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Chapter 7: Problem 31

The proportion of deaths due to lung cancer in males ages \(15-64\) in England and Wales during the period \(1970-\) 1972 was 12\%. Suppose that of 20 deaths that occur among male workers in this age group who have worked for at least 1 year in a chemical plant, 5 are due to lung cancer. We wish to determine whether there is a difference between the proportion of deaths from lung cancer in this plant and the proportion in the general population. Is a one-sided or two-sided test appropriate here?

Short Answer

Expert verified
A two-sided test is appropriate because we are testing for any difference, not a specific direction.

Step by step solution

01

Define the Hypotheses

First, we need to define the null and alternate hypotheses. The null hypothesis \(H_0\) states that the proportion of deaths due to lung cancer in the chemical plant \(p\) is equal to the proportion in the general population \(p_0 = 0.12\). The alternate hypothesis \(H_1\) suggests that \(p\) is different from \(p_0\), thus: \[ H_0: p = 0.12 \] \[ H_1: p eq 0.12 \] Since we are comparing whether the actual proportion is different (not specifically less or more), we consider a two-sided test.
02

Determine Test Type

A two-sided test is appropriate for this scenario. We are interested in determining any difference in the proportions, not a specific direction of difference (i.e., we are not specifying if the proportion in the plant is strictly higher or lower than that in the general population, just that it is different). This scenario implies deviations can be in either direction, thus a two-sided test is needed.

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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 the starting point in statistical hypothesis testing. It is a statement that assumes there is no effect or no difference between groups or variables in a study. In this exercise, the null hypothesis (\( H_0 \)) claims that the proportion of deaths due to lung cancer among male workers in a chemical plant is the same as the general population in England and Wales, which was 12% during 1970-1972.
So, mathematically:
  • Null Hypothesis (\( H_0 \)): \( p = 0.12 \)
This hypothesis presents a baseline we test against to see if the data provides enough evidence to reject it. If the data significantly differs from the null hypothesis, we can reject it in favor of the alternate hypothesis. To make these determinations, a test statistic is calculated, and its results tell us about the likelihood of the null hypothesis being true.
Alternate Hypothesis
The alternate hypothesis (\( H_1 \)) contradicts the null hypothesis. It is a statement that proposes a difference or effect exists. In contexts like this, we are looking to see if there's convincing evidence to support a proportion different from the specified null hypothesis.
In this exercise, the alternate hypothesis asserts that the proportion of lung cancer deaths in the chemical plant is different from the 12% of the general population. This leads us to write it as:
  • Alternate Hypothesis (\( H_1 \)): \( p eq 0.12 \)
Serving as the change-hypothesis, it indicates that the observed data will show significant evidence for a difference, or deviation, from what is assumed under our null hypothesis. It's crucial for determining the structure of statistical tests.
Two-Sided Test
A two-sided test is a specific method in statistical hypothesis testing used to measure if there is any difference, without indicating a particular direction. In our exercise, we apply a two-sided test because we are not simply looking for evidence that the proportion of lung cancer deaths in the chemical plant is higher or lower than the general level. Instead, we want to see if there is any difference at all.
Using a two-sided test involves determining whether the observed instances fall outside the range of values predicted by the null hypothesis on either side. It's employed when differences can occur in either direction, thus testing for the possibility that the parameter is either higher or lower than the specified value. This makes it a broader and more comprehensive statistical testing method.
Proportion Analysis
Proportion analysis plays a crucial role in comparing the ratios or proportions of particular outcomes between two datasets. Here, we analyze the proportion of lung cancer deaths in the context of the chemical plant versus the broader population.
First, you compute the proportion of observed cases. For instance, the plant has 5 out of 20 deaths related to lung cancer, which results in a proportion (\( \hat{p} \)) of 0.25 or 25%.
Analyzing the data involves:
  • Comparing the sample proportion (\( \hat{p} \)) to the population proportion (\( p_0 \)).
  • Using hypothesis testing to decide if the observed data is statistically significant.
  • Determining if the difference is due to random variation or an actual effect.
Through proportion analysis, we can intellectually assert if our sample represents a deviation from the expected norm, given the conditions set forth by our null hypothesis.

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Most popular questions from this chapter

Suppose we wish to test the hypothesis \(H_{0}: \mu=2\) vs. \(H_{1}: \mu \neq 2 .\) We find a two-sided \(p\) -value of .03 and \(a 95 \%\) Cl for \(\mu\) of \((1.5,4.0) .\) Are these two results possibly compatible? Why or why not?

Use a computer program to compute the probability that a \(t\) distribution with 36 df exceeds 2.5.

The mortality experience of 8146 male employees of a research, engineering, and metal-fabrication plant in Tonawanda, New York, was studied from 1946 to 1981 [2]. Potential workplace exposures included welding fumes, cutting oils, asbestos, organic solvents, and environmental ionizing radiation, as a result of waste disposal during the Manhattan Project of World War II. Comparisons were made for specific causes of death between mortality rates in workers and U.S. white-male mortality rates from 1950 to 1978 . Suppose that 17 deaths from cirrhosis of the liver were observed among workers who were hired prior to 1946 and who had worked in the plant for 10 or more years, whereas 6.3 were expected based on U.S. white-male mortality rates. What is the SMR for this group?

A clinical epidemiologic study was conducted to determine the long-term health effects of workplace exposure to the process of manufacturing the herbicide ( 2,4,5 trichlorophenoxy) acetic acid \((2,4,5-\mathrm{T}),\) which contains the contaminant dioxin [7]. This study was conducted among active and retired workers of a Nitro, West Virginia, plant who were exposed to the \(2,4,5-T\) process between 1948 and 1969 . It is well known that workers exposed to 2,4,5 -T have high rates of chloracne (a generalized acneiform eruption). Less well known are other potential effects of \(2,4,5-T\) exposure. One of the variables studied was pulmonary function. Suppose the researchers expect from general population estimates that \(5 \%\) of workers have an abnormal forced expiratory volume (FEV); defined as less than \(80 \%\) of predicted, based on their age and height. They found that 32 of 203 men who were exposed to \(2,4,5-T\) while working at the plant had an abnormal FEV. What hypothesis test can be used to test the hypothesis that the percentage of abnormal FEV values among exposed men differs from the general-population estimates?

Researchers have reported that the incidence rate of cataracts may be elevated among people with excessive exposure to sunlight. To confirm this, a pilot study is conducted among 200 people ages \(65-69\) who report an excessive tendency to bum on exposure to sunlight. Of the 200 people, 4 develop cataracts over a 1-year period. Suppose the expected incidence rate of cataracts among \(65-\) to 69 -year olds is \(1 \%\) over a 1 -year period. What test procedure can be used to compare the 1-year rate of cataracts in this population with that in the general population?
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