BUSS 200 QUIZ 2
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Questions and Answers

What is the main difference between a point estimate and a confidence interval estimate?

  • A point estimate provides a single value, while a confidence interval provides a range of values that is likely to contain the true population parameter. (correct)
  • A point estimate is always more accurate than a confidence interval.
  • A point estimate provides a range of values, while a confidence interval provides a single value that is likely to contain the true population parameter.
  • A confidence interval is always more accurate than a point estimate.

    • Can we eliminate sampling error completely?

    False (B)

    A confidence interval is used to quantify the uncertainty associated with a point estimate.

    True (A)

    The general formula for all confidence intervals includes a point estimate, a critical value and a standard error.

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

    The confidence level is a percentage that represents the probability that the true population parameter will be within the confidence interval.

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

    How is the confidence level related to the interval's width?

    <p>A higher confidence level leads to a wider confidence interval.</p> Signup and view all the answers

    Match the following terms with their descriptions:

    <p>Point Estimate = A single value used to estimate a population parameter Confidence Interval = A range of values that is likely to contain the true population parameter Confidence Level = A percentage representing the confidence that the true population parameter lies within the confidence interval Standard Error = The standard deviation of the sampling distribution</p> Signup and view all the answers

    What is the margin of error for a confidence interval?

    <p>The margin of error is the amount added and subtracted to the point estimate to form the confidence interval. It represents the level of uncertainty in the point estimate.</p> Signup and view all the answers

    The margin of error increases as the sample size increases.

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

    The margin of error decreases as the confidence level increases.

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

    What is the formula for calculating the required sample size when the population standard deviation is known?

    <p>n = (Zα/2 * σ / e)²</p> Signup and view all the answers

    When is it appropriate to consider a large sample approximation for the confidence interval for the mean?

    <p>When the sample size is greater than or equal to 30.</p> Signup and view all the answers

    The Student's t-distribution is used when the population standard deviation is known.

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

    What are the degrees of freedom for the Student's t-distribution?

    <p>The degrees of freedom (df) for the Student's t-distribution are calculated as n - 1, where n is the sample size.</p> Signup and view all the answers

    The Student's t-distribution has a smaller spread than the standard normal distribution for the same degrees of freedom.

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

    As the sample size increases, the shape of the Student's t-distribution becomes more similar to the standard normal distribution.

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

    How is a confidence interval for a population proportion calculated?

    <p>The confidence interval for a population proportion is calculated by adding an allowance for uncertainty to the sample proportion. This allowance is based on the standard error and the desired level of confidence.</p> Signup and view all the answers

    A larger sample size will lead to a narrower confidence interval for a population proportion.

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

    You should always round down the required sample size when calculating it.

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

    What does the null hypothesis (H0) generally assume?

    <p>It states that there is no effect or no difference. (D)</p> Signup and view all the answers

    Which of the following statements correctly describes the alternative hypothesis (H1 or HA)?

    <p>It is usually the hypothesis that needs to be tested against. (B)</p> Signup and view all the answers

    What is the main purpose of formulating a decision rule in hypothesis testing?

    <p>To specify how to reject or fail to reject the null hypothesis. (B)</p> Signup and view all the answers

    Which of the following best describes a Type I error in hypothesis testing?

    <p>Rejecting the null hypothesis when it is actually true. (C)</p> Signup and view all the answers

    What is the significance of the critical value in hypothesis testing?

    <p>It indicates the cutoff point for rejecting the null hypothesis. (D)</p> Signup and view all the answers

    What does the p-value in hypothesis testing signify?

    <p>The extremity of the observed results under the null hypothesis. (A)</p> Signup and view all the answers

    How can the probability of a Type II error be characterized?

    <p>It represents failing to reject the null hypothesis when it is actually false. (A)</p> Signup and view all the answers

    What is one feature that is always included in the null hypothesis?

    <p>An equality sign (=), less than or equal sign (≤), or greater than or equal sign (≥). (A)</p> Signup and view all the answers

    What does the level of significance, denoted as α, represent in hypothesis testing?

    <p>The probability of rejecting the null hypothesis when it is true (A)</p> Signup and view all the answers

    In a two-tailed test, what are the critical values denoted by?

    <p>zα/2 and -zα/2 (C)</p> Signup and view all the answers

    When performing a lower tail test, how is the cutoff value often designated?

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

    What is the primary action taken when the test statistic falls within the rejection region?

    <p>Reject the null hypothesis (C)</p> Signup and view all the answers

    What is the relationship between sample size and the impact on the test statistic when the population standard deviation is unknown?

    <p>A larger sample tends to stabilize the test statistic (D)</p> Signup and view all the answers

    In hypothesis testing for a mean, which statistic is primarily used when the sample size is large?

    <p>Z statistic (D)</p> Signup and view all the answers

    What is indicated by a critical value in hypothesis testing?

    <p>The threshold to determine rejection of the null hypothesis (C)</p> Signup and view all the answers

    How does the choice of level of significance affect the testing process?

    <p>It establishes the cutoff for making decisions about H0 (A)</p> Signup and view all the answers

    What type of test is appropriate when the null hypothesis states that the mean is equal to a specific value?

    <p>Two-tailed test (A)</p> Signup and view all the answers

    What happens to the rejection region as the level of significance α increases?

    <p>It expands, allowing for more extreme values (C)</p> Signup and view all the answers

    What role does the critical value play in hypothesis testing?

