Confidence Intervals and Hypothesis Testing
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Questions and Answers

What typically happens to the confidence interval when the confidence level increases?

  • It becomes narrower.
  • It becomes wider. (correct)
  • It disappears.
  • It remains the same.

What symbol is commonly used to represent the population standard deviation when it is known?

  • σ (correct)
  • μ
  • s
  • x̄

In statistics, what Greek letter is often used to denote the population mean?

  • μ (correct)
  • σ
  • x̄
  • s

What is the primary goal when constructing a confidence interval?

<p>To estimate a population parameter. (A)</p> Signup and view all the answers

When is it most appropriate to calculate a confidence interval for a mean?

<p>When you want to estimate the average value of a population. (B)</p> Signup and view all the answers

What is the primary purpose of hypothesis testing?

<p>To determine if there is enough evidence to reject a null hypothesis (B)</p> Signup and view all the answers

What does the null hypothesis typically state?

<p>There is no effect or no difference (D)</p> Signup and view all the answers

What is the significance level (alpha, denoted as α) used for?

<p>To set the standard for rejecting the null hypothesis (B)</p> Signup and view all the answers

What does a P-value measure?

<p>The probability of observing results as extreme as, or more extreme than, the observed results if the null hypothesis is true (A)</p> Signup and view all the answers

In hypothesis testing, If the P-value is less than or equal to the significance level, what decision should be made?

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

What does 'evidence to support the claim' aim to do?

<p>Strengthen the claim's validity (D)</p> Signup and view all the answers

What is a plan designed to do?

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

What are 'detailed solutions' intended to serve as?

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

What are students expected to use to serve the population?

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

What should students provide to support a claim in college?

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

What is the purpose of collecting definitive information?

<p>To serve the population (D)</p> Signup and view all the answers

What is the role of evidence to support a claim?

<p>To validate information (B)</p> Signup and view all the answers

What does 'detailed solutions' provide?

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

Which of the following is needed to find the confidence interval for the mean?

<p>All of the above (D)</p> Signup and view all the answers

What does α represent in the context of confidence intervals?

<p>The significance level (B)</p> Signup and view all the answers

What mathematical operation is represented by the symbol '/'?

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

What happens to the width of the confidence interval as the sample size increases?

<p>It becomes narrower (C)</p> Signup and view all the answers

In the expression $_X_ − \frac{659}{\sqrt{36}}$, what does $\sqrt{36}$ represent?

<p>The square root of 36 (C)</p> Signup and view all the answers

When is the t-distribution typically used instead of the standard normal distribution?

<p>When the population standard deviation is unknown (D)</p> Signup and view all the answers

What is the value of $\sqrt{36}$?

<p>6 (B)</p> Signup and view all the answers

In the equation $\frac{700}{5} = 2.28$, what operation does the fraction bar represent?

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

What value of z corresponds to a 95% confidence level?

<p>1.96 (A)</p> Signup and view all the answers

What is the result of the operation $\frac{700}{5}$?

<p>140 (A)</p> Signup and view all the answers

What does 'σ' usually represent in statistical formulas?

<p>Population standard deviation (C)</p> Signup and view all the answers

If you increase the confidence level, what happens to the width of the confidence interval?

<p>The width increases (D)</p> Signup and view all the answers

In the context of hypothesis testing and confidence intervals, what is 'power'?

<p>The probability of correctly rejecting a false null hypothesis (C)</p> Signup and view all the answers

What does a small P-value typically indicate?

<p>Strong evidence against the null hypothesis (D)</p> Signup and view all the answers

Which of the following is a common significance level (alpha)?

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

What is a P-value used for in hypothesis testing?

<p>To decide whether the null hypothesis should be rejected (B)</p> Signup and view all the answers

In the context of hypothesis testing, what is the null hypothesis?

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

What does the significance level (alpha) represent in hypothesis testing?

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

If a P-value is greater than the significance level (alpha), what is the typical conclusion?

<p>Fail to reject the null hypothesis (C)</p> Signup and view all the answers

What is the alternative hypothesis?

<p>The hypothesis that contradicts the null hypothesis (C)</p> Signup and view all the answers

What information is needed to perform a hypothesis test?

<p>Sample data and a hypothesis (A)</p> Signup and view all the answers

What does rejecting the null hypothesis imply?

<p>There is sufficient evidence to support the alternative hypothesis (D)</p> Signup and view all the answers

Which of the following directly influences the P-value in a hypothesis test?

<p>The sample data (D)</p> Signup and view all the answers

Flashcards

Confidence Interval for the Mean

Range of estimates for an unknown population mean.

