BUSS 200 QUIZ 2

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

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

Flashcards

Point Estimate

A single value used to estimate an unknown population parameter.

Confidence Interval

A range of values that is likely to contain the unknown population parameter.

Confidence Level

The percentage of confidence that a confidence interval will contain the true population parameter.

Sampling Error

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

Signup and view all the flashcards

Point Estimate for Mean

Sample mean (x̄) is the point estimate for the population mean (μ).

Signup and view all the flashcards

Point Estimate for Proportion

Sample proportion (p̂) is the point estimate for the population proportion (p).

Signup and view all the flashcards

Standard Error

The standard deviation of a sampling distribution.

Signup and view all the flashcards

Margin of Error

The amount added and subtracted to the point estimate to form the confidence interval.

Signup and view all the flashcards

Confidence Interval for Population Mean (σ known)

Calculating a range to estimate the true population mean when the population standard deviation is known.

Signup and view all the flashcards

Confidence Interval for Population Mean (σ unknown)

Calculating a range to estimate the true population mean when the population standard deviation is unknown.

Signup and view all the flashcards

Standard Normal Distribution

A normal distribution with a mean of 0 and a standard deviation of 1.

Signup and view all the flashcards

Student's t Distribution

A family of probability distributions used for estimating population means when the population standard deviation is unknown.

Signup and view all the flashcards

Degrees of Freedom (df)

The number of independent pieces of information used to estimate a parameter in a sample.

Signup and view all the flashcards

Critical Value

The value from the z or t distribution that determines the boundaries of a confidence interval.

Signup and view all the flashcards

Confidence Interval for Population Proportion

A range that estimates the true proportion of a certain characteristic in a population.

Signup and view all the flashcards

Required Sample Size

The minimum number of observations needed to achieve a desired level of precision for the estimate of a parameter.

Signup and view all the flashcards

Sample Proportion

The proportion of a specific characteristic in a sample.

Signup and view all the flashcards

Pilot Sample

A small preliminary sample used to estimate the population standard deviation, or proportion.

Signup and view all the flashcards

Hypothesis Test

A statistical procedure used to determine whether there is enough evidence to reject a null hypothesis about a population parameter.

Signup and view all the flashcards

Null Hypothesis (H0)

A statement about a population parameter that is assumed to be true.

Signup and view all the flashcards

Alternative Hypothesis (HA)

A statement that contradicts the null hypothesis.

Signup and view all the flashcards

Significance Level (α)

The probability of rejecting a true null hypothesis.

Signup and view all the flashcards

Type I Error

Rejecting a true null hypothesis.

Signup and view all the flashcards

Type II Error

Failing to reject a false null hypothesis.

Signup and view all the flashcards

P-value

The probability of observing a sample statistic as extreme as the one obtained, assuming the null hypothesis is true.

Signup and view all the flashcards

Reject H0

There is enough evidence to reject the null hypothesis.

Signup and view all the flashcards

What is a statistical hypothesis?

It's a claim or assumption about a population parameter. This parameter could be the population mean or proportion.

Signup and view all the flashcards

What's the purpose of hypothesis testing?

To determine if there's enough evidence to reject the null hypothesis and support the alternative hypothesis.

Signup and view all the flashcards

What are the steps involved in hypothesis testing?

Formulate null and alternative hypotheses, calculate test statistics, determine the p-value or critical value, make a decision and interpret the results.

Signup and view all the flashcards

Why is controlling error probability important?

It helps minimize the risk of making incorrect conclusions. This is essential for drawing meaningful interpretations from your data.

Signup and view all the flashcards

Level of Significance (α)

The probability of rejecting the null hypothesis when it is actually true. It defines the rejection region of the sampling distribution.

Signup and view all the flashcards

Rejection Region

The area under the sampling distribution where the test statistic falls if the null hypothesis is rejected.

Signup and view all the flashcards

Lower Tail Test

A hypothesis test where the rejection region is entirely in the left tail of the sampling distribution. The alternative hypothesis is that the population parameter is less than the null value.

Signup and view all the flashcards

Upper Tail Test

A hypothesis test where the rejection region is entirely in the right tail of the sampling distribution. The alternative hypothesis is that the population parameter is greater than the null value.

Signup and view all the flashcards

Two-Tailed Test

A hypothesis test where the rejection region is divided into two tails of the sampling distribution. The alternative hypothesis is that the population parameter is not equal to the null value.

Signup and view all the flashcards

Test Statistic

A value calculated from the sample data that is used to test the null hypothesis. This value tells you how far your sample data deviates from the null assumption.

Signup and view all the flashcards

z-statistic

A test statistic used when the population standard deviation is known and the sample size is large.

Signup and view all the flashcards

t-statistic

A test statistic used when the population standard deviation is unknown and the sample size is small.

Signup and view all the flashcards

How to Determine the Rejection Region

  1. Convert the sample statistic to a test statistic (z or t). 2. Find the critical value(s) from a table or computer program for the chosen level of significance. 3. Determine if the test statistic falls in the rejection region.
Signup and view all the flashcards

Not Rejecting H0

The conclusion we reach when the calculated test statistic does not fall within the rejection region, indicating that the sample data does not provide enough evidence against the null hypothesis.

Signup and view all the flashcards

What is the Relationship Between Level of Significance and the Rejection Region?

A higher level of significance (α) indicates a larger rejection region. This means you are more likely to reject the null hypothesis, potentially leading to a Type I error (rejecting a true null hypothesis).

Signup and view all the flashcards

Why is it Important to Choose a Level of Significance Before Conducting the Test?

Choosing the level of significance beforehand ensures that the decision to reject or not reject the null hypothesis is not influenced by the results. It avoids bias in the analysis.

Signup and view all the flashcards

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 (xÌ„) acts as a point estimate for the population mean (μ).
  • Example for a proportion: the sample proportion (pÌ‚) 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: xÌ„ ± zα/2 (σ/√n), where:
    • xÌ„ 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: xÌ„ ± tα/2 (s/√n), where:
    • xÌ„ 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-xÌ„)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: pÌ‚ ± zα/2 √(pÌ‚(1-pÌ‚)/n), where: - pÌ‚ is the sample proportion - zα/2 is the critical z-value for the desired confidence level - n is the sample size
  • The sample proportion (pÌ‚) is an important component of the estimate.
  • Margin of error: zα/2 √(pÌ‚(1-pÌ‚)/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.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team

Related Documents

Estimating Single Population Parameters PDF
Hypothesis Testing PDF

More Like This

Use Quizgecko on...
Browser
Open
Browser
Browser