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Chapter 17: Problem 25

Researchers investigated the effectiveness of oral zinc, as compared to a placebo, in reducing the duration of the common cold when taken within 24 hours of the onset of symptoms. The researchers found those taking oral zinc had a statistically significantly shorter duration \((P<0.05)\) than those taking the placebo. \({ }^{5}\) This means that (a) the probability that the null hypothesis is true is less than \(0.05\). (b) the value of the test statistic, the mean reduction in duration of the cold, is large. (c) neither of the above is true.

Short Answer

Expert verified
(c) neither of the above is true.

Step by step solution

01

Understanding the Hypothesis Test

The researchers conducted a hypothesis test to compare the effectiveness of oral zinc versus a placebo. The null hypothesis (\(H_0\)) typically assumes no difference in effectiveness (i.e., zinc does not reduce the duration of the cold more than the placebo). The alternative hypothesis (\(H_a\)) suggests that zinc does reduce the duration significantly when compared to a placebo.
02

Interpretation of P-Value

A \(P\)-value indicates the probability of observing the data, or something more extreme, under the null hypothesis. A \(P\)-value less than 0.05 is typically considered statistically significant, indicating sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.
03

Analyzing the Options

Option (a) suggests the probability that the null hypothesis is true is less than 0.05, which is incorrect. A \(P\)-value does not measure the probability of the null hypothesis being true; instead, it measures the probability of observing the data if the null hypothesis were true.Option (b) suggests the value of the test statistic is large, which can be misleading. A small \(P\)-value typically indicates the test statistic is in a critical region far from the null hypothesis, but we cannot directly infer the size of the test statistic from a \(P\)-value.Option (c) states that neither of the above is true, which aligns with the correct interpretation of the \(P\)-value.
04

Final Conclusion

Since neither option (a) nor (b) accurately interprets the \(P\)-value, option (c) is the correct choice based on the definitions and interpretations of statistical significance and \(P\)-values.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis is the starting point for any statistical test. It is often denoted as \(H_0\) and usually posits that there is no effect or no difference between groups. For example, when testing the effectiveness of oral zinc compared to a placebo, the null hypothesis would state that taking zinc does not reduce the duration of a cold any more than taking a placebo does.
When we conduct a test, the null hypothesis serves as the default position that we assume to be true until we have enough evidence to suggest otherwise. It acts like a benchmark or baseline for comparison.
  • The null hypothesis is not necessarily what researchers believe; rather, it provides a testable statement.
  • It is crucial to have a clear and precise null hypothesis before analyzing data since this guides the entire hypothesis testing process.
P-Value Interpretation
The concept of a P-value is pivotal in understanding statistical tests. The P-value quantifies the probability of obtaining the observed experimental results, or even more extreme ones, assuming the null hypothesis is true.
Here's a simplified breakdown:
  • A low P-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting that the observed data is unlikely under the null hypothesis.
  • It does not measure the probability that the null hypothesis is true or false itself. Instead, it assesses the evidence provided by the data.
  • Interpreting P-values correctly is essential for making informed conclusions from statistical tests.
In the zinc experiment, a P-value less than 0.05 would lead researchers to reject the null hypothesis, implying zinc potentially shortens cold duration compared to placebo.
Statistical Significance
Statistical significance answers a critical question: Is the effect observed in the sample data unlikely to have occurred under the null hypothesis? It is quantified using the P-value.
  • When researchers say results are statistically significant, it typically means the P-value is below a pre-determined threshold (commonly 0.05).
  • Significance does not imply importance or practical relevance. It strictly concerns the probability of obtaining the observed results by chance.
  • It is possible to have statistically significant results that, while unlikely by mere chance, may not be practically significant or meaningful.
In the case of zinc affecting cold duration, a statistically significant result suggests that zinc is likely more effective than placebo at reducing cold duration, though other factors should also be considered for practical application.
Alternative Hypothesis
The alternative hypothesis, denoted as \(H_a\), is the statement you aim to support in hypothesis testing. It posits that there is an effect or a difference contrary to the null hypothesis.
  • An alternative hypothesis in the zinc experiment would be that oral zinc does indeed reduce the duration of the cold more than the placebo.
  • This hypothesis is what researchers aim to demonstrate through their data analysis.
  • Acceptance of the alternative hypothesis occurs only when there is sufficient evidence to reject the null hypothesis.
Understanding the relationship between the null and alternative hypotheses is crucial for correctly conducting and interpreting hypothesis tests.

