/*! 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 2 You are testing that the mean sp... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 2

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

Short Answer

Expert verified
H_0: \mu \leq 3\; Mbps, \quad H_a: \mu > 3\; Mbps

Step by step solution

01

Identify the Problem Context

In hypothesis testing, we want to prove something about a population parameter using sample data. Here, we are interested in testing whether the mean speed of a cable Internet connection is more than 3 Megabits per second.
02

Define the Null Hypothesis

The null hypothesis (H_0) is a statement of no effect or no difference. We use it as a starting point for testing. In this case, the null hypothesis is that the mean speed of the Internet connection is less than or equal to 3 Megabits per second. Mathematically, it can be represented as:\(H_0: \mu \leq 3 \text{ Megabits per second}\)
03

Define the Alternative Hypothesis

The alternative hypothesis (H_a) is what you want to prove. It is the opposite of the null hypothesis. Here, we want to show that the mean speed is more than 3 Megabits per second. The alternative hypothesis is:\(H_a: \mu > 3 \text{ Megabits per second}\)

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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 a cornerstone in the process of hypothesis testing. It鈥檚 where we start our journey into understanding whether an assumption about a population parameter is valid or not. Imagine it as the hypothesis that claims "No, nothing's different or changed here." In this context, it's a presumption that there is no significant effect or relationship concerning the phenomenon you're examining.
  • The null hypothesis is commonly denoted by the symbol \(H_0\), establishing a starting point for statistical testing.
  • In our example with Internet speed, the null hypothesis states that the mean speed is 3 Megabits per second or less. This is pivotal because it forms the basis from which we test and compare the results of our sample data.
The essential role of the null hypothesis is to provide a claim that can either be rejected or not rejected based on the data. The aim of hypothesis testing is generally to challenge this status quo, testing sample data against this default explanation.
Alternative Hypothesis
The alternative hypothesis is the statement that we aim to find evidence for. When conducting hypothesis testing, this is the statement reflecting the change or difference we are investigating. Think of it as the challenger鈥攖he new idea we're testing to see if it holds water against the foundation of the null hypothesis.
  • Represented by \(H_a\), the alternative hypothesis is often what researchers or scientists hope is true. It aligns with the newer, bolder theory that鈥檚 being examined.
  • In our example situation, the alternative hypothesis posits that the mean Internet speed is greater than 3 Megabits per second, suggesting a significant improvement or change over the standard expectation.
Selecting the appropriate alternative hypothesis often aligns with the objectives of your research. It should be established before examining your data to avoid bias and add credibility to the study.
Population Parameter
A population parameter is a number that describes something about an entire group or population. It鈥檚 a critical aspect in statistics because it helps form connections between a sample that we can feasibly study and the larger population. In hypothesis testing, the population parameter is what you are trying to draw conclusions about using your sample data.
  • The population parameter might be a mean, proportion, variance, or another statistical measure, depending on what detail is crucial for your analysis.
  • In the testing of Internet speed, our parameter of interest is the mean speed of greater than 3 Megabits per second. This parameter sets the stage for framing our null and alternative hypotheses.
By focusing on the population parameter, hypothesis testing translates sample data into broader conclusions. It empowers decisions about whether the hypothesis related to the whole population should be rejected or accepted based on sample insights.

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Most popular questions from this chapter

The probability of winning the grand prize at a particular carnival game is 0.005. Is the outcome of winning very likely or very unlikely?

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"Macaroni and Cheese, please!!" by Nedda Misherghi and Rachelle Hall As a poor starving student I don't have much money to spend for even the bare necessities. So my favorite and main staple food is macaroni and cheese. It's high in taste and low in cost and nutritional value. One day, as I sat down to determine the meaning of life, I got a serious craving for this, oh, so important, food of my life. So I went down the street to Great way to get a box of macaroni and cheese, but it was SO expensive! 2.02 dollar !!! Can you believe it? It made me stop and think. The world is changing fast. I had thought that the mean cost of a box (the normal size, not some super-gigantic- family-value-pack) was at most 1 dollar, but now I wasn't so sure. However, I was determined to find out. I went to 53 of the closest grocery stores and surveyed the prices of macaroni and cheese. Here are the data I wrote in my notebook: Price per box of Mac and Cheese: \- 5 stores @ 2.02 dollar \- 15 stores @ 0.25 dollar \- 3 stores @ 1.29 dollar \- 6 stores @ 0.35 dollar \- 4 stores @ 2.27 dollar \- 7 stores @ 1.50 \- 5 stores @ 1.89 dollar \- 8 stores @ 0.75 . I could see that the cost varied but I had to sit down to figure out whether or not I was right. If it does turn out that this mouth-watering dish is at most 1 dollar, then I'll throw a big cheesy party in our next statistics lab, with enough macaroni and cheese for just me. (After all, as a poor starving student I can't be expected to feed our class of animals!)

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. At a significance level of \(a=0.05,\) what is the correct conclusion? a. There is enough evidence to conclude that the mumber of hours is more than 4.75 b. There is enough evidence to conclude that the mean number of hore than 4.5 c. There is not enough evidence to conclude that the mumber of hours is more than 4.5 d. There is not enough evidence to conclude that the mean number of hours is more than 4.75

The mean work week for engineers in a start-up company is believed to be about 60 hours. A newly hired engineer hopes that it's shorter. She asks ten engineering friends in start-ups for the lengths of their mean work weeks. Based on the results that follow, should she count on the mean work week to be shorter than 60 hours? Data (length of mean work week): 70; 45; 55; 60; 65; 55; 55; 60; 50; 55.
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