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Chapter 9: Problem 1

\(\mathbf{H}_{0}\) or \(\mathbf{H}_{a}\) ? For parts a and \(\mathrm{b},\) is the statement a null hypothesis, or an alternative hypothesis? a. In Canada, the proportion of adults who favor legalized gambling equals \(0.50 .\) b. The proportion of all Canadian college students who are regular smokers is less than 0.24 , the value it was 10 years ago. c. Introducing notation for a parameter, state the hypotheses in parts a and \(\mathrm{b}\) in terms of the parameter values.

Short Answer

Expert verified
a. Null hypothesis (\( H_0 \)). b. Alternative hypothesis (\( H_a \)). c. \( H_0: p = 0.50 \); \( H_a: p < 0.24 \).

Step by step solution

01

Understanding Null vs. Alternative Hypothesis

The null hypothesis, denoted as \( H_0 \), is a statement that there is no effect or no difference and is usually a statement of equality (e.g., \( p = 0.50 \)). The alternative hypothesis, denoted as \( H_a \), is the statement that there is an effect or a difference, often expressed as an inequality (e.g., \( p eq 0.50 \) or \( p < 0.24 \)).
02

Identify Hypothesis for Part a

In part a, the statement "the proportion of adults who favor legalized gambling equals \(0.50\)" is saying that the proportion is exactly 0.50. This statement indicates no change or difference from this proportion, making it a null hypothesis \( H_0: p = 0.50 \).
03

Identify Hypothesis for Part b

In part b, the statement specifies that the proportion "is less than 0.24". This indicates an expectation of a change or difference from the value it was 10 years ago, making it an alternative hypothesis \( H_a: p < 0.24 \).
04

State Hypotheses in Terms of Parameters for Part a

For part a, define the parameter \( p \) as the proportion of adults who favor legalized gambling. The null hypothesis is \( H_0: p = 0.50 \).
05

State Hypotheses in Terms of Parameters for Part b

For part b, let \( p \) represent the proportion of Canadian college students who are regular smokers. The alternative hypothesis is \( H_a: p < 0.24 \).

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis is the foundational statement that is tested to determine if there is enough evidence to reject it. It is symbolized as \( H_0 \) and typically implies that there is no significant effect or difference. In simpler terms, it is a claim that an existing condition remains unchanged.

For example, if you want to test if a new teaching method has made any difference in students' performance, \( H_0 \) would state that the mean test score under this new method is the same as the previous method, mathematically expressed as \( \mu_{new} = \mu_{old} \).

In the original exercise, part a states "the proportion of adults who favor legalized gambling equals \(0.50\)". This statement is typical of a null hypothesis because it asserts equality. Thus, it is expressed as \( H_0: p = 0.50 \), claiming no change in the proportion.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_a \), provides a statement that contradicts the null hypothesis. It suggests that there is an effect or a difference and it often includes inequality. This is the statement we try to prove to be true through statistical testing.

Consider a situation where a company wants to check if a new marketing strategy has increased its sales. The \( H_a \) could be stated as \( \mu_{new} > \mu_{old} \), indicating that the sales are greater with the new strategy.

In the context of the original exercise, part b suggests "the proportion of all Canadian college students who are regular smokers is less than 0.24". This indicates a change from what it was 10 years ago and aligns with typical forms of alternative hypotheses. Hence, \( H_a: p < 0.24 \) is concluded, meaning there's a belief that the proportion of smokers has decreased.
Proportion in Statistics
Proportion in statistics refers to the fraction or percentage of a whole that exhibits a certain attribute. It's a type of measure that shows how many parts of the total population or sample possess a particular characteristic.

When conducting hypothesis testing on proportions, the goal is to assess whether a sample's proportion mirrors that of the population. This can be expressed mathematically through the parameter \( p \).

For instance, if a researcher claims that 60% of high school students own a smartphone, and this is stated as \( p = 0.60 \), hypothesis testing might be conducted to determine if this claim holds true in the population.

In the provided exercise, proportions are used as the parameter under investigation. For example, the proportion of adults favoring legalized gambling is posited to be 0.50, expressed as \( p = 0.50 \). Hypothesis testing will help to validate or refute these claims about population proportions.

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Most popular questions from this chapter

Gender bias in selecting managers Exercise 9.19 tested the claim that female employees were passed over for management training in favor of their male colleagues. Statewide, the large pool of more than 1000 eligible employees who can be tapped for management training is \(40 \%\) female and \(60 \%\) male. Let \(p\) be the probability of selecting a female for any given selection. For testing \(\mathrm{H}_{0}: p=0.40\) against \(\mathrm{H}_{a}: p<0.40\) based on a random sample of 50 selections, using the 0.05 significance level, verify that: a. A Type II error occurs if the sample proportion falls less than 1.645 standard errors below the null hypothesis value, which means that \(\hat{p}>0.286\). b. When \(p=0.20,\) a Type II error has probability 0.06 .

Alabama GPA Suppose the mean GPA of all students graduating from the University of Alabama in 1985 was 3.05. The registrar plans to look at records of students graduating in 2011 to see if mean GPA has changed. Define notation and state the null and alternative hypotheses for this investigation.

Men at work When the 636 male workers in the 2008 GSS were asked how many hours they worked in the previous week, the mean was 45.5 with a standard deviation of 15.16 . Does this suggest that the population mean work week for men exceeds 40 hours? Answer by: a. Identifying the relevant variable and parameter. b. Stating null and alternative hypotheses. c. Reporting and interpreting the P-value for the test statistic value of \(t=9.15\). d. Explaining how to make a decision for the significance level of 0.01

Fishing for significance A marketing study conducts 60 significance tests about means and proportions for several groups. Of them, 3 tests are statistically significant at the 0.05 level. The study's final report stresses only the tests with significant results, not mentioning the other 57 tests. What is misleading about this?

Dogs and cancer A recent study \(^{6}\) considered whether dogs could be trained to detect if a person has lung cancer or breast cancer by smelling the subject's breath. The researchers trained five ordinary household dogs to distinguish, by scent alone, exhaled breath samples of 55 lung and 31 breast cancer patients from those of 83 healthy controls. A dog gave a correct indication of a cancer sample by sitting in front of that sample when it was randomly placed among four control samples. Once trained, the dogs' ability to distinguish cancer patients from controls was tested using breath samples from subjects not previously encountered by the dogs. (The researchers blinded both dog handlers and experimental observers to the identity of breath samples.) Let \(p\) denote the probability a dog correctly detects a cancer sample placed among five samples, when the other four are controls. a. Set up the null hypothesis that the dog's predictions correspond to random guessing. b. Set up the alternative hypothesis to test whether the probability of a correct selection differs from random guessing. c. Set up the alternative hypothesis to test whether the probability of a correct selection is greater than with random guessing. d. In one test with 83 Stage I lung cancer samples, the dogs correctly identified the cancer sample 81 times. The test statistic for the alternative hypothesis in part \(c\) was \(z=17.7\). Report the P-value to three decimal places, and interpret. (The success of dogs in this study made researchers wonder whether dogs can detect cancer at an earlier stage than conventional methods such as MRI scans.)
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