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Chapter 4: Problem 11

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 the percentage of a population who watch the Home Shopping Network is less than \(20 \%\)

Short Answer

Expert verified
The parameter needed for the problem is the proportion of the population that watches the Home Shopping Network, denoted as \(p\). The null hypothesis \(H_0 : p = 0.20\) states that 20% of the population watches the Home Shopping Network while the alternative hypothesis \(H_a : p < 0.20\) states that less than 20% of the population watches the Home Shopping Network.

Step by step solution

01

Identify the parameter

The parameter for this problem is the proportion of the population who watches the Home Shopping Network, denoted here as \(p\).
02

State the Null Hypothesis

The null hypothesis, \(H_0\), often claims that there is no effect or difference in the population. In this case, the null hypothesis is that the percentage of the population who watches the Home Shopping Network is 20%. Mathematically, its defined as \(H_0 : p = 0.20\).
03

State the Alternative Hypothesis

The alternative hypothesis, \(H_a\), is a claim about the population that is contradictory to \(H_0\) and what we conclude when the data provides sufficient evidence against \(H_0\). In this case, the alternative hypothesis is that the percentage of the population who watches the Home Shopping Network is less than 20%. So, it can be represented as \(H_a : p < 0.20\).

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Key Concepts

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

Null Hypothesis
In statistical hypothesis testing, the null hypothesis is a fundamental starting point for any analysis. It represents a default position that there is no effect or no difference and serves as a benchmark for evaluating evidence from data.
The null hypothesis, usually denoted as \(H_0\), assumes that any observed effect is due to chance until proven otherwise. In our scenario with the Home Shopping Network:
  • The null hypothesis assumes that 20% of the population watches this network. Mathematically, this assumption is expressed as \(H_0: p = 0.20\).
  • This implies that without further evidence, we would work under the assumption that the proportion remains the same.
Deciding on a null hypothesis is crucial as it directly impacts how statistical tests are interpreted and the conclusions that arise from the data analysis.
Alternative Hypothesis
The alternative hypothesis represents what we are looking to prove or find evidence for in our statistical test. When evidence suggests that the null hypothesis is not plausible, we consider the alternative hypothesis, denoted as \(H_a\).
It is the claim that researchers hope to support with data.
In the Home Shopping Network example:
  • The alternative hypothesis states that fewer people, less than 20%, watch the Home Shopping Network, symbolized by \(H_a: p < 0.20\).
  • It offers a direction for the research, often positing that an effect does indeed exist.
Understanding the alternative hypothesis is essential because it defines the potential changes and effects we want to test and eventually affirm through our data analysis.
Population Parameter
A population parameter is a value that gives information about an entire population, often estimated using sample data. It provides a means to summarize a population's characteristic in a single number.
In our statistical test, the population parameter of focus is the proportion \(p\), which reflects the percentage of people in the population watching the Home Shopping Network.
Here's why this is important:
  • It allows a robust definition of what we're studying, in this case, a particular viewing habit in a population.
  • Knowing the population parameter is critical for constructing accurate hypotheses.
  • It guides the choice of statistical test and influences how the results will be interpreted with respect to wider population conclusions.
Thus, the population parameter forms the numerical foundation upon which all subsequent analysis in hypothesis testing is built.

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Most popular questions from this chapter

Radiation from Cell Phones and Brain Activity Does heavy cell phone use affect brain activity? There is some concern about possible negative effects of radiofrequency signals delivered to the brain. In a randomized matched-pairs study, \(^{24}\) 47 healthy participants had cell phones placed on the left and right ears. Brain glucose metabolism (a measure of brain activity) was measured for all participants under two conditions: with one cell phone turned on for 50 minutes (the "on" condition) and with both cell phones off (the "off" condition). The amplitude of radiofrequency waves emitted by the cell phones during the "on" condition was also measured. (a) Is this an experiment or an observational study? Explain what it means to say that this was a "matched-pairs" study. (b) How was randomization likely used in the study? Why did participants have cell phones on their ears during the "off" condition? (c) The investigators were interested in seeing whether average brain glucose metabolism was different based on whether the cell phones were turned on or off. State the null and alternative hypotheses for this test. (d) The p-value for the test in part (c) is 0.004 . State the conclusion of this test in context. (e) The investigators were also interested in seeing if brain glucose metabolism was significantly correlated with the amplitude of the radiofrequency waves. What graph might we use to visualize this relationship? (f) State the null and alternative hypotheses for the test in part (e). (g) The article states that the p-value for the test in part (e) satisfies \(p<0.001\). State the conclusion of this test in context.

In Exercises 4.107 to \(4.111,\) null and alternative hypotheses for a test are given. Give the notation \((\bar{x},\) for example) for a sample statistic we might record for each simulated sample to create the randomization distribution. $$ H_{0}: \rho=0 \text { vs } H_{a}: \rho \neq 0 $$

In Exercises 4.150 to \(4.152,\) a confidence interval for a sample is given, followed by several hypotheses to test using that sample. In each case, use the confidence interval to give a conclusion of the test (if possible) and also state the significance level you are using. A \(95 \%\) confidence interval for \(p: 0.48\) to 0.57 (a) \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (b) \(H_{0}: p=0.75\) vs \(H_{a}: p \neq 0.75\) (c) \(H_{0}: p=0.4\) vs \(H_{a}: p \neq 0.4\)

Exercises 4.117 to 4.122 give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.7\) vs \(H_{a}: p<0.7\) Sample data: \(\hat{p}=125 / 200=0.625\) with \(n=200\)

Indicate whether the analysis involves a statistical test. If it does involve a statistical test, state the population parameter(s) of interest and the null and alternative hypotheses. Polling 1000 people in a large community to determine if there is evidence for the claim that the percentage of people in the community living in a mobile home is greater than \(10 \%\)
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