Hypothesis Testing in Business Statistics

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Questions and Answers

What is the null hypothesis for the fertilizer company's claim about the yield of wheat per hectare?

  • The mean yield is equal to 35 quintals. (correct)
  • The mean yield is greater than 35 quintals.
  • The mean yield is less than 35 quintals.
  • The mean yield is not equal to 35 quintals.

At a 1% significance level, what conclusion can be drawn if the sample mean salary of teachers is Birr 9,400?

  • The sample mean is significantly lower than the claimed average.
  • The union's claim can be rejected. (correct)
  • The union's claim can be accepted.
  • There is insufficient evidence to conclude anything.

What is the primary role of a null hypothesis in hypothesis testing?

  • To prove a hypothesis definitively true
  • To provide an alternative claim to the tested hypothesis
  • To serve as a default assumption that can be tested for rejection (correct)
  • To represent the researcher's desired outcome

Which statement correctly describes a Type I error?

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

In the hypothesis testing of the contractor's claim about idle construction workers, what was the sample mean found?

<p>84 minutes (B)</p> Signup and view all the answers

What is the population standard deviation used in the teacher salary case?

<p>Birr 3000 (A)</p> Signup and view all the answers

What is the significance level used to test the claim about the magazine's readers being college students?

<p>10% (B)</p> Signup and view all the answers

What does the alternative hypothesis represent in hypothesis testing?

<p>The potential change or effect that the researcher aims to demonstrate (A)</p> Signup and view all the answers

In the example analyzing shoppers' trip times, what was the sample mean shopping trip time observed?

<p>30 minutes (A)</p> Signup and view all the answers

In business statistics, why is hypothesis testing crucial?

<p>It provides a method for determining the likelihood of a hypothesis being valid (B)</p> Signup and view all the answers

When testing the claim made by the director about typing speed, what is the sample variance found?

<p>720 (C)</p> Signup and view all the answers

When setting up a hypothesis test, what must be identified first?

<p>The population parameter to be tested (D)</p> Signup and view all the answers

What sample size was used in the testing of the contractor's assumption about idle time?

<p>25 (C)</p> Signup and view all the answers

What is the null hypothesis (Ho) when testing the economist's claim about unemployment in Addis?

<p>P &gt; 0.35 (B)</p> Signup and view all the answers

For the fertilizer company's yield claim, what was the sample mean yield obtained?

<p>34 quintals (A)</p> Signup and view all the answers

Which of the following best describes a Type II error?

<p>Accepting a null hypothesis that is false (B)</p> Signup and view all the answers

What is the standard deviation of the sample taken for the supermarket shoppers' trip time?

<p>10 minutes (A)</p> Signup and view all the answers

In hypothesis testing, what is typically assumed at the beginning of the process?

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

What level of significance was used to test the tire company's claim?

<p>5% (B)</p> Signup and view all the answers

Using a significance level of 0.01, what is being tested regarding the primary breakfast beverage for Addis residents?

<p>If more than 60% prefer tea (C)</p> Signup and view all the answers

What is the conclusion if the test statistic falls beyond the critical region at a 1% level of significance for the teachers' union case?

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

What is true about the formulation of hypotheses in hypothesis testing?

<p>The hypotheses should be clearly defined and oppositional (B)</p> Signup and view all the answers

What is the sample size for the analysis of college students among the magazine's readers?

<p>200 readers (A)</p> Signup and view all the answers

Which statistical method is primarily being utilized in the examples presented?

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

What proportion of the sampled population in Addis claimed tea as their primary beverage according to the survey?

<p>325 out of 500 (A)</p> Signup and view all the answers

What is a key characteristic of a population parameter that must be understood before hypothesis testing?

<p>It is a specific measurable characteristic of the population (C)</p> Signup and view all the answers

What assumption can be made when analyzing the results of a hypothesis test?

<p>A hypothesis cannot be tested without sample data (D)</p> Signup and view all the answers

In hypothesis testing, what alternative hypothesis (Ha) is appropriate if we want to know if the unemployment rate exceeds 35%?

<p>Ha: P &gt; 0.35 (A)</p> Signup and view all the answers

In the context of hypothesis testing, a 2% level of significance means what about the probability of a Type I error?

<p>0.02 (A)</p> Signup and view all the answers

What aspect is being analyzed when querying whether more than 35% of Addis's labor force is unemployed?

<p>Hypothesis testing about a proportion (B)</p> Signup and view all the answers

What is the correct interpretation of a Type II error in hypothesis testing?

