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Chapter 19: Problem 3

Which of the following are true? If false, explain briefly. a. A very high P-value is strong evidence that the null hypothesis is false. b. A very low P-value proves that the null hypothesis is false. c. A high P-value shows that the null hypothesis is true. d. A P-value below 0.05 is always considered sufficient evidence to reject a null hypothesis.

Short Answer

Expert verified
All the statements are false. High P-values suggest the data are likely under the null hypothesis, not that the null hypothesis is false or true. Low P-values suggest the observed data are unlikely under the null hypothesis, but don't prove it's false. A P-value below 0.05 is not always sufficient to reject the null hypothesis; the threshold for significance can vary.

Step by step solution

01

Interpretation of each statement

Go over each statement a, b, c and d individually and interpret whether they're true or false, providing explanations in case of false statements.
02

Statement a

This statement is false. A high P-value actually means that the data observed are highly likely under the null hypothesis, not that the null hypothesis itself is false.
03

Statement b

This statement is also false. A low P-value indicates that the observed data are very unlikely under the null hypothesis. It does not necessarily prove that the null hypothesis is false, just that it's less likely given the data observed.
04

Statement c

This statement is also false. A high P-value signifies that the observed data are likely under the null hypothesis, but it doesn't prove the null hypothesis is true.
05

Statement d

This statement is also false. Although a P-value below 0.05 is generally considered strong evidence against the null hypothesis, it is not universally decisive. The threshold for 'significance' can vary depending on the field of study or the specific experiment.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

P-value Interpretation
Interpreting the P-value is a critical aspect of hypothesis testing. The P-value helps us understand the probability of observing the results given that the null hypothesis is true. In simple terms, the P-value indicates how likely the observed data would occur if the null hypothesis holds true.
  • A low P-value means that the observed data are unlikely under the null hypothesis. This might suggest evidence against the null hypothesis. However, it does not prove that the null hypothesis is false.
  • A high P-value indicates that the data observed are fairly likely under the null hypothesis, suggesting no strong evidence against it. But, this does not confirm that the null hypothesis is true either.
It's important to remember that P-values do not prove a hypothesis; they only indicate the level of compatibility of the data with the null hypothesis. Their interpretation requires careful contextual consideration.
Null Hypothesis
The null hypothesis, often symbolized as \( H_0 \), is a fundamental concept in hypothesis testing. It usually represents a statement of no effect, no difference, or a status quo condition that you wish to test against an alternative hypothesis.
  • The null hypothesis articulates a default or standard position that there is no relationship between two measured phenomena.
  • In hypothesis testing, you aim to gather evidence against the null hypothesis to support its rejection.
  • A common misconception is that a high P-value confirms the null hypothesis is true, which is not the case. It merely suggests that the data is not inconsistent with the null hypothesis.
The null hypothesis is a cornerstone of statistical inference, and understanding its role helps in making informed conclusions from data analysis.
Statistical Significance
Statistical significance relates to the likelihood that an effect observed in a study is due to chance or random variation rather than an actual effect. It is the threshold we use to decide when to reject the null hypothesis.
  • Typically, a statistical test is considered significant if the P-value is below a pre-determined threshold, often 0.05.
  • Achieving statistical significance suggests enough evidence exists to reject the null hypothesis.
  • But, statistical significance does not measure the size or importance of an effect. It only indicates that the effect is unlikely to be zero.
In summary, statistical significance is crucial for making decisions based on data, but it should not be confused with practical or scientific significance.
Significance Threshold
The significance threshold, also known as the alpha level (\( \alpha \)), is a pre-set standard used in hypothesis testing to determine whether a result is statistically significant. It is the threshold against which we compare our P-value to decide on the rejection of the null hypothesis.
  • The conventional threshold for significance is often set at 0.05, meaning there is a 5% risk of rejecting the null hypothesis when it is actually true (Type I error).
  • The choice of significance threshold can vary based on the field of study or specific experiment. For example, more stringent fields may use thresholds like 0.01.
  • An understanding of the significance threshold is necessary for interpreting the results of hypothesis testing correctly.
Choosing the right threshold requires a balance between the risk of false positives and the need for evidence to reach a conclusion in a study.

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Most popular questions from this chapter

A basketball player with a poor foul-shot record practices intensively during the off-season. He tells the coach that he has raised his proficiency from \(60 \%\) to \(80 \%\). Dubious, the coach asks him to take 10 shots, and is surprised when the player hits 9 out of 10. Did the player prove that he has improved? a. Suppose the player really is no better than before-still a \(60 \%\) shooter. What's the probability he can hit at least 9 of 10 shots anyway? (Hint: Use a Binomial model.) b. If that is what happened, now the coach thinks the player has improved when he has not. Which type of error is that? c. If the player really can hit \(80 \%\) now, and it takes at least 9 out of 10 successful shots to convince the coach, what's the power of the test? d. List two ways the coach and player could increase the power to detect any improvement.

In a drawer are two coins. They look the same, but one coin produces heads \(90 \%\) of the time when spun while the other one produces heads only \(30 \%\) of the time. You select one of the coins. You are allowed to spin it once and then must decide whether the coin is the \(90 \%\) - or the \(30 \%\) -head coin. Your null hypothesis is that your coin produces \(90 \%\) heads. a. What is the alternative hypothesis? b. Given that the outcome of your spin is tails, what would you decide? What if it were heads? c. How large is \(\alpha\) in this case? d. How large is the power of this test? (Hint: How many possibilities are in the alternative hypothesis?) e. How could you lower the probability of a Type I error and increase the power of the test at the same time?

Environmentalists concerned about the impact of high-frequency radio transmissions on birds found that there was no evidence of a higher mortality rate among hatchlings in nests near cell towers. They based this conclusion on a test using \(\alpha=0.05\). Would they have made the same decision at \(\alpha=0.10 ?\) How about \(\alpha=0.01 ?\) Explain.

Which of the following statements are true? If false, explain briefly. a. It is better to use an alpha level of 0.05 than an alpha level of 0.01 . b. If we use an alpha level of 0.01 , then a P-value of 0.001 is statistically significant. c. If we use an alpha level of \(0.01,\) then we reject the null hypothesis if the \(\mathrm{P}\) -value is 0.001 d. If the P-value is 0.01 , we reject the null hypothesis for any alpha level greater than 0.01 .

A company is sued for job discrimination because only \(19 \%\) of the newly hired candidates were minorities when \(27 \%\) of all applicants were minorities. Is this strong evidence that the company's hiring practices are discriminatory? a. Is this a one-tailed or a two-tailed test? Why? b. In this context, what would a Type I error be? c. In this context, what would a Type II error be? d. In this context, what is meant by the power of the test? e. If the hypothesis is tested at the \(5 \%\) level of significance instead of \(1 \%\), how will this affect the power of the test? \(\mathrm{f}\). The lawsuit is based on the hiring of 37 employees. Is the power of the test higher than, lower than, or the same as it would be if it were based on 87 hires?
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