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Chapter 4: Problem 10

A situation is described for a statistical test. In each case, define the relevant parameter(s) and state the null and alternative hypotheses. Testing to see if there is evidence that a correlation between height and salary is significant (that is, different than zero ).

Short Answer

Expert verified
The relevant parameters are height and salary. The null hypothesis (H0) is that there is no correlation between height and salary (\( \rho = 0 \)). The alternative hypothesis (H1) is that the correlation between height and salary is significantly different than zero (\( \rho \neq 0 \)).

Step by step solution

01

Definition of Parameters

The parameters in this case study are height and salary. They both constitute a correlation relationship.
02

Setting up the Null Hypothesis

The null hypothesis (H0) is often a statement of no effect or no relationship. So in this case, the null hypothesis would be: There is no correlation between height and salary, which translates to a correlation coefficient of 0.
03

Setting up the Alternative Hypothesis

The alternative hypothesis (H1) is the statement that directly contradicts the null hypothesis. What we want to prove is that a correlation between height and salary exists (meaning the correlation is significantly different than zero). Hence, the alternative hypothesis would be that there is a non-zero correlation.

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Key Concepts

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

Null Hypothesis
In any statistical analysis, the null hypothesis is critical as it is the claim being tested. The null hypothesis asserts that there is no association or effect between the variables of interest. In the context of the correlation between height and salary, the null hypothesis (\( H_0 \)) posits that the correlation coefficient, which measures the strength and direction of this relationship, is equal to zero. This means there is no evidence to suggest any relationship between height and salary in the population.

It's like a base assumption that we start with, and our objective in hypothesis testing is to challenge this assumption. It places the burden of proof on those who claim that a relationship exists, requiring them to demonstrate this with sufficient evidence from data. Setting up a clear null hypothesis is crucial for a strong foundation in any statistical test.
Alternative Hypothesis
The alternative hypothesis (\( H_1 \) or (\( H_a \)) serves as the counter-claim to the null hypothesis. It's what you suspect might be true and are seeking evidence to support. In the scenario where we're investigating a relationship between height and salary, the alternative hypothesis posits that there exists a non-zero correlation: essentially, height does have some effect on salary.

This hypothesis doesn't necessarily specify whether the effect is positive or negative, only that there's some statistically significant relationship to be observed. The alternative hypothesis fosters the investigative nature of statistical testing, where we're set to either reject the null hypothesis in favor of the alternative or fail to find enough evidence to do so.
Correlation Coefficient
The correlation coefficient is a statistical measure that calculates the strength of the relationship between two variables. It ranges from -1 to 1, where 1 represents a perfect positive correlation, 0 represents no correlation, and -1 represents a perfect negative correlation.

This coefficient is the crux of our statistical test in the given exercise. If it significantly deviates from zero (in either a positive or negative direction), it suggests that as one variable increases, the other variable also tends to increase (or decrease, if the correlation is negative). It's this value that is tested against the null hypothesis when we assess the significance of the relationship between height and salary.
Parameters in Statistics
Parameters in statistics refer to the defining characteristics or measurable factors of a population that help in concluding from sampled data. For example, the mean of a population is a parameter that summarizes the average value of that population.

In the exercise at hand, height and salary are the key parameters as they define the aspects of the population we're examining for a correlation. Identifying the correct parameters is essential in setting up the hypotheses and conducting an accurate test. In practice, we often estimate parameters through a statistic, a measure computed from sample data, since having data for an entire population is rarely possible.