<p>Failing to identify a significant effect when one actually exists. (B)</p> Signup and view all the answers

In Example 2, what are the null and alternative hypotheses regarding mean productivity?

<p>H₀: μ1 = μ2; H₁: μ1 ≠ μ2 (A)</p> Signup and view all the answers

What is the null hypothesis for the hotel guest bill scenario?

<p>The mean guest bill is less than or equal to Birr 400. (D)</p> Signup and view all the answers

Which of the following hypotheses in Example 3 suggests employee dishonesty?

<p>H₁: The employees are not honest. (C)</p> Signup and view all the answers

In hypothesis testing, what does failing to reject the null hypothesis imply?

<p>There is insufficient evidence to support the alternative hypothesis. (A)</p> Signup and view all the answers

If a drugstore manager assumes employee honesty but shortages occur, what must be assumed if the opposite hypothesis is accepted?

<p>Some shortages are due to employee dishonesty. (C)</p> Signup and view all the answers

For the marketing campaign, what will failing to reject the null hypothesis imply?

<p>There may be a positive effect on sales, but it remains unproven. (A)</p> Signup and view all the answers

What does the null hypothesis represent in statistical testing?

<p>The hypothesis that there is no effect or difference. (B)</p> Signup and view all the answers

In defining a one-tailed test, which of the following statements is correct?

<p>It tests for an increase or decrease in one direction only. (B)</p> Signup and view all the answers

In Example 1, what does the null hypothesis involve regarding tire life claims?

<p>The mean life of tires is 35,000 miles or less. (B)</p> Signup and view all the answers

Flashcards

What is a hypothesis?

A statement about a population parameter (like its mean, proportion, or variance) that we want to test.

What is the null hypothesis (H0)?

The default assumption or claim about a population parameter that we test for possible rejection. We assume it TRUE until proven false.

What is the alternative hypothesis (H1)?

The opposite of the null hypothesis. It represents what the researcher is trying to prove.

What is the purpose of hypothesis testing?

The goal of hypothesis testing is to determine whether the sample data supports the null hypothesis or the alternative hypothesis.

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What are the steps in hypothesis testing?

  1. State the hypotheses (H0 and H1). 2. Collect and analyze data. 3. Calculate a test statistic. 4. Determine the p-value. 5. Make a decision (reject or fail to reject H0).
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What is a Type I error?

Rejecting a true null hypothesis. Think of it as a false alarm.

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What is a Type II error?

Failing to reject a false null hypothesis. Think of it as missing a real difference.

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Give an example of a Type I error.

For example, concluding a drug is effective (reject H0), when it is not, is a Type I error.

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Give an example of a Type II error.

For example, concluding a drug is ineffective (failing to reject H0), when it is actually effective is a Type II error.

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Type II Error

Failing to reject the null hypothesis when it is actually false; missing a real effect, like concluding a marketing campaign didn't increase sales when it actually did.

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Hypothesis

A statement about a population parameter that we want to test.

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Alternative Hypothesis (H₁)

A statement that contradicts the null hypothesis; it's what we are trying to prove.

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Null Hypothesis (H₀)

A statement that there is no effect or difference in the population; we aim to disprove it.

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Two-Tailed Test

A hypothesis test that is used to determine whether there is a significant difference between two population means, where the alternative hypothesis is that the means are not equal.

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One-Tailed Test

A hypothesis test where the alternative hypothesis specifies a direction of the effect. It tests whether the population mean is greater than or less than a specific value.

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Example 1: Null Hypothesis

The mean guest bill for a weekend is Birr 400 or less.

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Example 1: Alternative Hypothesis

The mean guest bill for a weekend is greater than Birr 400.

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Example 2: Null Hypothesis

There is no difference in the mean productivity of workers trained in the two programs.

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Example 2: Alternative Hypothesis

There is a difference in the mean productivity of workers trained in the two programs.

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t-test for a population mean

A statistical hypothesis testing procedure used to determine whether there is enough evidence to reject a null hypothesis about the population mean when the population standard deviation is unknown and the sample size is small.

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Null Hypothesis (H0)

The hypothesis that there is no difference between the population mean and the hypothesized value.

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Alternative Hypothesis (H1)

The hypothesis that there is a difference between the population mean and the hypothesized value.

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Significance Level (alpha)

The level of significance, usually set at 0.05, represents the maximum probability of committing a Type I error.

