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Chapter 4: Problem 147

In Exercises 4.146 to \(4.149,\) hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (a) \(95 \%\) confidence interval for \(p: \quad 0.53\) to 0.57 (b) \(95 \%\) confidence interval for \(p: \quad 0.41\) to 0.52 (c) 99\% confidence interval for \(p: \quad 0.35\) to 0.55

Short Answer

Expert verified
For sample A, the null hypothesis that \(p=0.5\) is rejected with a 5% significance level. For samples B and C, the null hypothesis that \(p=0.5\) cannot be rejected with a 5% and 1% significance level, respectively.

Step by step solution

01

Analyze Confidence Interval for Sample A

The 95% confidence interval for sample A is between 0.53 and 0.57. Since the value from the null hypothesis (0.5) does not fall within this interval, the null hypothesis \(H_{0}: p=0.5\) is rejected for sample A with a 5% significance level.
02

Analyze Confidence Interval for Sample B

The 95% confidence interval for sample B is between 0.41 and 0.52. Since the value from the null hypothesis (0.5) does fall within this interval, the null hypothesis \(H_{0}: p=0.5\) cannot be rejected for sample B with a 5% significance level.
03

Analyze Confidence Interval for Sample C

The 99% confidence interval for sample C is between 0.35 and 0.55. Since the value from the null hypothesis (0.5) does fall within this interval, the null hypothesis \(H_{0}: p=0.5\) cannot be rejected for sample C with a 1% significance level.

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Key Concepts

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

Confidence Intervals
Understanding confidence intervals is fundamental in the realm of statistics, as they offer a range of plausible values for a population parameter (like a mean or proportion). A confidence interval (CI) tells us that we can be certain to some degree - typically 95% or 99% - that the true parameter falls within this range.

Let's consider the example from the exercise where hypotheses are tested using confidence intervals for different samples. Here, CIs are used to decide whether to reject the null hypothesis. If the interval does not contain the null hypothesis value, the hypothesis is rejected, assuming that our sample is representative of the population.

For instance, a 95% CI of 0.53 to 0.57 for a population proportion means that we can be 95% confident that the true proportion lies within this range. When the null hypothesis value of 0.5 is not within this interval, it allows us to conclude that there is a significant difference from the hypothesized value at a 5% significance level.
Null Hypothesis
A null hypothesis, denoted by H0, is a statement used in statistics that there is no effect or no difference, and it serves as the starting point for any statistical hypothesis testing. In hypothesis testing, it's the hypothesis that researchers aim to test against the alternative hypothesis, denoted by Ha or H1, which suggests that there is an effect or a difference.

In the given exercise, the null hypothesis is that the population proportion (p) is equal to 0.5, expressed as H0: p=0.5. It is used as a standard to measure against the provided confidence intervals. If the confidence interval includes the value of 0.5, we do not have sufficient evidence to reject the null hypothesis. Conversely, if the value of 0.5 does not reside within the interval, it implies that there might be a significant difference and leads to rejecting the null hypothesis.
Significance Level
The significance level, often denoted by alpha (伪), is a threshold chosen by the researcher that determines when to reject the null hypothesis. It is the probability of rejecting the null hypothesis when in fact it is true, also known as a Type I error.

Common levels include 0.05 (5%) and 0.01 (1%), corresponding to confidence levels of 95% and 99%, respectively. In our textbook example, a 95% confidence interval corresponds to a significance level of 0.05. In other words, if the confidence interval were to be calculated from numerous samples, the true proportion would fall outside of the interval 5% of the time. When an observed statistic falls outside the chosen confidence interval, we say it is 'statistically significant' at the given significance level.

To decide whether to reject the null hypothesis, one compares the significance level to the p-value of the test. If the p-value is less than or equal to the significance level, the null hypothesis is rejected, indicating significant evidence against it.

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Most popular questions from this chapter

In Exercises 4.112 to \(4.116,\) the null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) Sample: \(\hat{p}_{1}=0.3, n_{1}=20\) and \(\hat{p}_{2}=0.167, n_{2}=12\) Randomization statistic \(=\hat{p}_{1}-\hat{p}_{2}\)

Multiple Sclerosis and Sunlight It is believed that sunlight offers some protection against multiple sclerosis (MS) since the disease is rare near the equator and more prevalent at high latitudes. What is it about sunlight that offers this protection? To find out, researchers \({ }^{15}\) injected mice with proteins that induce a condition in mice comparable to MS in humans. The control mice got only the injection, while a second group of mice were exposed to UV light before and after the injection, and a third group of mice received vitamin D supplements before and after the injection. In the test comparing UV light to the control group, evidence was found that the mice exposed to UV suppressed the MS-like disease significantly better than the control mice. In the test comparing mice getting vitamin D supplements to the control group, the mice given the vitamin \(\mathrm{D}\) did not fare significantly better than the control group. If the p-values for the two tests are 0.472 and 0.002 , which p-value goes with which test?

In Exercises 4.71 to \(4.74,\) using the p-value given, are the results significant at a \(10 \%\) level? At a \(5 \%\) level? At a 1\% level? $$ \text { p-value }=0.0320 $$

State the null and alternative hvpotheses for the statistical test described. Testing to see if there is evidence that a proportion is greater than 0.3

A situation is described for a statistical test and some hypothetical sample results are given. In each case: (a) State which of the possible sample results provides the most significant evidence for the claim. (b) State which (if any) of the possible results provide no evidence for the claim. Testing to see if there is evidence that the proportion of US citizens who can name the capital city of Canada is greater than \(0.75 .\) Use the following possible sample results: Sample A: \(\quad 31\) successes out of 40 Sample B: \(\quad 34\) successes out of 40 Sample C: \(\quad 27\) successes out of 40 Sample \(\mathrm{D}: \quad 38\) successes out of 40
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