Hypothesis Testing: Notations and Definitions

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Questions and Answers

What is the primary goal of hypothesis testing in statistics?

  • To determine the accuracy of the sample data
  • To determine the value of a population parameter based on some preconceived ideas (correct)
  • To calculate the probability of a null hypothesis
  • To select the most representative sample for the population

What is the assumption about a population parameter that may or may not be true?

  • Null Hypothesis
  • Alternative Hypothesis
  • Hypothesis (correct)
  • Hypothesis Testing

In a criminal trial, what is the hypothesis that the jury initially assumes?

  • The jury is undecided
  • The defendant is innocent (correct)
  • The defendant is guilty
  • The evidence is insufficient

What is the term used to describe the testing of a hypothesis to determine whether the data obtained are indicative of the general or future behavior of the phenomenon being studied?

<p>Hypothesis Testing (C)</p> Signup and view all the answers

What is the alternative hypothesis in a criminal trial?

<p>The defendant is guilty (B)</p> Signup and view all the answers

What is the purpose of looking at the context of a criminal trial in understanding hypothesis testing?

<p>To understand the ideas of hypotheses testing (D)</p> Signup and view all the answers

What is the outcome if the jury rejects the null hypothesis in a trial?

<p>The defendant is guilty beyond reasonable doubt. (A)</p> Signup and view all the answers

What does acquitting the defendant mean in a trial?

<p>The evidence is insufficient to prove guilt. (B)</p> Signup and view all the answers

What is the purpose of stating a null hypothesis in hypothesis testing?

<p>To formulate a statement with the hope of it being rejected. (B)</p> Signup and view all the answers

What is the first step in hypothesis testing?

<p>Stating the null and alternative hypotheses. (D)</p> Signup and view all the answers

What is the definition of β in hypothesis testing?

<p>Power of a test. (A)</p> Signup and view all the answers

What is the goal of hypothesis testing?

<p>To make decisions about the population based on the sample characteristics. (C)</p> Signup and view all the answers

What is the purpose of selecting a level of significance in a hypothesis test?

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

What is the term for the set of values of the test statistic that result in the rejection of the null hypothesis?

<p>Critical region (C)</p> Signup and view all the answers

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

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

What is the term for a test hypothesis where the alternative hypothesis is one-sided?

<p>One-tailed test (D)</p> Signup and view all the answers

What is the purpose of computing the value of the test statistic in a hypothesis test?

<p>To evaluate the sample data using the test statistic formula (A)</p> Signup and view all the answers

What does it mean if the value of the test statistic falls within the critical region?

<p>The null hypothesis is rejected in favor of the alternative hypothesis (A)</p> Signup and view all the answers

Which test is used when the data are normally distributed, the sample size is greater than or equal to 30, and the population variance or population standard deviation is known?

<p>z-test (B)</p> Signup and view all the answers

What is the reason for using the sample variance in the t-test?

<p>The population variance is unknown (C)</p> Signup and view all the answers

A survey was conducted among 25 employees, and the sample mean is 24000. If the population standard deviation is unknown, which test should be used to determine if there is a significant difference between the population mean and sample mean?

<p>t-test (D)</p> Signup and view all the answers

A sample of 8 measurements was randomly selected from a normally distributed population. Which test should be used to test for significant difference between the sample mean and the population mean?

<p>t-test (B)</p> Signup and view all the answers

Which of the following conditions is not a requirement for using the z-test?

<p>The population variance is unknown (B)</p> Signup and view all the answers

A factory owner claims that the average capacity of their bottled juice is 250 ml and a standard deviation of 5 ml. If a sample of 30 measurements is taken, which test should be used to test for significant difference between the sample mean and the population mean?

<p>z-test (D)</p> Signup and view all the answers

Flashcards

Hypothesis Testing Goal

To evaluate a preconceived idea about a population parameter.

What is a Hypothesis?

An assumption about a population parameter that may or may not be true.

Initial assumption in court

The defendant is innocent until proven guilty.

Hypothesis Testing Defined

Assessing a hypothesis to see if data reflects the future behavior of a phenomenon.

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Alternative hypothesis in court

The defendant is guilty.

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Rejecting the null hypothesis

The jury finds the defendant guilty beyond a reasonable doubt.

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Acquitting the defendant means?

Lack of sufficient evidence to prove guilt.

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Purpose of stating a null hypothesis

To create a statement hoping it will be proven wrong.

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First step in hypothesis testing

Stating the null and alternative hypotheses.

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What is β?

Power of a test.

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Goal of Hypothesis Testing

Making decisions about the population based on sample data.

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Selecting a level of significance

To define the probability of incorrectly rejecting a true null hypothesis.

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Critical region

The set of test statistic values leading to rejection of the null hypothesis.

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Formulating the decision rule

To decide when to reject or not reject the null hypothesis.

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One-tailed test

Alternative hypothesis focuses on one direction.

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Computing the test statistic

To assess sample data using the test statistic formula.

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Test statistic falls in critical region

The null hypothesis is rejected in favor of the alternative hypothesis.

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Conditions for z-test

Data is normally distributed, sample size >= 30, population variance known.

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Why use sample variance in t-test?

Population variance is unknown.

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Test without population SD

t-test

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Small normally distributed sample test

t-test

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Not a z-test requirment

The population variance is unknown

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Bottled juice capacity test

z-test

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Study Notes

Hypothesis Testing

  • Hypothesis: an assumption or conjecture about a population parameter, which may or may not be true
  • Examples of hypotheses:
    • The average of the final grades of Grade 11 Liceo Male Students is higher than those of the Female Students
    • Coca-Cola contains 80% sugar
    • A radio station selects the music it plays based on the assumption that the average age of its listening audience is 30 years old

Notations and Definitions

  • H0: Null Hypothesis
  • Ha: Alternative Hypothesis
  • α: Significance Level (e.g. 5%, 1%)
  • β: Power of a Test, Probability of a Type II error
  • N: Population Size
  • n: Sample Size
  • μ: Population Mean
  • σ: Population Standard Deviation
  • σ2: Population Variance
  • XÌ„: Sample Mean
  • s: Sample Standard Deviation
  • s2: Sample Variance
  • z: z-value
  • t: t-value
  • c: Critical Value

Steps in Hypothesis Testing

  • STEP 1: State the null and alternative hypotheses
  • STEP 2: Select a Level of Significance (α)
  • STEP 3: Select the Test Statistic
  • STEP 4: Formulate the decision rule
  • STEP 5: Compute the value of the test statistic
  • STEP 6: Make a decision (reject or do not reject the null hypothesis)

Types of Tests

  • One-tailed test: when the alternative hypothesis is one-sided
  • Two-tailed test: when the alternative hypothesis is two-sided
  • z-test: used when data are normally distributed, sample size ≥ 30, and population variance is known
  • t-test: used when z-test cannot be used (e.g. sample size < 30, population variance unknown)

Examples of Hypothesis Testing

  • Example 1: Testing the average income of customer service representatives (use t-test)
  • Example 2: Testing the sample mean and population mean (use t-test)
  • Example 3: Testing the average capacity of bottled juice (use t-test)

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