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Use the following information to answer next five exercises. A study was conducted to test the effectiveness of a juggling class. Before the class started, six subjects juggled as many balls as they could at once. After the class, the same six subjects juggled as many balls as they could. The differences in the number of balls are calculated. The differences have a normal distribution. Test at the 1% significance level. $$ \begin{array}{|c|c|c|c|c|c|}\hline \text { Subject } & {\mathbf{A}} & {\mathbf{B}} & {\mathbf{c}} & {\mathbf{D}} & {\mathbf{E}} & {\mathbf{F}} \\\ \hline \text { Before } & {3} & {4} & {3} & {2} & {4} & {5} \\ \hline \text { After } & {4} & {5} & {6} & {4} & {5} & {7} \\ \hline\end{array} $$ State the null and alternative hypotheses.

Step by step solution

01

Identify the populations involved

In this problem, we are dealing with two different groups of data for the same subjects: the number of balls juggled before the class and the number of balls juggled after the class. These two sets of data will be compared to understand the impact of the juggling class.

02

Define the parameter of interest

The parameter of interest is the mean difference in the number of balls juggled by the subjects before and after taking the juggling class. Let this mean difference be denoted as \( \mu_d \), where \( d = \text{after} - \text{before} \). We are interested in testing whether \( \mu_d \) is equal to zero.

03

State the null hypothesis

The null hypothesis \( H_0 \) is a statement of no effect or no difference. For this problem, it states that the juggling class has no effect on the number of balls juggled. Mathematically, this can be expressed as: \[ H_0: \mu_d = 0 \]

04

State the alternative hypothesis

The alternative hypothesis \( H_a \) is what we aim to support with evidence. Since we expect that the juggling class might lead to an increase in the number of balls juggled, the alternative hypothesis is that the mean difference is greater than zero. Mathematically, this is: \[ H_a: \mu_d > 0 \]

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Key Concepts

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

Null Hypothesis

The null hypothesis is one of the cornerstones of hypothesis testing. It refers to a general statement or default position that there is no relationship or no effect between two measured phenomena. In our juggling study, the null hypothesis (\( H_0 \)) claims that taking the class doesn't change the number of balls a person can juggle. Essentially, it suggests that any observed difference is purely due to random chance.

When conducting a test, determining the null hypothesis is the initial step. If we are interested in seeing whether a technique works 鈥� like this juggling class 鈥� we start by assuming it doesn't work and then look for evidence to prove otherwise. Remember:

• The null hypothesis can often be seen as the "status quo."
• It is typically expressed in mathematical terms such as \( \mu_d = 0 \).
• Rejecting the null means you have sufficient evidence to support an alternative story.

Alternative Hypothesis

The alternative hypothesis is the statement that suggests a potential outcome which we think might be true or want to prove true. In the juggling class scenario, the alternative hypothesis (\( H_a \)) posits that the class actually helps participants juggle more balls, meaning the mean difference in the balls juggled is greater than zero.

The alternative hypothesis is crucial because it represents the actual claim or effect that the study aims to demonstrate. Here's what you need to know:

• It opposes the null hypothesis.
• In statistical notation, our problem's alternative hypothesis is \( \mu_d > 0 \).
• The goal is to gather enough evidence to reject the null in favor of the alternative.

It essentially drives the research and explains what the study seeks to reveal. Without it, there would be no direction for the analysis.

Significance Level

In hypothesis testing, the significance level is a critical value that helps determine how confident we need to be when drawing conclusions. It is denoted by the Greek letter alpha (\( \alpha \)) and often set at levels such as 0.05 (5%) or 0.01 (1%), depending on how stringent the test needs to be. Here, we're testing at the 1% significance level, which means we are adopting a more stringent criterion for our test.

The significance level tells us the probability of rejecting the null hypothesis when it is actually true, known as Type I error.

• In our situation, a 1% significance level means we have only a 1% risk of incorrectly saying the class improves juggling skills when it doesn't.
• The lower the significance level, the more evidence we need to reject \( H_0 \).
• A stringent level like 1% offers stronger evidence when the null hypothesis is rejected.

This level assures the reliability of our eventual findings.

Mean Difference

The mean difference (\( \mu_d \)) represents the average change between two sets of data. In our juggler's experiment, it's calculated as the number of balls juggled after the class minus those juggled before. Essentially, it shows the real effect of the juggling class.

Let's break it down:

• If the mean difference is zero (\( \mu_d = 0 \)), it implies the class had no effect.
• A positive mean difference (\( \mu_d > 0 \)) suggests an improvement in juggling skills.
• Mean differences can also be negative, indicating a decrease in performance, but that's not expected in this context.

The mean difference serves as the parameter we are testing against, allowing us to understand the impact of the class, if any.