Alta Chapter 9 - Hypothesis Testing

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Questions and Answers

What is the formula to find the degrees of freedom?

n - 1

What are the critical values for a two-tailed test with 15 degrees of freedom at a significance level of 0.005?

  • 2.160 and -2.160
  • 3.12
  • 2.947 and -2.947 (correct)
  • 1.697 and -1.697

What is the significance level used by Vae for her hypothesis test?

2.5%

The test statistic for Isabella's hypothesis test is 1.489.

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

In Isabella's test, what conclusion can be drawn based on the test statistic and critical values?

<p>Fail to reject H0 (B)</p> Signup and view all the answers

What is the calculated p-value for a right-tailed test with t = 3.12 and 60 degrees of freedom?

<p>less than 0.005</p> Signup and view all the answers

Should Rebecca reject or fail to reject the null hypothesis based on her sample data?

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

What is the sample mean market value of houses found by Rebecca?

<p>$259,860</p> Signup and view all the answers

Which of the following results in a null hypothesis p=0.61 and alternative hypothesis p>0.61?

<p>The null hypothesis corresponds to the claim in the study. (A), The alternative hypothesis corresponds to what the researcher is trying to show. (B), The researcher believes more than 61% of students study less than 5 hours per week. (C), A study claims at most 61% of students study less than 5 hours per week. (D)</p> Signup and view all the answers

What is the test statistic (z-score) of Olivia's one-mean hypothesis test?

<p>3.87</p> Signup and view all the answers

What is the test statistic (z-score) of Jolyn's hypothesis test?

<p>-0.95</p> Signup and view all the answers

What is the test statistic (z-score) of the bowler's one-mean hypothesis test?

<p>3.74</p> Signup and view all the answers

What is the test statistic (z-score) of the baker's one-mean hypothesis test?

<p>12.65</p> Signup and view all the answers

What is the test statistic (z-score) of William's one-mean hypothesis test?

<p>5.30</p> Signup and view all the answers

What is the test statistic (z-score) for Floretta's hypothesis test?

<p>Not provided</p> Signup and view all the answers

What is the test statistic for the knee surgery hypothesis test?

<p>-2.90</p> Signup and view all the answers

What is the test statistic for the unemployment rate hypothesis test?

<p>3.06</p> Signup and view all the answers

What are the null and alternative hypotheses for Kylie's seed germination test?

<p>H0:p=0.93; Ha:p&lt;0.93</p> Signup and view all the answers

A t-test is appropriate if the sample size is less than 30.

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

What conclusion did Olivia reach regarding the percentage of tips received by waitstaff?

<p>Do not reject the null hypothesis.</p> Signup and view all the answers

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Study Notes

Hypothesis Testing Overview

  • Null Hypothesis (H0): Represents a statement of no effect or no difference; often reflects a status quo.
  • Alternative Hypothesis (Ha): Represents what the researcher aims to prove; suggests a difference or effect.
  • Significance Level (α): Defines the probability of rejecting the null hypothesis when it is true, often set at 0.05 or 0.10.

Hypothesis Testing Procedure

  • Identify the null and alternative hypotheses.
  • Determine the significance level (α) for the test.
  • Collect data to compute the test statistic.
  • Compare the test statistic against a critical value to decide whether to reject H0.

Test Statistics

  • Z-Score Calculation:

    • Used when population standard deviation is known.
    • Formula: ( z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} )
    • Example: Olivia's test yields a z-score of 3.87 when examining golf drive distances.
  • Sample Mean and Standard Deviation:

    • Critical for calculating the test statistic.
    • Understanding sample sizes and their impacts on the results (larger samples lead to more reliable estimates).

Common Scenarios

  • Different Claims:
    • A golfer (Olivia) hypothesizes her drive distance is greater than an established average; specific statistical calculations were completed for her case using standard deviation and sample calculations.
  • Baker's Claim:
    • William's hypothesis about bread height's average was tested; obtained a z-score revealing whether to accept or reject his claims based on a threshold.

Example Outcomes

  • Successful and Unsuccessful Tests:
    • Results may lead to rejecting or failing to reject H0 based on whether the calculated test statistics fell within the rejection region defined by the critical values.
    • Example of a failed rejection in the context of waitstaff tips indicating that average tips do not differ significantly from established norms.

Special Cases for Proportions

  • Situations may arise requiring hypothesis testing for proportions, analyzed through different z-scores.
  • Formulation includes determining sample proportions, successful events, and failure rates.

Selecting the Right Test

  • T-Tests vs. Z-Tests:
    • T-tests are appropriate when the population standard deviation is unknown and the sample size is small.
    • Require knowledge of the sample mean, sample size, and sample standard deviation.

Two-Tailed vs. One-Tailed Tests

  • Understanding when to apply left-tail, right-tail, or two-tailed tests based on the nature of the hypotheses.
  • Example from a linguist’s study about study time for languages showcases the importance of tail direction.

Conclusions from Testing

  • Deciding whether to reject or accept the null hypothesis based on calculated p-values and corresponding significance levels plays a critical role in hypothesis testing outcomes.

Key Calculations

  • Results include detailed calculations for various scenarios such as:
    • z-score examples for Olivia (3.87), baker (some higher z-scores) leading to evidence rejection or acceptance of claims.
    • Various calculated test statistics rounded to two decimal places for clarity and communication of results.

Summary

  • Hypothesis testing is a systematic approach to validate claims using statistical evidence, critical for decision-making in research and practical applications across various fields.### Hypothesis Testing Overview
  • Null hypothesis (H0) posits that the population mean is a specific value (e.g., $250,000 for Rebecca).
  • Alternative hypothesis (Ha) suggests that the population mean is greater than that specific value.

Conclusion for Card 70

  • Do not reject null hypothesis due to calculated test statistic and p-value considerations.
  • Sample mean salary from 61 employees is $56,500 with a standard deviation of $3,750.
  • Test statistic calculated as 3.12 with 60 degrees of freedom signifies a right-tailed test.
  • The p-value associated with t = 3.12 is less than 0.005, indicating strong evidence against null hypothesis if true.

Conclusion for Card 74

  • Rebecca tests whether the mean market value of houses is greater than $250,000.
  • Sample of 35 houses gives a mean market value of $259,860 with a standard deviation of $24,922.
  • t-statistic calculated as approximately 2.34 with 34 degrees of freedom.
  • At a significance level (α) of 0.05, Rebecca should assess whether to reject or fail to reject the null hypothesis based on p-value interpretation, which should be obtained using the t-distribution table.

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