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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

Weight Loss Program Suppose that a weight loss company advertises that people using its program lose an average of 8 pounds the first month and that the Federal Trade Commission (the main government agency responsible for truth in advertising) is gathering evidence to see if this advertising claim is accurate. If the FTC finds evidence that the average is less than 8 pounds, the agency will file a lawsuit against the company for false advertising. (a) What are the null and alternative hypotheses the FTC should use? (b) Suppose that the FTC gathers information from a very large random sample of patrons and finds that the average weight loss during the first month in the program is \(\bar{x}=7.9\) pounds with a p-value for this result of \(0.006 .\) What is the conclusion of the test? Are the results statistically significant? (c) Do you think the results of the test are practically significant? In other words, do you think patrons of the weight loss program will care that the average is 7.9 pounds lost rather than 8.0 pounds lost? Discuss the difference between practical significance and statistical significance in this context.

Multiple Sclerosis and Sunlight It is believed that sunlight offers some protection against multiple sclerosis (MS) since the disease is rare near the equator and more prevalent at high latitudes. What is it about sunlight that offers this protection? To find out, researchers \({ }^{15}\) injected mice with proteins that induce a condition in mice comparable to MS in humans. The control mice got only the injection, while a second group of mice were exposed to UV light before and after the injection, and a third group of mice received vitamin D supplements before and after the injection. In the test comparing UV light to the control group, evidence was found that the mice exposed to UV suppressed the MS-like disease significantly better than the control mice. In the test comparing mice getting vitamin D supplements to the control group, the mice given the vitamin \(\mathrm{D}\) did not fare significantly better than the control group. If the p-values for the two tests are 0.472 and 0.002 , which p-value goes with which test?

Mercury Levels in Fish Figure 4.26 shows a scatterplot of the acidity (pH) for a sample of \(n=53\) Florida lakes vs the average mercury level (ppm) found in fish taken from each lake. The full dataset is introduced in Data 2.4 on page 68 and is available in FloridaLakes. There appears to be a negative trend in the scatterplot, and we wish to test whether there is significant evidence of a negative association between \(\mathrm{pH}\) and mercury levels. (a) What are the null and alternative hypotheses? (b) For these data, a statistical software package produces the following output: $$ r=-0.575 \quad p \text { -value }=0.000017 $$ Use the p-value to give the conclusion of the test. Include an assessment of the strength of the evidence and state your result in terms of rejecting or failing to reject \(H_{0}\) and in terms of \(\mathrm{pH}\) and mercury. (c) Is this convincing evidence that low pH causes the average mercury level in fish to increase? Why or why not?

Penalty Shots in Soccer A recent article noted that it may be possible to accurately predict which way a penalty-shot kicker in soccer will direct his shot. \({ }^{23}\) The study finds that certain types of body language by a soccer player-called "tells"-can be accurately read to predict whether the ball will go left or right. For a given body movement leading up to the kick, the question is whether there is strong evidence that the proportion of kicks that go right is significantly different from one-half. (a) What are the null and alternative hypotheses in this situation? (b) If sample results for one type of body movement give a p-value of \(0.3184,\) what is the conclusion of the test? Should a goalie learn to distinguish this movement? (c) If sample results for a different type of body movement give a p-value of \(0.0006,\) what is the conclusion of the test? Should a goalie learn to distinguish this movement?

Are You "In a Relationship"? A new study \(^{45}\) shows that relationship status on Facebook matters to couples. The study included 58 college-age heterosexual couples who had been in a relationship for an average of 19 months. In 45 of the 58 couples, both partners reported being in a relationship on Facebook. In 31 of the 58 couples, both partners showed their dating partner in their Facebook profile picture. Men were somewhat more likely to include their partner in the picture than vice versa. However, the study states: "Females' indication that they are in a relationship was not as important to their male partners compared with how females felt about male partners indicating they are in a relationship." Using a population of college-age heterosexual couples who have been in a relationship for an average of 19 months: (a) A \(95 \%\) confidence interval for the proportion with both partners reporting being in a relationship on Facebook is about 0.66 to \(0.88 .\) What is the conclusion in a hypothesis test to see if the proportion is different from \(0.5 ?\) What significance level is being used? (b) A 95\% confidence interval for the proportion with both partners showing their dating partner in their Facebook profile picture is about 0.40 to 0.66. What is the conclusion in a hypothesis test to see if the proportion is different from \(0.5 ?\) What significance level is being used?
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