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t-statistic

A statistic used to test the null hypothesis in a t-test. It represents the difference between the sample mean and the hypothesized population mean, divided by the estimated standard error of the mean.

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Critical Value

The critical value of the t-distribution, which depends on the degrees of freedom and the significance level, separates the rejection region from the non-rejection region.

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p-value

The area under the t-distribution curve that corresponds to the p-value. It represents the probability of observing a t-statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true.

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Statistical Inference

The process of using sample data to make inferences about a population. It involves testing hypotheses, estimating parameters, and constructing confidence intervals.

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Hypothesis Testing

A statistical test used to determine whether there is enough evidence to reject the null hypothesis, which states that there is no significant difference between the population parameter and the hypothesized value.

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Significance Level (α)

The predetermined threshold for rejecting the null hypothesis. It represents the maximum allowable risk of rejecting the null hypothesis when it is actually true.

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Decision

A decision based on the p-value and the significance level. If the p-value is less than the significance level, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

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Hypothesis Testing for Population Proportion

A statistical test used to determine whether there is enough evidence to reject the null hypothesis, which states that the population proportion is equal to a hypothesized value.

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Hypothesis Testing for Population Mean

A statistical test used to determine whether there is enough evidence to reject the null hypothesis, which states that the population mean is equal to a hypothesized value.

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Study Notes

Hypothesis Testing

  • Hypothesis testing is a statistical method used to evaluate a claim about a population parameter.
  • A hypothesis is an assumption about a population parameter (e.g., population mean, proportion, or variance).
  • The parameter must be identified prior to analysis.
  • In business statistics, a hypothesis is a formal statement about a population parameter which needs to be tested.

What is a Hypothesis?

  • A hypothesis is an assumption about a population parameter.
  • A parameter is a characteristic of a population (e.g., mean, proportion, variance).
  • The parameter must be identified before conducting an analysis.
  • In business statistics, a hypothesis is a formal statement about a population parameter that is tested.

Hypothesis Testing Steps

  • State the Hypothesis: Define the null and alternative hypotheses.
    • Null Hypothesis (H₀): This is the default assumption, often that there is no effect or difference.
    • Alternative Hypothesis (H₁): This is the claim to be tested.
  • Set the Significance Level: Determine the acceptable risk of rejecting a true null hypothesis (e.g., 5% or 1%).
  • Collect Data: Gather a relevant sample of data to test the hypothesis.
  • Calculate the Test Statistic: Use an appropriate test statistic (e.g., t-test, z-test, chi-square) based on the collected data and hypotheses.
  • Make a Decision: Compare the calculated test statistic to a critical value, or calculate a p-value.
    • Reject H₀ if the test statistic falls within the rejection region (or if the p-value is less than the significance level).
    • Otherwise, fail to reject H₀.

Type I and Type II Errors

  • Type I Error (False Positive): Rejecting a true null hypothesis.
  • Type II Error (False Negative): Failing to reject a false null hypothesis.

One-Tailed and Two-Tailed Tests

  • Two-Tailed Test: The hypothesis about the population parameter is rejected when the sample statistic falls into either tail of the distribution.
  • One-Tailed Test: The hypothesis about the population parameter is rejected when the sample statistic falls into one side (tail) of the distribution.
    • A right-tailed test has rejection in the right tail.
    • A left-tailed test has rejection in the left tail.

Hypothesis Testing: Population Proportion (p)

  • Similar to testing population means, testing population proportions also follows three types of hypotheses:
    • H₀: p = y
    • H₀: p ≥ y
    • H₀: p ≤ y and their corresponding alternatives Ha: p ≠ y, Ha: p < y, and Ha: p > y.

Examples of Formulating Hypotheses

  • Example 1: The mean guest bill for a weekend is Birr 400 or less.
    • H₀: μ ≤ 400
    • H₁: μ > 400
  • Example 2: Production workers trained in two different programs—no difference in mean productivity.
    • H₀: μ₁ = μ₂
    • H₁: μ₁ ≠ μ₂
  • Example 3: Employees are honest; no shortages due to dishonesty.
    • H₀: Employees are honest (or no shortages due to dishonesty)
    • H₁: Employees are dishonest (or shortages are due to dishonesty).

Hypothesis Testing about μ: Population Normal, σ known, n small.

  • (Examples and solutions for testing claims about population means.)

Hypothesis Testing about μ: Population Normal, σ unknown, n small.

  • (Examples and solutions for testing claims about population means.)

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