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Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would be rejected if \(P\) -value \(=.0003\) b. Why \(H_{0}\) would not be rejected if \(P\) -value \(=.350\)

Short Answer

Expert verified
The P-value quantifies the strength of evidence against the null hypothesis \(H_{0}\). If the P-value is below a predetermined significance level (often 0.05), then \(H_{0}\) is rejected. Thus, for a P-value of 0.0003 (< 0.05), \(H_{0}\) would be rejected. Conversely, for a P-value of 0.350 (> 0.05), there is not sufficient evidence to reject \(H_{0}\), so \(H_{0}\) is not rejected.

Step by step solution

01

Understanding the P-value

The P-value in a hypothesis testing scenario represents the probability of obtaining a test statistic result at least as extreme as the one that was actually observed, given that the null hypothesis is true. If this probability (the P-value) is smaller than a predetermined significance level (typically 0.05), then there is strong evidence against the null hypothesis, so it is rejected.
02

Application of P-value interpretation to part a

When the P-value is 0.0003, it means there's only a 0.03% chance of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true. This is a very low probability, far below the typical 5% significance level. As such, the evidence against the null hypothesis is considered strong, and hence \(H_{0}\) would be rejected.
03

Application of P-value interpretation to part b

On the other hand, when the P-value is 0.350, it means there's a 35% chance of obtaining a test statistic as extreme as the observed one if the null hypothesis is true. This is a far higher probability, above the common 5% significance level. Therefore, the evidence against the null hypothesis is considered not strong enough to reject it, so \(H_{0}\) would not be rejected.

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Key Concepts

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

Null Hypothesis Rejection
The null hypothesis, often denoted by H_{0}, is a general statement or default position that there is no relationship between two measured phenomena or no association among groups. Rejecting the null hypothesis is a crucial step in hypothesis testing. It essentially means that you have sufficient evidence to conclude that the null hypothesis is not likely to be true.

In the context of the exercise, if the P-value is .0003, it indicates that the results observed in the study (or more extreme results) would occur only 0.03% of the time if the null hypothesis were true. Since this chance is extremely low, scientists have agreed on a threshold below which they conclude the null hypothesis is unlikely. This leads to its rejection, and it's typically done with a high level of confidence in the result.
Significance Level
The significance level, denoted by \(\beta\), is a threshold chosen by the researcher to decide whether to reject the null hypothesis. The most commonly used significance level is 0.05 or 5%. This standard has been established as a balance between being too lenient and too strict when it comes to identifying a statistically significant result.

The significance level is a critical value in statistical tests and represents the probability of rejecting the null hypothesis when it is actually true, an error known as a Type I error. When your P-value is lower than the significance level, it implies that the observed data are inconsistent with the assumption that the null hypothesis is true, favoring the alternative hypothesis.
Test Statistic
A test statistic is a value calculated from the sample data during a hypothesis test. Its purpose is to help determine the likelihood of the null hypothesis given the data. The test statistic can be a t-value, z-score, chi-squared statistic, or any number of other statistics, depending on the type of test being performed.

In practical terms, the test statistic compares your data to what is expected under the null hypothesis. The more the test statistic diverges from what we'd expect if \(H_{0}\) were true, the less likely the null hypothesis is accurate. The P-value directly relates to the test statistic, as it tells you the probability of seeing a value as extreme as or more extreme than the test statistic, assuming the null hypothesis is true.
Probability
In hypothesis testing, probability measures the likelihood of a certain event occurring. It ranges from 0 to 1, with lower values indicating a lower likelihood and higher values indicating a greater likelihood. When you calculate a P-value in the context of hypothesis testing, you're computing the probability of obtaining results at least as extreme as the observed results, under the assumption that the null hypothesis is correct.

The concept of probability is foundational when interpreting P-values. A P-value of .350 means there is a 35% chance of obtaining a test statistic as extreme as the observed one if the null hypothesis were true—a relatively high probability, which suggests that such an outcome isn't particularly unusual and, thus, doesn't provide strong evidence against the null hypothesis.

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Most popular questions from this chapter

The paper "Debt Literacy, Financial Experiences and Over-Indebtedness" (Social Science Research Network, Working paper W14808, 2008 ) included analysis of data from a national sample of 1000 Americans. One question on the survey was: "You owe \(\$ 3000\) on your credit card. You pay a minimum payment of \(\$ 30\) each month. At an Annual Percentage Rate of \(12 \%\) (or \(1 \%\) per month), how many years would it take to eliminate your credit card debt if you made no additional charges?" Answer options for this question were: (a) less than 5 years; (b) between 5 and 10 years; (c) between 10 and 15 years; (d) never-you will continue to be in debt; (e) don't know; and (f) prefer not to answer. a. Only 354 of the 1000 respondents chose the correct answer of never. For purposes of this exercise, you can assume that the sample is representative of adult Americans. Is there convincing evidence that the proportion of adult Americans who can answer this question correctly is less than \(.40(40 \%) ?\) Use \(\alpha=.05\) to test the appropriate hypotheses. b. The paper also reported that \(37.8 \%\) of those in the sample chose one of the wrong answers \((a, b,\) and \(c)\) as their response to this question. Is it reasonable to conclude that more than one-third of adult Americans would select a wrong answer to this question? Use \(\alpha=.05\).

The power of a test is influenced by the sample size and the choice of significance level. a. Explain how increasing the sample size affects the power (when significance level is held fixed). b. Explain how increasing the significance level affects the power (when sample size is held fixed).

Many consumers pay careful attention to stated nutritional contents on packaged foods when making purchases. It is therefore important that the information on packages be accurate. A random sample of \(n=12\) frozen dinners of a certain type was selected from production during a particular period, and the calorie content of each one was determined. (This determination entails destroying the product, so a census would certainly not be desirable!) Here are the resulting observations, along with a boxplot and normal probability plot: \(\begin{array}{llllllll}255 & 244 & 239 & 242 & 265 & 245 & 259 & 248\end{array}\) \(\begin{array}{llll}225 & 226 & 251 & 233\end{array}\) a. Is it reasonable to test hypotheses about mean calorie content \(\mu\) by using a \(t\) test? Explain why or why not. b. The stated calorie content is \(240 .\) Does the boxplot suggest that true average content differs from the stated value? Explain your reasoning. c. Carry out a formal test of the hypotheses suggested in Part (b).

Explain why the statement \(\bar{x}=50\) is not a legitimate hypothesis.

Consider the following quote from the article "Review Finds No Link Between Vaccine and Autism" (San Luis Obispo Tribune, October 19,2005 ): "'We found no evidence that giving MMR causes Crohn's disease and/or autism in the children that get the MMR, said Tom Jefferson, one of the authors of The Cochrane Review. 'That does not mean it doesn't cause it. It means we could find no evidence of it." (MMR is a measlesmumps-rubella vaccine.) In the context of a hypothesis test with the null hypothesis being that MMR does not cause autism, explain why the author could not conclude that the MMR vaccine does not cause autism.

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