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Most popular questions from this chapter

Experiments on learning in animals sometimes measure how long it takes mice to find their way through a maze. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of 10 mice takes with a noise as stimulus. The sample mean is \(\mathrm{x}^{-} \bar{x}=16.5\) seconds. The null hypothesis for the test of significance is (a) \(H_{0}: \mu=18\) (b) \(H_{0}: \mu=16.5\). (c) \(H_{0}: \mu<18\).

A test of \(H_{0}: \mu=0\) against \(H_{a}: \mu>0\) has test statistic \(z=1.65\). Is this test statistically significant at the \(5 \%\) level \((\alpha=0.05)\) ? Is it statistically significant at the \(1 \%\) level \((\alpha=0.01)\) ?

A randomized comparative experiment examined the effect of the attractiveness of an instructor on the performance of students on a quiz given by the instructor. The researchers found a statistically significant difference in quiz scores between students in a class with an instructor rated as attractive and students in a class with an instructor rated as unattractive \((P=\) \(0.005) \cdot{ }^{7}\) When asked to explain the meaning of " \(P=0.005\)," a student says, "This means there is only probability of \(0.005\) that the null hypothesis is true." Explain what \(P=0.005\) really means in a way that makes it clear that the student's explanation is wrong.

The National Institute of Standards and Technology (NIST) supplies "standard copper samples" whose melting point is supposed to be exactly \(1084.80^{\circ} \mathrm{C}\). To do so, NIST must check that samples that they intend to supply meet this condition. Is there reason to think that the true melting point of a new copper sample is not \(1084.80^{\circ} \mathrm{C}\) ? To find out, NIST measures the melting point of this sample six times. Repeated measurements of the same thing vary, which is why NIST makes six measurements. These measurements are an SRS from the population of all possible measurements. This population has Normal distribution with mean \(\mu\) equal to the true melting point and standard deviation \(\sigma=0.25^{\circ} \mathrm{C}\). (a) We seek evidence against the claim that \(\mu=1084.80\). What is the sampling distribution of the mean \(x^{-} \bar{x}\) in many samples of six measurements of one sample if the claim is true? Make a sketch of the Normal curve for this distribution. (Draw a Normal curve, then mark on the axis the values of the mean and 1, 2, and 3 standard deviations on either side of the mean.) (b) Suppose that the sample mean is \(\mathrm{x}^{-} \bar{x}=1084.90\). Mark this value on the axis of your sketch. Another copper sample has \(\mathrm{x}^{-} \bar{x}=1084.50\) for six measurements. Mark this value on the axis as well. Explain in simple language why one result is good evidence that the true melting point differs from \(1084.80\) and why the other result gives no reason to doubt that \(1084.80\) is correct.

Every society has its own marks of wealth and prestige. In ancient China, it appears that owning pigs was such a mark. Evidence comes from examining burial sites. The skulls of sacrificed pigs tend to appear along with expensive ornaments, which suggests that the pigs, like the ornaments, signal the wealth and prestige of the person buried. A study of burials from around 3500 B.C. concluded that "there are striking differences in grave goods between burials with pig skulls and burials without them . . . A test indicates that the two samples of total artifacts are statistically significantly different at the \(0.01\) level."8 Explain clearly why "statistically significantly different at the \(0.01\) level" gives good reason to think that there really is a systematic difference between burials that contain pig skulls and those that lack them.
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