Confidence Level

Desired degree of assurance that the true population parameter lies within the calculated interval.

Confidence Interval Width

As confidence increases, the interval widens; as confidence decreases, the interval narrows.

When to use t-distribution?

Use a t-distribution instead of a z-distribution when the population standard deviation (σ) is unknown and the sample size is small.

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Consequence of incorrect distribution?

Using incorrect distribution (e.g., z instead of t when σ unknown) leads to underestimation of uncertainty.

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Hypothesis

A claim or statement about a population parameter that we aim to test.

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Hypothesis Testing

A statistical method used to evaluate a claim about a population based on sample data.

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Null Hypothesis

The assumption that there is no effect or no difference; the starting point for testing.

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P-value

The probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is correct.

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Alpha (α)

The level of significance used in hypothesis testing; the probability of rejecting the null hypothesis when it is actually true.

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What is a T-Test?

A statistical test to determine if there is a significant difference between the means of two groups

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T-Test Formula

The formula used to calculate the t-statistic. It involves the difference between the sample means, and the standard error.

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Interpret Results

The final step to determine if the results are statistically meaningful.

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Confidence Interval (σ known)

The confidence interval for the population mean (μ) when the population standard deviation (σ) is known.

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Confidence Interval

The value that defines the interval in which the true population parameter is estimated to lie.

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α (Alpha)

The probability of rejecting the null hypothesis when it is actually true (Type I error).

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zα/2

Given a fixed α, it stands for the z-score that cuts off the upper α/2 of a standard normal distribution.

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Minimum Sample Size

The minimum number of samples needed to achieve a desired level of precision and confidence.

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1 - α

The confidence level is 1 - α, where α is the significance level.

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z0.005 value

z-score corresponding to a α of 0.005 is 2.57 or 2.58.

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Support the Claim

Presenting reasons and evidence to support a particular point of view or assertion.

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College Evidence

Evidence, facts, or data used to back up a statement or belief.

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Detailed Steps

Detailed, step-by-step explanations of a process or procedure.

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Solutions

Possible answers or methods to tackle a problem.

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Serve as Models

Acting as examples for others to follow or imitate.

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Theoretical

Connected with a theory.

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Students Use

Used in a classroom setting.

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Null Hypothesis (H0)

A statement that there is no effect or no difference.

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Wind Speed in a city

Average wind speed in a city: 8.2 miles per hour.

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Decision Rule (P-value)

Reject the null hypothesis if the P-value is less than or equal to the set significance level.

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What is the value of the statistic alpha?

α = 0.05

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Study Notes

Confidence Intervals for the Mean

  • Confidence intervals for the mean can be found when σ is known.
  • When σ is known, a z value is used.
  • The confidence interval for the mean can be found when σ is unknown.
  • When σ is unknown, the t distribution table is instead used.
  • 95% Confidence (α = 0.05) is most often used.
  • 99% Confidence (α = 0.01) is also often used, and uses a z value of 2.58.
  • 90% Confidence (α = 0.10) is less often used.

Hypothesis Testing

  • Hypothesis testing includes stating claims, finding the P-value, and using either the traditional or P-value method.
  • The P-value method involves five steps: stating the hypotheses, selecting α, computing the test statistic, finding the P-value, making the decision, and summarizing the results.

Example Hypothesis Test Scenario: College Costs

  • A researcher wants to test if the average cost of college is greater than $5700.
  • The population mean is believed to be $5900.
  • The selected α = 0.05.
  • Public colleges were surveyed to collect data to support the claim from the College Board.
  • Hundreds of examples with detailed solutions can be found in the College Board report.
  • Evidence will either support or reject the claim.
  • Summarizing the results is the final step.

Hypothesis Testing Calculation Example

  • To compute the test statistic, the formula (X - μ) / (σ / √n) = z is used.
  • Using values from a four-year college tuition and fees study: (5900 - 5700)/(659/√36) = 2.28
  • The P-value (0.0113) is greater than α = 0.05 in the example scenario.
  • The null hypothesis is rejected since the P-value is less than α.

Example Hypothesis Test Scenario: Wind Speed

  • A claim is made that the average wind speed in a certain city is 8 miles per hour.
  • The study shows the average wind speed to be 8.2 miles per hour.
  • α = 0.05

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Description

Explore confidence intervals for the mean when σ is known or unknown, using z values or t-distribution. Learn about hypothesis testing, including stating claims and finding P-values. Review a hypothesis test scenario related to college costs.

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