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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

Arsenic in Chicken Data 4.5 on page 228 discusses a test to determine if the mean level of arsenic in chicken meat is above 80 ppb. If a restaurant chain finds significant evidence that the mean arsenic level is above \(80,\) the chain will stop using that supplier of chicken meat. The hypotheses are $$ \begin{array}{ll} H_{0}: & \mu=80 \\ H_{a}: & \mu>80 \end{array} $$ where \(\mu\) represents the mean arsenic level in all chicken meat from that supplier. Samples from two different suppliers are analyzed, and the resulting p-values are given: Sample from Supplier A: p-value is 0.0003 Sample from Supplier B: p-value is 0.3500 (a) Interpret each p-value in terms of the probability of the results happening by random chance. (b) Which p-value shows stronger evidence for the alternative hypothesis? What does this mean in terms of arsenic and chickens? (c) Which supplier, \(\mathrm{A}\) or \(\mathrm{B}\), should the chain get chickens from in order to avoid too high a level of arsenic?

Watch Out for Lions after a Full Moon Scientists studying lion attacks on humans in Tanzania \(^{34}\) found that 95 lion attacks happened between \(6 \mathrm{pm}\) and \(10 \mathrm{pm}\) within either five days before a full moon or five days after a full moon. Of these, 71 happened during the five days after the full moon while the other 24 happened during the five days before the full moon. Does this sample of lion attacks provide evidence that attacks are more likely after a full moon? In other words, is there evidence that attacks are not equally split between the two five-day periods? Use StatKey or other technology to find the p-value, and be sure to show all details of the test. (Note that this is a test for a single proportion since the data come from one sample.)

Testing for a Home Field Advantage in Soccer In Exercise 3.108 on page \(215,\) we see that the home team was victorious in 70 games out of a sample of 120 games in the FA premier league, a football (soccer) league in Great Britain. We wish to investigate the proportion \(p\) of all games won by the home team in this league. (a) Use StatKey or other technology to find and interpret a \(90 \%\) confidence interval for the proportion of games won by the home team. (b) State the null and alternative hypotheses for a test to see if there is evidence that the proportion is different from 0.5 . (c) Use the confidence interval from part (a) to make a conclusion in the test from part (b). State the confidence level used. (d) Use StatKey or other technology to create a randomization distribution and find the p-value for the test in part (b). (e) Clearly interpret the result of the test using the p-value and using a \(10 \%\) significance level. Does your answer match your answer from part (c)? (f) What information does the confidence interval give that the p-value doesn't? What information does the p-value give that the confidence interval doesn't? (g) What's the main difference between the bootstrap distribution of part (a) and the randomization distribution of part (d)?

Guilty Verdicts in Court Cases A reporter on cnn.com stated in July 2010 that \(95 \%\) of all court cases that go to trial result in a guilty verdict. To test the accuracy of this claim, we collect a random sample of 2000 court cases that went to trial and record the proportion that resulted in a guilty verdict. (a) What is/are the relevant parameter(s)? What sample statistic(s) is/are used to conduct the test? (b) State the null and alternative hypotheses. (c) We assess evidence by considering how likely our sample results are when \(H_{0}\) is true. What does that mean in this case?

Hockey Malevolence Data 4.3 on page 224 describes a study of a possible relationship between the perceived malevolence of a team's uniforms and penalties called against the team. In Example 4.31 on page 270 we construct a randomization distribution to test whether there is evidence of a positive correlation between these two variables for NFL teams. The data in MalevolentUniformsNHL has information on uniform malevolence and penalty minutes (standardized as z-scores) for National Hockey League (NHL) teams. Use StatKey or other technology to perform a test similar to the one in Example 4.31 using the NHL hockey data. Use a \(5 \%\) significance level and be sure to show all details of the test.
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