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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

A reporter on cnn.com stated in July 2010 that \(95 \%\) of all court cases that go to trial result in a guilty verdict. To test the accuracy of this claim, we collect a random sample of 2000 court cases that went to trial and record the proportion that resulted in a guilty verdict. (a) What is/are the relevant parameter(s)? What sample statistic(s) is/are used to conduct the test? (b) State the null and alternative hypotheses. (c) We assess evidence by considering how likely our sample results are when \(H_{0}\) is true. What does that mean in this case?

Euchre One of the authors and some statistician friends have an ongoing series of Euchre games that will stop when one of the two teams is deemed to be statistically significantly better than the other team. Euchre is a card game and each game results in a win for one team and a loss for the other. Only two teams are competing in this series, which we'll call team A and team B. (a) Define the parameter(s) of interest. (b) What are the null and alternative hypotheses if the goal is to determine if either team is statistically significantly better than the other at winning Euchre? (c) What sample statistic(s) would they need to measure as the games go on? (d) Could the winner be determined after one or two games? Why or why not? (e) Which significance level, \(5 \%\) or \(1 \%,\) will make the game last longer?

By some accounts, the first formal hypothesis test to use statistics involved the claim of a lady tasting tea. \({ }^{11}\) In the 1920 's Muriel Bristol- Roach, a British biological scientist, was at a tea party where she claimed to be able to tell whether milk was poured into a cup before or after the tea. R.A. Fisher, an eminent statistician, was also attending the party. As a natural skeptic, Fisher assumed that Muriel had no ability to distinguish whether the milk or tea was poured first, and decided to test her claim. An experiment was designed in which Muriel would be presented with some cups of tea with the milk poured first, and some cups with the tea poured first. (a) In plain English (no symbols), describe the null and alternative hypotheses for this scenario. (b) Let \(p\) be the true proportion of times Muriel can guess correctly. State the null and alternative hypothesis in terms of \(p\).

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: \(H_{0}: \mu=10\) vs \(H_{a}: \mu>10\) Sample: \(\bar{x}=12, s=3.8, n=40\)

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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