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1. Many baseball players believe that "corking" their bats will help them hit t...

Jun 6, 2024

Solution by Steps

step 1

An experiment allows for control over variables, ensuring that the only difference between groups is the treatment (corked vs. noncorked bats)

step 2

Using available data may include confounding variables that could affect the outcome, making it difficult to determine causality

step 3

Randomization in an experiment helps to eliminate bias and ensures that the results are due to the treatment and not other factors

Answer

Investigating this question using an experiment is better because it allows for control over variables, eliminates confounding factors, and ensures that the results are due to the treatment.

Question b) What are the explanatory and response variables in this experiment?

step 1

The explanatory variable is the type of bat used (corked or noncorked)

step 2

The response variable is the distance the ball is hit

Answer

The explanatory variable is the type of bat used (corked or noncorked), and the response variable is the distance the ball is hit.

Question c) What are the treatments in the experiment?

step 1

The treatments in the experiment are the use of a corked bat and the use of a noncorked bat

Answer

The treatments are the use of a corked bat and the use of a noncorked bat.

Question d) Why is it important to randomize the order of the bats?

step 1

Randomizing the order of the bats helps to eliminate any potential order effects or biases that could influence the results

step 2

It ensures that the results are due to the treatment and not the order in which the bats are used

Answer

Randomizing the order of the bats is important to eliminate order effects and ensure that the results are due to the treatment.

Question e) Explain the concept of control and how it should be used in this experiment.

step 1

Control involves keeping all other variables constant except for the treatment being tested

step 2

In this experiment, control can be achieved by using the same player, same pitching conditions, and same environment for both types of bats

Answer

Control involves keeping all other variables constant except for the treatment. In this experiment, it can be achieved by using the same player, same pitching conditions, and same environment for both types of bats.

Question f) Will you be blind in this experiment? Explain how this could limit the conclusions we can make.

step 1

Blinding means that the participant does not know which treatment they are receiving

step 2

In this experiment, it may be difficult to blind the player to the type of bat being used

step 3

Lack of blinding could introduce bias, as the player's performance might be influenced by their knowledge of the bat type

Answer

Blinding means the participant does not know which treatment they are receiving. In this experiment, it may be difficult to blind the player, which could introduce bias and limit the conclusions we can make.

Question g) State the hypotheses we are interested in testing.

step 1

The null hypothesis (

H_0

) is that there is no difference in the mean distance hit between corked and noncorked bats

step 2

The alternative hypothesis (

H_a

) is that there is a difference in the mean distance hit between corked and noncorked bats

Answer

The null hypothesis ($H_0$) is that there is no difference in the mean distance hit between corked and noncorked bats. The alternative hypothesis ($H_a$) is that there is a difference.

Question h) Describe a Type I and a Type II error in the context of these hypotheses.

step 1

A Type I error occurs when we reject the null hypothesis when it is actually true

step 2

In this context, a Type I error would mean concluding that corking the bat makes a difference when it actually does not

step 3

A Type II error occurs when we fail to reject the null hypothesis when it is actually false

step 4

In this context, a Type II error would mean concluding that corking the bat does not make a difference when it actually does

Answer

A Type I error would mean concluding that corking the bat makes a difference when it actually does not. A Type II error would mean concluding that corking the bat does not make a difference when it actually does.

Question i) Calculate the value of the test statistic (difference in means).

step 1

The mean distance for the corked bat is 225 feet

step 2

The mean distance for the noncorked bat is 221 feet

step 3

The test statistic is the difference in means:

225 - 221 = 4

feet

Answer

The value of the test statistic (difference in means) is 4 feet.

Question j) Describe how to simulate the distribution of the difference in means, assuming you have the same ABILITY to hit with each type of bat.

step 1

Combine all the distances hit with both types of bats into one dataset

step 2

Randomly split this combined dataset into two groups of 20 distances each

step 3

Calculate the difference in means between the two groups

step 4

Repeat this process 100 times to create a distribution of the difference in means

Answer

To simulate the distribution of the difference in means, combine all distances into one dataset, randomly split into two groups of 20, calculate the difference in means, and repeat 100 times.

Question k) Describe what information is provided by the dotplot.

step 1

The dotplot shows the distribution of the simulated differences in means from 100 trials

step 2

It provides a visual representation of the variability and range of the differences in means

step 3

It helps to determine if the observed difference in means is unusual compared to the simulated differences

Answer

The dotplot shows the distribution of the simulated differences in means from 100 trials, providing a visual representation of variability and helping to determine if the observed difference is unusual.

Question l) Use the dotplot to estimate and interpret the

p

-value.

step 1

The

p

-value is the proportion of simulated differences that are as extreme or more extreme than the observed difference

step 2

Count the number of dots in the dotplot that are as extreme or more extreme than the observed difference of 4 feet

step 3

Divide this count by the total number of trials (100) to estimate the

p

-value

Answer

The $p$-value is estimated by counting the number of dots as extreme or more extreme than 4 feet and dividing by 100.

Question m) Based on this

p

-value, what would be an appropriate conclusion?

step 1

If the

p

-value is less than the significance level (e.g., 0.05), we reject the null hypothesis

step 2

If the

p

-value is greater than the significance level, we fail to reject the null hypothesis

Answer

Based on the $p$-value, if it is less than 0.05, we reject the null hypothesis and conclude that corking the bat makes a difference. If it is greater than 0.05, we fail to reject the null hypothesis.

Question n) If your conclusion was in error, which type of error did you make? Explain. What can you do to reduce the chances of making this type of error?

step 1

If we rejected the null hypothesis incorrectly, we made a Type I error

step 2

If we failed to reject the null hypothesis incorrectly, we made a Type II error

step 3

To reduce Type I error, we can lower the significance level

step 4

To reduce Type II error, we can increase the sample size

Answer

If we rejected the null hypothesis incorrectly, we made a Type I error. If we failed to reject it incorrectly, we made a Type II error. To reduce Type I error, lower the significance level; to reduce Type II error, increase the sample size.

Key Concept

Hypothesis Testing

Explanation

Hypothesis testing involves making decisions about population parameters based on sample data, with potential errors being Type I (false positive) and Type II (false negative).