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Chapter 10: Problem 86

When a published article reports the results of many hypothesis tests, the \(P\) -values are not usually given. Instead, the following type of coding scheme is frequently used: \({ }^{*} p=.05,{ }^{* *} p=.01,{ }^{* * *} p=.001,{ }^{* * * *} p=.0001\). Which of the symbols would be used to code for each of the following \(P\) -values? a. \(.037\) c. 072 b. \(.0026\) d. \(.0003\)

Short Answer

Expert verified
The symbols for the P-values .037, .072, .0026, and .0003 are \(*\), no symbol, \(**\), and \(***\) respectively.

Step by step solution

01

Identify Relevant Symbol for P-value a

If we look at the P-value of .037, it's less than .05 but greater than .01. Therefore, according to the code, the symbol for P-value .037 is \(*\).
02

Identify Relevant Symbol for P-value c

The P-value .072 is greater than .05 so it doesn't get any of the symbols associated. Thus, for the P-value .072 there would be no symbol.
03

Identify Relevant Symbol for P-value b

For the given P-value .0026, it's less than .01 but greater than .001. Therefore, according to the code, the symbol for P-value .0026 is \(**\).
04

Identify Relevant Symbol for P-value d

Looking at the provided P-value .0003, it's less than .001 but greater than .0001. Therefore, the relevant symbol for this P-value would be \(***\).

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Key Concepts

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

P-value interpretation
Understanding P-values is crucial in hypothesis testing. A P-value helps you determine the strength of your evidence against the null hypothesis. Essentially, it is the probability of observing a test statistic as extreme as the one observed, under the assumption that the null hypothesis is true.

Here鈥檚 a simplified way to interpret P-values:
  • If the P-value is small (typically less than 0.05), it suggests strong evidence against the null hypothesis, prompting us to reject it.
  • A large P-value indicates weak evidence against the null hypothesis, so we fail to reject it.
  • It does not measure the probability that the null hypothesis is true.
It鈥檚 important to note that while P-values are useful, they are just part of the statistical toolbox and should be considered alongside other metrics.
Statistical significance
Statistical significance is a way to measure if the results of your data are likely not due to chance. It helps determine if your findings are meaningful.

Here's how it works:
  • You set a threshold, known as the significance level ( \(\alpha\)), commonly set at 0.05. This is the probability of rejecting the null hypothesis when it is actually true.
  • If the P-value of your test is less than the \(\alpha\), the result is considered statistically significant.
Significance doesn't imply importance. A statistically significant result may be insignificant practically. The context of the study should always be considered when interpreting statistically significant results.
Coding scheme for P-values
A coding scheme for P-values provides a shorthand method to report significance in studies. Instead of listing P-values for each test, symbols are used to represent them.

A common scheme might include:
  • \( * \) for P-values less than 0.05
  • \( ** \) for P-values less than 0.01
  • \( *** \) for P-values less than 0.001
  • \( **** \) for P-values less than 0.0001
This approach simplifies data presentation but can sometimes obscure details. Including actual P-values is helpful for those who want more depth. Always pay attention to the scheme's key in the report, as it can vary across studies.

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Most popular questions from this chapter

Suppose that you are an inspector for the Fish and Game Department and that you are given the task of determining whether to prohibit fishing along part of the Oregon coast. You will close an area to fishing if it is determined that fish in that region have an unacceptably high mercury content. a. Assuming that a mercury concentration of \(5 \mathrm{ppm}\) is considered the maximum safe concentration, which of the following pairs of hypotheses would you test: $$ H_{0}: \mu=5 \text { versus } H_{a}: \mu>5 $$ or $$ H_{0}: \mu=5 \text { versus } H_{a}: \mu<5 $$ Give the reasons for your choice. b. Would you prefer a significance level of \(.1\) or \(.01\) for your test? Explain.

White remains the most popular car color in the United States, but its popularity appears to be slipping. According to an annual survey by DuPont (Los Angeles Times, February 22,1994 ), white was the color of \(20 \%\) of the vehicles purchased during 1993 , a decline of \(4 \%\) from the previous year. (According to a DuPont spokesperson, white represents "innocence, purity, honesty, and cleanliness.") A random sample of 400 cars purchased during this period in a certain metropolitan area resulted in 100 cars that were white. Does the proportion of all cars purchased in this area that are white appear to differ from the national percentage? Test the relevant hypotheses using \(\alpha=.05\). Does your conclusion change if \(\alpha=.01\) is used?

Pizza Hut, after test-marketing a new product called the Bigfoot Pizza, concluded that introduction of the Bigfoot nationwide would increase its sales by more than \(14 \%\) (USA Today, April 2, 1993). This conclusion was based on recording sales information for a random sample of Pizza Hut restaurants selected for the marketing trial. With \(\mu\) denoting the mean percentage increase in sales for all Pizza Hut restaurants, consider using the sample data to decide between \(H_{0}: \mu=14\) and \(H_{a}: \mu>14\). a. Is Pizza Hut's conclusion consistent with a decision to reject \(H_{0}\) or to fail to reject \(H_{0}\) ? b. If Pizza Hut is incorrect in its conclusion, is the company making a Type I or a Type II error?

A comprehensive study conducted by the National Institute of Child Health and Human Development tracked more than 1000 children from an early age through elementary school (New York Times, November 1, 2005). The study concluded that children who spent more than 30 hours a week in child care before entering school tended to score higher in math and reading when they were in the third grade. The researchers cautioned that the findings should not be a cause for alarm because the effects of child care were found to be small. Explain how the difference between the mean math score for third graders who spent long hours in child care and the overall mean for thirdgraders could be small but the researchers could still reach the conclusion that the mean for the child care group is significantly higher than the overall mean for third-graders.

A number of initiatives on the topic of legalized gambling have appeared on state ballots. Suppose that a political candidate has decided to support legalization of casino gambling if he is convinced that more than twothirds of U.S. adults approve of casino gambling. USA Today (June 17,1999 ) reported the results of a Gallup poll in which 1523 adults (selected at random from households with telephones) were asked whether they approved of casino gambling. The number in the sample who approved was 1035 . Does the sample provide convincing evidence that more than two-thirds approve?
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