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Chapter 4: Problem 148

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. 4.148 Hypotheses: \(H_{0}: \rho=0\) vs \(H_{a}: \rho \neq 0 .\) In addition, in each case for which the results are significant, give the sign of the correlation. (a) \(95 \%\) confidence interval for \(\rho: \quad 0.07\) to 0.15 (b) \(90 \%\) confidence interval for \(\rho: \quad-0.39\) to -0.78 (c) \(99 \%\) confidence interval for \(\rho:-0.06\) to 0.03

Short Answer

Expert verified
For sub-question (a), there's positive significant correlation at 5% significance level. For sub-question (b), there's negative significant correlation at 10% significance level. For sub-question (c), there's no significant correlation at 1% significance level.

Step by step solution

01

Analyzing Confidence Interval for sub-question (a)

The confidence interval for \(\rho\) for sub-question (a) is from 0.07 to 0.15. This interval does not include 0, which is the hypothesized value under \(H_{0}\). This implies the data is significantly different from the null hypothesis, i.e., there's evidence to reject the null hypothesis at a significance level of 5%.\nThe sign of the correlation is positive as all values in the interval are positive.
02

Analyzing Confidence Interval for sub-question (b)

The confidence interval for \(\rho\) for sub-question (b) is from -0.39 to -0.78. Again, 0 is not included in this interval, indicating significant evidence to reject the null hypothesis, this time at a 10% significance level.\nThe sign of the correlation is negative, indicated by all values in the interval being negative.
03

Analyzing Confidence Interval for sub-question (c)

\nThe confidence interval for this question (c) is from -0.06 to 0.03. This interval includes 0, which means we can't reject the null hypothesis. There's no significant evidence to claim that the correlation is different from 0 at the 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
A confidence interval offers a range of values within which we expect the true parameter of a population (like a correlation coefficient \( \rho \)) to lie. These intervals are constructed with a certain level of confidence, often set at 90%, 95%, or 99%. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, 95% of those intervals would contain the true parameter. Each interval gives us insight into how confident we are regarding the sample statistics representing the actual population parameters. The width of these intervals is influenced by sample size: larger samples typically produce narrower intervals, implying more precise estimates. In context, analyzing these intervals helps us determine whether a null hypothesis can be rejected.
Null Hypothesis
The null hypothesis, denoted as \( H_0 \), is a foundational concept in hypothesis testing. It represents a statement of no effect or no difference. For instance, in the exercise given, \( H_0: \rho = 0 \) suggests that there's no correlation in the population. It's our starting hypothesis that we aim to test against an alternative hypothesis (\( H_a \)), which in this case suggests \( \rho eq 0 \), meaning there is some correlation. Utilizing statistical tests and confidence intervals, we gather evidence to either reject or fail to reject the null hypothesis. If a confidence interval does not include the null hypothesis value, it indicates enough evidence to reject \( H_0 \) and support the alternative hypothesis.
Significance Level
The significance level, often represented as \( \alpha \), is the threshold set by the researcher to decide whether to reject the null hypothesis. Common significance levels are 0.05, 0.10, or 0.01, corresponding to 95%, 90%, or 99% confidence levels, respectively. These levels signify the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true. In practice, if the results are significant at a given \( \alpha \), it means the findings are unlikely to have occurred purely by chance. In the provided exercise, for instance, sub-question (a) indicates rejection of \( H_0 \) at a 5% significance level because the interval does not include zero, suggesting that there is a statistically significant positive correlation.

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Most popular questions from this chapter

Exercises 4.117 to 4.122 give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=42 / 100=0.42\) with \(n=100\)

Print vs E-books Suppose you want to find out if reading speed is any different between a print book and an e-book. (a) Clearly describe how you might set up an experiment to test this. Give details. (b) Why is a hypothesis test valuable here? What additional information does a hypothesis test give us beyond the descriptive statistics we discussed in Chapter \(2 ?\) (c) Why is a confidence interval valuable here? What additional information does a confidence interval give us beyond the descriptive statistics of Chapter 2 and the results of a hypothesis test described in part (b)? (d) A similar study" has been conducted and reports that "the difference between Kindle and the book was significant at the \(p<.01\) level, and the difference between the iPad and the book was marginally significant at \(p=.06\)." The report also stated that "the iPad measured at \(6.2 \%\) slower reading speed than the printed book, whereas the Kindle measured at \(10.7 \%\) slower than print. However, the difference between the two devices [iPad and Kindle] was not statistically significant because of the data's fairly high variability." Can you tell from the first quotation which method of reading (print or e-book) was faster in the sample or do you need the second quotation for that? Explain the results in your own words.

In Exercises 4.5 to 4.8 , state the null and alternative hvpotheses for the statistical test described. Testing to see if there is evidence that the mean of group \(\mathrm{A}\) is not the same as the mean of group \(\mathrm{B}\).

Determining Statistical Significance How small would a p-value have to be in order for you to consider results statistically significant? Explain. (There is no correct answer! This is just asking for your personal opinion. We'll study this in more detail in the next section.)

In Exercises 4.107 to \(4.111,\) null and alternative hypotheses for a test are given. Give the notation \((\bar{x},\) for example) for a sample statistic we might record for each simulated sample to create the randomization distribution. $$ H_{0}: \mu_{1}=\mu_{2} \operatorname{vs} H_{a}: \mu_{1}>\mu_{2} $$
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