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Questions and Answers
What is the definition of the alternative hypothesis?
What is the definition of the alternative hypothesis?
Which of the following symbols can be used in the alternative hypothesis?
Which of the following symbols can be used in the alternative hypothesis?
What is the correct form of the null hypothesis when testing if the mean weight is at most 195 lb?
What is the correct form of the null hypothesis when testing if the mean weight is at most 195 lb?
What conclusion can be drawn if the null hypothesis is rejected?
What conclusion can be drawn if the null hypothesis is rejected?
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What does the critical region represent in hypothesis testing?
What does the critical region represent in hypothesis testing?
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In a hypothesis test, what does failing to reject H0 imply?
In a hypothesis test, what does failing to reject H0 imply?
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Which statement is true regarding the symbols used in hypotheses?
Which statement is true regarding the symbols used in hypotheses?
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When should the null and alternative hypotheses be stated?
When should the null and alternative hypotheses be stated?
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What type of hypothesis test is used when testing if a mean has either increased or decreased?
What type of hypothesis test is used when testing if a mean has either increased or decreased?
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In a right-tailed hypothesis test, where is the significance level α placed?
In a right-tailed hypothesis test, where is the significance level α placed?
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What are the null and alternative hypotheses in the given example about backpack weights?
What are the null and alternative hypotheses in the given example about backpack weights?
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Which of the following statements is true about a left-tailed test?
Which of the following statements is true about a left-tailed test?
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What does the symbol H0 signify in hypothesis testing?
What does the symbol H0 signify in hypothesis testing?
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If a hypothesis test is two-tailed, how is the α divided?
If a hypothesis test is two-tailed, how is the α divided?
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In hypothesis testing, what does the alternative hypothesis (H1) indicate?
In hypothesis testing, what does the alternative hypothesis (H1) indicate?
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Which hypothesis test would you use to test if the average weight of backpacks has decreased since a given mean?
Which hypothesis test would you use to test if the average weight of backpacks has decreased since a given mean?
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What is the value of the test statistic z in the hypothesis test?
What is the value of the test statistic z in the hypothesis test?
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What is the corresponding critical value z for α = 0.05?
What is the corresponding critical value z for α = 0.05?
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What conclusion is made when the test statistic does not fall in the critical region?
What conclusion is made when the test statistic does not fall in the critical region?
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What is the margin of error calculated using the z value and sample size?
What is the margin of error calculated using the z value and sample size?
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What does the P-value of 0.0643 indicate about the sample mean of 172.55 lb?
What does the P-value of 0.0643 indicate about the sample mean of 172.55 lb?
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What is the 90% confidence interval constructed from the sample mean and margin of error?
What is the 90% confidence interval constructed from the sample mean and margin of error?
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What is the population mean assumption in this hypothesis test?
What is the population mean assumption in this hypothesis test?
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What method is used to test the claim about the population mean in the example?
What method is used to test the claim about the population mean in the example?
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What does a P-value of 0.0014 indicate if the significance level is α = 0.01?
What does a P-value of 0.0014 indicate if the significance level is α = 0.01?
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Which condition is necessary when testing a claim about a mean with an unknown standard deviation?
Which condition is necessary when testing a claim about a mean with an unknown standard deviation?
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What is the formula to find degrees of freedom when testing a claim about a mean?
What is the formula to find degrees of freedom when testing a claim about a mean?
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What does the Student t distribution reflect about variability?
What does the Student t distribution reflect about variability?
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As the sample size increases, what happens to the Student t distribution?
As the sample size increases, what happens to the Student t distribution?
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What is the mean of the Student t distribution?
What is the mean of the Student t distribution?
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What is a requirement for the population when testing claims about a mean?
What is a requirement for the population when testing claims about a mean?
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What is a characteristic of the Student t distribution for different sample sizes?
What is a characteristic of the Student t distribution for different sample sizes?
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What is a Type I error in hypothesis testing?
What is a Type I error in hypothesis testing?
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What symbol represents the probability of a Type I error?
What symbol represents the probability of a Type I error?
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Which step involves stating the original claim in symbolic form?
Which step involves stating the original claim in symbolic form?
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What do you do if the P-value is greater than α?
What do you do if the P-value is greater than α?
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When should you select a smaller significance level α?
When should you select a smaller significance level α?
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In hypothesis testing, a Type II error is defined as:
In hypothesis testing, a Type II error is defined as:
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What is the first step in the P-value method?
What is the first step in the P-value method?
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What must be true when the original claim is false?
What must be true when the original claim is false?
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What type of test is used when the alternative hypothesis suggests the mean is less than a specified value?
What type of test is used when the alternative hypothesis suggests the mean is less than a specified value?
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Given the null hypothesis $H_0 : µ = 12$ ounces, what is the alternative hypothesis for this scenario?
Given the null hypothesis $H_0 : µ = 12$ ounces, what is the alternative hypothesis for this scenario?
