preliminary sample taken to estimate parameters for a larger sample.

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    Interpretations of Confidence Intervals

    Understanding what a confidence interval tells us about the population parameter.

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    Sample Size for Margin of Error

    Sample size needed to achieve a desired level of precision in estimates.

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    Study Notes

    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.

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    Description

    Explore the fundamentals of business statistics in Chapter 8, which focuses on confidence intervals and point estimates. This chapter covers the calculation of confidence intervals for means and proportions using real data from the Bureau of Labor Statistics. Understand the uncertainty of statistical sampling and learn how to interpret point estimates effectively.

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