    <p>It indicates where the test statistic must fall to reject H0 (B)</p> Signup and view all the answers

    What is one consequence of choosing a very low level of significance?

    <p>It makes it harder to reject the null hypothesis (C)</p> Signup and view all the answers

    Which calculation is utilized to determine the test statistic when the population standard deviation is known?

    <p>z = (x̄ - μ) / σ (D)</p> Signup and view all the answers

    What does it signify if the test statistic is less than the cutoff value in a hypothesis test?

    <p>The null hypothesis is accepted (B)</p> Signup and view all the answers

    In the context of hypothesis testing, what conclusion can be drawn from a p-value of 0.0136 when $eta$ is set at 0.05?

    <p>There is sufficient evidence to reject the null hypothesis. (B)</p> Signup and view all the answers

    Which of the following statements accurately describes the consequences of a Type I error?

    <p>Rejecting a true null hypothesis. (C)</p> Signup and view all the answers

    What is the level of significance, denoted as α, generally set by the researcher?

    <p>0.05 (D)</p> Signup and view all the answers

    When conducting the Z test for proportion, what values indicate rejection of the null hypothesis?

    <p>Values beyond ± 1.96 (D)</p> Signup and view all the answers

    If a researcher fails to reject a false null hypothesis, which type of error is made?

    <p>Type II Error (A)</p> Signup and view all the answers

    What does the critical region represent in hypothesis testing?

    <p>Values indicating rejection of the null hypothesis. (B)</p> Signup and view all the answers

    How is the probability of making a Type I error expressed in hypothesis testing?

    <p>As $α$ (A)</p> Signup and view all the answers

    Study Notes

    Confidence Intervals

    • Confidence intervals provide a range of values likely to contain a population parameter. They offer more information than a point estimate, which is a single value.
    • Data variability, sample size, and confidence level influence interval width. A larger sample size and a higher confidence level lead to wider intervals.
    • Intervals are calculated using sample statistics and a critical value to reflect the uncertainty in estimating the true population parameter.
    • Using a critical value quantifies this uncertainty.

    Point Estimates

    • Point estimates are single values that estimate a population parameter (e.g., mean or proportion).
    • Example for a mean: the sample mean (x̄) acts as a point estimate for the population mean (μ).
    • Example for a proportion: the sample proportion (p̂) acts as a point estimate for the population proportion (p).
    • Point estimates are used in conjunction with confidence intervals to obtain more reliable estimates of population characteristics.

    Confidence Intervals - Population Mean (σ known)

    • Assumptions: population standard deviation (σ) is known, and the population is normally distributed. A large sample can be used if the population isn't normally distributed.
    • Formula: x̄ ± zα/2 (σ/√n), where:
      • x̄ is the sample mean
      • zα/2 is the critical z-value for the desired confidence level.
      • σ is the population standard deviation
      • n is the sample size
    • Margin of error: zα/2 (σ/√n) – the amount added and subtracted to the point estimate to create the interval.

    Confidence Intervals - Population Mean (σ unknown)

    • Assumptions: population standard deviation (σ) is unknown, and the population is normally distributed. Use a large sample if the population is not normally distributed.
    • Use the Student's t-distribution instead of the standard normal distribution. The t-distribution's shape varies with the degrees of freedom (df = n-1).
    • Formula: x̄ ± tα/2 (s/√n), where:
      • x̄ is the sample mean
      • tα/2 is the critical t-value for the desired confidence level and degrees of freedom (n-1)
      • s is the sample standard deviation; calculated as Σ(xi-x̄)2/ (n-1).
      • n is the sample size
    • Margin of error: tα/2 (s/√n) – this is the amount added and subtracted to the point estimate.

    Confidence Intervals - Population Proportion

    • Formula: p̂ ± zα/2 √(p̂(1-p̂)/n), where: - p̂ is the sample proportion - zα/2 is the critical z-value for the desired confidence level - n is the sample size
    • The sample proportion (p̂) is an important component of the estimate.
    • Margin of error: zα/2 √(p̂(1-p̂)/n)

    Determining Sample Size

    • To estimate the required sample size, the desired margin of error (e) and level of confidence (1 – α) must be provided.
    • If σ is known, formula is: n = (zα/2σ/e)2
    • If σ is unknown, use a pilot sample to estimate σ or use a conservative estimate for p.

    Student's t-Distribution

    • A family of symmetric, bell-shaped distributions, different from the standard normal distribution (z).
    • Its spread varies depending on the degrees of freedom (df = sample size(n) − 1).
      • As sample size (n) increases, the t-distribution approaches the standard normal distribution (z).
    • Used for estimating population means when the population standard deviation (σ) is unknown.
    • Critical values (tα/2) for the t-distribution are found in t-tables based on degrees of freedom and confidence levels.

    Degrees of Freedom

    • The number of independent observations minus the number of estimated parameters in the calculation.
    • For population mean calculations where the standard deviation (σ) is unknown, the degrees of freedom equals n – 1, where n is the sample size
    • Finding critical values (tα/2) in t-tables depends on the degrees of freedom (df).

    Approximations

    • When sample size (n) is ≥ 30, the t-distribution can be approximated with the standard normal distribution (z) when estimating population means.

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    Related Documents

    Estimating Single Population Parameters PDF
    Hypothesis Testing PDF

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