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Chapter 8: Problem 56

If we reject the null hypothesis, can we claim to have proved that the null hypothesis is false? Why or why not?

Short Answer

Expert verified
No, rejecting the null hypothesis does not prove it false. It only suggests that there is strong evidence, based on the sample data and the chosen significance level, to support the alternative hypothesis. This is because hypothesis testing works with sample data and probability, and there is always a non-zero chance that the observed effect happened due to random chance.

Step by step solution

01

Understanding Hypothesis Testing

In statistical analysis, a hypothesis is assumed for a hypothesis testing. This assumed hypothesis is known as the null hypothesis, usually denoted by H0. The null hypothesis represents a statement of no effect or no difference, and is assumed true until statistical evidence suggests otherwise. When we perform a hypothesis test, we compare the evidence provided by our data sample with this null hypothesis.
02

Rejection of Null Hypothesis

During hypothesis testing, if the sample data provides strong evidence against the null hypothesis, we reject the null hypothesis in favor of an alternative hypothesis. This decision is based on a probability measure known as the P-value. If the P-value is smaller than our significance level (commonly set at 0.05), we reject the null hypothesis.
03

Interpretation of Rejecting Null Hypothesis

Rejecting the null hypothesis means that, given the data sample and the chosen significance level, there is strong evidence to suggest that the null hypothesis is not true, and the effect stated in the alternative hypothesis exists. However, rejecting the null hypothesis does not prove it false. This is because hypothesis testing works with sample data and probability, and there is always a non-zero probability that the observed effect in the sample happened due to random chance.

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Most popular questions from this chapter

In problem \(8.16\), a college chemistry instructor thinks the use of embedded tutors will improve the success rate in introductory chemistry courses. The instructor carried out a hypothesis test and found that the observed value of the test statistic was \(2.33 .\) The \(\mathrm{p}\) -value associated with this test statistic is \(0.0099 .\) Explain the meaning of the p-value in this context. Based on this result, should the instructor believe the success rate has improved?

According to a 2018 survey by Timex reported in Shape magazine, \(73 \%\) of Americans report working out one or more times each week. A nutritionist is interested in whether this percentage has increased. A random sample of 200 Americans found 160 reported working out one or more times each week. Carry out the first two steps of a hypothesis test to determine whether the proportion has increased. Explain how you would fill in the required TI calculator entries for \(p_{0}, x\), and \(n .\)

Choosing a Test and Naming the Population(s) For each of the following, state whether a one-proportion \(z\) -test or a two-proportion \(z\) -test would be appropriate, and name the population(s). a. A polling agency takes a random sample of voters in California to determine if a ballot proposition will pass. b. A researcher asks a random sample of residents from coastal states and a random sample of residents of non-coastal states whether they favor increased offshore oil drilling. The researcher wants to determine if there is a difference in the proportion of residents who support off-shore drilling in the two regions.

p-Values (Example 11) A researcher carried out a hypothesis test using a two- sided alternative hypothesis. Which of the following \(z\) -scores is associated with the smallest p-value? Explain. i. \(z=0.50\) ii. \(z=1.00\) iii. \(z=2.00\) iv. \(z=3.00\)

Student Loans According to a 2016 report from the Institute for College Access and Success \(66 \%\) of all graduates from public colleges and universities had student loans. A public college surveyed a random sample of 400 graduates and found that \(62 \%\) had student loans. a. Test the hypothesis that the percentage of graduates with student loans from this college is different from the national percentage. Use a significance level of \(0.05\). b. After conducting the hypothesis test, a further question one might ask is what proportion of graduates from this college have student loans? Use the sample data to find a \(95 \%\) confidence interval for the proportion of graduates from the college who have student loans. How does this confidence interval support the hypothesis test conclusion?
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