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How is the P-value interpreted in hypothesis testing?
How is the P-value interpreted in hypothesis testing?
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What happens if the P-value is less than 0.05 in hypothesis testing?
What happens if the P-value is less than 0.05 in hypothesis testing?
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In the context of hypothesis testing, what is the critical region?
In the context of hypothesis testing, what is the critical region?
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For a left-tailed test, how is the P-value determined?
For a left-tailed test, how is the P-value determined?
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Which hypothesis test would involve a P-value calculated as twice the area in the tail beyond the test statistic?
Which hypothesis test would involve a P-value calculated as twice the area in the tail beyond the test statistic?
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What identifies a left-tailed test in the hypothesis testing framework?
What identifies a left-tailed test in the hypothesis testing framework?
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Study Notes
Hypothesis Testing Overview
- Hypothesis testing is a procedure used to test claims about a population parameter.
- Population parameters are often estimated, but hypotheses can be formed.
Hypothesis and Hypothesis Test
- A hypothesis is a claim or statement about a population property.
- A hypothesis test is used to test a claim about a population property.
Null Hypothesis (H₀)
- The null hypothesis is the hypothesis to be tested.
- It's often denoted as H₀.
- Typically states that a population parameter (e.g., proportion, mean, standard deviation) is equal to a specific value.
- It is assumed to be true initially.
- The conclusion is either to reject or fail to reject the null hypothesis.
Alternative Hypothesis (H₁)
- The alternative hypothesis (denoted by H₁, Hₐ, or Hₐ) represents the opposite of the null hypothesis.
- It suggests that the parameter's value differs from the null hypothesis in some way (≠, <, or >).
- The symbolic alternative hypothesis must contain one of these signs: ≠, <, >
Stating H₀ and H₁
- The null hypothesis is usually stated with an equal sign (=).
- The alternative hypothesis is stated without an equal sign (≠, <, or >).
- The true value of the population parameter should be included in a set specified by H₀ or in a set specified by H₁.
Critical Region and Significance Level (α)
- The critical region is the set of all test statistic values that lead to rejecting the null hypothesis.
- The significance level (α) is the probability of rejecting the null hypothesis when it's actually true (making the mistake of rejecting the null when it's true).
- Common values for α include 0.05, 0.01, and 0.10.
Test Statistic
- The test statistic is a value used to make decisions about the null hypothesis. It's derived by converting a sample statistic to a score assuming the null hypothesis is true.
- Different formulas for different scenarios (mean, proportion).
Critical Value
- A critical value separates the critical region from the values of the test statistic that do not lead to rejection.
- The critical values depend on the null hypothesis, the sampling distribution, and the significance level (α).
Types of Hypothesis Tests
- Two-tailed: critical region is in both tails of the distribution.
- Left-tailed: critical region is in the left tail.
- Right-tailed: critical region is in the right tail.
P-value
- The probability of obtaining a test statistic as extreme or more extreme than the one representing the sample data, assuming the null hypothesis is true.
- If the p-value is very small (e.g., less than 0.05), the null hypothesis is often rejected.
Decisions and Conclusions
- Do not use "Accept" for "Fail to Reject."
- Avoid multiple negatives in the conclusion.
Errors in Hypothesis Tests
- Type I Error: Rejecting the null hypothesis when it's true (probability denoted by α).
- Type II Error: Failing to reject the null hypothesis when it's false (probability denoted by β).
Confidence Method
- Constructing a confidence interval to test a claim about a population parameter.
- A claim is rejected if the population parameter value is not included in the confidence interval.
Testing a Claim About a Mean (σ Known)
- Methods for testing claims about a population mean when the population standard deviation is known.
- Requirements: a simple random sample, known population standard deviation, and a normally distributed population or a large enough sample size (n ≥ 30).
Testing a Claim About a Mean (σ Not Known)
- Methods for testing claims about a population mean when the population standard deviation is not known.
- Requirements: a simple random sample, unknown population standard deviation, and a normally distributed population or large enough sample size (n ≥ 30).
Important Properties of the Student t Distribution
- It differs depending on the sample size.
- It has the same shape as a normal distribution but is wider, reflecting that sample standard deviation has more variability than population standard deviation.
- It has a mean of zero.
- Its standard deviation varies with sample size.
- As the sample size increases, it approaches the standard normal distribution.
Finding P-Values with the Student t Distribution
- Determining a range of p-values using the t-table, not exact values.
- Locate the relevant row of the t-table using degrees of freedom.
- Determine where the test statistic falls relative to the t-values in that row.
- Estimate the area using area values in the t-table to get the range of the p-value.
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Description
This quiz covers the essential concepts of hypothesis testing, including the definitions and roles of null and alternative hypotheses. Understand how to formulate claims regarding population parameters and the importance of hypothesis testing in statistical analysis.
