/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 In Problems \(15-22,(a)\) determ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 21

In Problems \(15-22,(a)\) determine the null and alternative hypotheses, (b) explain what it would mean to make a Type I error, and (c) explain what it would mean to make a Type II error. According to the CTIA-The Wireless Association, the mean monthly revenue per cell phone was \(\$ 48.79\) in 2014 . A researcher suspects the mean monthly revenue per cell phone is different today.

Short Answer

Expert verified
H鈧�: \( \mu = 48.79 \), H鈧�: \( \mu eq 48.79 \). Type I error: reject true H鈧�. Type II error: fail to reject false H鈧�.

Step by step solution

01

- Specify the Null and Alternative Hypotheses

Define the null and alternative hypotheses for this problem. The null hypothesis (H鈧�) represents the status quo or a statement of no effect. The alternative hypothesis (H鈧�) represents a statement of effect or difference. In this case, we test whether the mean monthly revenue per cell phone is different from \(\$ 48.79\) today. H鈧�: \( \mu = 48.79 \) H鈧�: \( \mu eq 48.79 \)
02

- Explain Type I Error

A Type I error occurs when the null hypothesis is true, but we reject it. In this context, a Type I error means concluding that the mean monthly revenue per cell phone is different from \(\$ 48.79\) when it actually is \(\$ 48.79\).
03

- Explain Type II Error

A Type II error occurs when the null hypothesis is false, but we fail to reject it. In this context, a Type II error means concluding that the mean monthly revenue per cell phone is \(\$ 48.79\) when it actually is different from \(\$ 48.79\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis (denoted as \(H_0\)) is a statement asserting that there is no effect or no difference. It's often referred to as the status quo. For the given exercise, our null hypothesis is that the mean monthly revenue per cell phone is still \$48.79 today. This means that the researcher does not expect any changes in the mean revenue since 2014.
If you were to write this formally: \[H_0: \, \mu = 48.79\]
Alternative Hypothesis
The alternative hypothesis (denoted as \(H_1\) or \(H_a\)) is the statement you are trying to find evidence for. It represents the possibility of an effect or difference. In the given exercise, the researcher suspects that the mean monthly revenue per cell phone is different from \$48.79 today.
This can be represented formally as: \[H_1: \,\mu eq 48.79\]
The alternative hypothesis is essentially the opposite of the null hypothesis, offering an alternate theory against the status quo.
Type I Error
A Type I error occurs when the null hypothesis is true, but we mistakenly reject it. In the context of the given exercise, a Type I error would happen if we concluded that the mean monthly revenue per cell phone is different from \$48.79, when in fact, it is still \$48.79.
Type I error is also known as a 'false positive' because it falsely indicates a difference. The probability of making a Type I error is denoted by \(\alpha\), often set at 0.05 (or 5%).
For example:
  • Claiming a new drug works when it doesn鈥檛
  • Concluding there鈥檚 an economic shift when there isn鈥檛
Type II Error
A Type II error occurs when the null hypothesis is false, but we fail to reject it. In this exercise, a Type II error would mean concluding that the mean monthly revenue per cell phone is \$48.79 when, in reality, it is different.
Type II error is also known as a 'false negative' because it fails to detect a real difference. The probability of making a Type II error is denoted by \(\beta\).
For example:
  • Believing a drug does not work when it actually does
  • Failing to detect an economic shift when there is one
Mean Monthly Revenue
Mean monthly revenue is essentially the average amount of money earned per month per cell phone user. For the given exercise, the historical mean monthly revenue was \$48.79 in 2014. This serves as a benchmark for the researcher who suspects there might be a change today.
To calculate mean revenue, you sum up all the revenues for the month and divide by the number of cell phone users. In mathematical terms, if you have revenues \(R_1, R_2, ..., R_n\) for \(n\) users, the mean revenue (\(\mu\)) is:
\[\mu = \frac{R_1 + R_2 + ... + R_n}{n}\]
Understanding mean revenue helps businesses and researchers make important decisions based on data trends.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose an acquaintance claims to have the ability to determine the birth month of randomly selected individuals. To test such a claim, you randomly select 80 individuals and ask the acquaintance to state the birth month of the individual. If the individual has the ability to determine birth month, then the proportion of correct birth months should exceed \(\frac{1}{12},\) the rate one would expect from simply guessing. (a) State the null and alternative hypotheses for this experiment. (b) Suppose the individual was able to guess nine correct birth months. The \(P\) -value for such results is \(0.1726 .\) Explain what this \(P\) -value means and write a conclusion for the test.

Reading at Bedtime It is well-documented that watching TV, working on a computer, or any other activity involving artificial light can be harmful to sleep patterns. Researchers wanted to determine if the artificial light from e-Readers also disrupted sleep. In the study, 12 young adults were given either an iPad or printed book for four hours before bedtime. Then, they switched reading devices. Whether the individual received the iPad or book first was determined randomly. Bedtime was \(10 \mathrm{P.M}\). and the time to fall asleep was measured each evening. It was found that participants took an average of 10 minutes longer to fall asleep after reading on an iPad. The \(P\) -value for the test was \(0.009 .\) (a) What is the research objective? (b) What is the response variable? It is quantitative or qualitative? (c) What is the treatment? (d) Is this a designed experiment or observational study? What type? (e) The null hypothesis for this test would be that there is no difference in time to fall asleep with an e-Reader and printed book. The alternative is that there is a difference. Interpret the \(P\) -value.

Test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test. $$\begin{array}{l}H_{0}: p=0.6 \text { versus } H_{1}: p<0.6 \\\n=250 ; x=124 ; \alpha=0.01\end{array}$$

In April 2009 , the Gallup organization surveyed 676 adults aged 18 and older and found that 352 believed they would not have enough money to live comfortably in retirement. The folks at Gallup want to know if this represents sufficient evidence to conclude a majority (more than \(50 \%\) ) of adults in the United States believe they will not have enough money in retirement. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, determine the probability of making a Type II error if the true population proportion is 0.53. What is the power of the test? (c) Redo part (b) if the true proportion is 0.55 .

According to menstuff.org, \(22 \%\) of married men have "strayed" at least once during their married lives. (a) Describe how you might go about administering a survey to assess the accuracy of this statement. (b) A survey of 500 married men indicated that 122 have "strayed" at least once during their married life. Construct a \(95 \%\) confidence interval for the population proportion of married men who have strayed. Use this interval to assess the accuracy of the statement made by menstuff.org.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.