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Most popular questions from this chapter

After exercise, massage is often used to relieve pain, and a recent study 33 shows that it also may relieve inflammation and help muscles heal. In the study, 11 male participants who had just strenuously exercised had 10 minutes of massage on one quadricep and no treatment on the other, with treatment randomly assigned. After 2.5 hours, muscle biopsies were taken and production of the inflammatory cytokine interleukin-6 was measured relative to the resting level. The differences (control minus massage) are given in Table 4.11 . $$ \begin{array}{lllllllllll} 0.6 & 4.7 & 3.8 & 0.4 & 1.5 & -1.2 & 2.8 & -0.4 & 1.4 & 3.5 & -2.8 \end{array} $$ (a) Is this an experiment or an observational study? Why is it not double blind? (b) What is the sample mean difference in inflammation between no massage and massage? (c) We want to test to see if the population mean difference \(\mu_{D}\) is greater than zero, meaning muscle with no treatment has more inflammation than muscle that has been massaged. State the null and alternative hypotheses. (d) Use Statkey or other technology to find the p-value from a randomization distribution. (e) Are the results significant at a \(5 \%\) level? At a \(1 \%\) level? State the conclusion of the test if we assume a \(5 \%\) significance level (as the authors of the study did).

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Does the airline you choose affect when you'll arrive at your destination? The dataset DecemberFlights contains the difference between actual and scheduled arrival time from 1000 randomly sampled December flights for two of the major North American airlines, Delta Air Lines and United Air Lines. A negative difference indicates a flight arrived early. We are interested in testing whether the average difference between actual and scheduled arrival time is different between the two airlines. (a) Define any relevant parameter(s) and state the null and alternative hypotheses. (b) Find the sample mean of each group, and calculate the difference in sample means. (c) Use StatKey or other technology to create a randomization distribution and find the p-value. (d) At a significance level of \(\alpha=0.01\), what is the conclusion of the test? Interpret the conclusion in context.

Mating Choice and Offspring Fitness Does the ability to choose a mate improve offspring fitness in fruit flies? Researchers have studied this by taking female fruit flies and randomly dividing them into two groups; one group is put into a cage with a large number of males and able to freely choose who to mate with, while flies in the other group are each put into individual vials, each with only one male, giving no choice in who to mate with. Females are then put into egg laying chambers, and a certain number of larvae collected. Do the larvae from the mate choice group exhibit higher survival rates? A study \(^{44}\) published in Nature found that mate choice does increase offspring fitness in fruit flies (with p-value \(<0.02\) ), yet this result went against conventional wisdom in genetics and was quite controversial. Researchers attempted to replicate this result with a series of related experiments, \({ }^{45}\) with data provided in MateChoice. (a) In the first replication experiment, using the same species of fruit fly as the original Nature study, 6067 of the 10000 larvae from the mate choice group survived and 5976 of the 10000 larvae from the no mate choice group survived. Calculate the p-value. (b) Using a significance level of \(\alpha=0.05\) and \(\mathrm{p}\) -value from (a), state the conclusion in context. (c) Actually, the 10,000 larvae in each group came from a series of 50 different runs of the experiment, with 200 larvae in each group for each run. The researchers believe that conditions dif- fer from run to run, and thus it makes sense to treat each \(\mathrm{run}\) as a case (rather than each fly). In this analysis, we are looking at paired data, and the response variable would be the difference in the number of larvae surviving between the choice group and the no choice group, for each of the 50 runs. The counts (Choice and NoChoice and difference (Choice \(-\) NoChoice) in number of surviving larva are stored in MateChoice. Using the single variable of differences, calculate the p-value for testing whether the average difference is greater than \(0 .\) (Hint: this is a single quantitative variable, so the corresponding test would be for a single mean.) (d) Using a significance level of \(\alpha=0.05\) and the p-value from (c), state the conclusion in context. (e) The experiment being tested in parts (a)-(d) was designed to mimic the experiment from the original study, yet the original study yielded significant results while this study did not. If mate choice really does improve offspring fitness in fruit flies, did the follow-up study being analyzed in parts (a)-(d) make a Type I, Type II, or no error? (f) If mate choice really does not improve offspring fitness in fruit flies, did the original Nature study make a Type I, Type II, or no error?

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: \(\hat{p}=28 / 40=0.70\) with \(n=40\)
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