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Researchers interested in lead exposure due to car exhaust sampled the blood of 52 police officers subjected to constant inhalation of automobile exhaust fumes while working traffic enforcement in a primarily urban environment. The blood samples of these officers had an average lead concentration of \(124.32 \mu \mathrm{g} / \mathrm{l}\) and a SD of \(37.74 \mu \mathrm{g} / \mathrm{l} ;\) a previous study of individuals from a nearby suburb, with no history of exposure, found an average blood level concentration of \(35 \mu \mathrm{g} / \mathrm{l} .\) (a) Write down the hypotheses that would be appropriate for testing if the police officers appear to have been exposed to a different concentration of lead. (b) Explicitly state and check all conditions necessary for inference on these data. (c) Regardless of your answers in part (b), test the hypothesis that the downtown police officers have a higher lead exposure than the group in the previous study. Interpret your results in context.

Step by step solution

01

Define Hypotheses

We need to define the null and alternative hypotheses. The null hypothesis \(H_0\) states that the lead concentration in police officers is equal to that in the suburb group, i.e., \(H_0: \mu = 35 \, \mu \text{g/L}\). The alternative hypothesis \(H_a\) suggests that police officers have a higher lead concentration, i.e., \(H_a: \mu > 35 \, \mu \text{g/L}\).

02

Check Conditions for Inference

Before conducting a hypothesis test, we need to check if the conditions for inference are satisfied. The sample size is 52 police officers, which is greater than 30, so we can apply the Central Limit Theorem. We assume that the data is approximately normally distributed or the sample size is sufficiently large.

03

Calculate the Test Statistic

The test statistic for a one-sample t-test is calculated as \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \), where \(\bar{x} = 124.32\), \(\mu_0 = 35\), \(s = 37.74\), and \(n = 52\). Compute this as: \[ t = \frac{124.32 - 35}{37.74/\sqrt{52}} \approx 18.508. \]

04

Determine the p-value

Using a t-distribution with \(n-1 = 51\) degrees of freedom, we find the p-value for the calculated t-statistic of 18.508. This value is extremely small, indicating a highly significant result.

05

Make a Decision

Given the p-value is much less than any common significance level (e.g., 0.05), we reject the null hypothesis \(H_0\). This indicates strong evidence that the police officers have a higher average lead concentration in their blood than the suburb group.

06

Interpret the Results

The analysis shows statistically significant evidence that the blood lead levels of police officers working in urban areas are higher than those from the nearby suburb. This suggests increased exposure to lead, likely due to automobile exhaust.

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Key Concepts

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

Lead Exposure

Lead exposure happens when we come in contact with or inhale lead particles. It's common in environments with heavy vehicular traffic, as lead was once a significant component of gasoline. Newer means of transport have reduced this, but traces still exist in areas with high traffic volume. The study focused on police officers in urban settings to examine the effects of ongoing exposure to lead.
Lead can gradually accumulate in the body, especially affecting individuals regularly exposed to environments rich in exhaust fumes. This highlights the importance of monitoring groups like traffic enforcement officers working alongside these pollutants.

• Lead accumulation can affect various bodily functions, including the nervous and circulatory systems.
• With prolonged exposure, the risk of serious health complications increases.
• Regular monitoring of blood lead levels in high-risk groups is essential for timely intervention.

Central Limit Theorem

The Central Limit Theorem (CLT) is pivotal in statistics and vital when analyzing datasets, like the officers' blood lead levels. Essentially, the CLT posits that the distribution of the sample means will approximate a normal distribution as the sample size becomes large, regardless of the original distribution of the data.
In this study, we have a sample size of 52 police officers. This size is essential because it surpasses the conventional threshold of 30, which often justifies the application of the CLT. Consequently, even if we don't know the population distribution, we can assume the distribution of their lead levels approaches normality, which is robust for making inferences.

• The CLT enables us to make predictions and inferences about the population based on sample data.
• It is a foundational principle that underlines much of statistical hypothesis testing.

T-Distribution

T-distribution is instrumental in hypothesis testing when dealing with small sample sizes or unknown population standard deviations. When we test for the mean difference in lead levels between police officers and the non-exposed group, we use a t-test because the population standard deviation is unknown, and our sample is finite.
The t-distribution resembles a normal distribution but has larger "tails," allowing for greater variability expected within smaller samples. For our study, we compute a t-statistic to see how far our sample mean deviates from the expected mean under the null hypothesis.

• The calculated t-statistic of 18.508 shows how significantly our sample mean deviates relative to our hypothesized mean.
• The degrees of freedom, 51 in this case, influence the t-distribution shape, informing the critical value for hypothesis testing.
• Tools like the t-distribution allow researchers to determine the probability of obtaining our data under the null hypothesis.

Blood Lead Levels

Blood lead levels indicate the concentration of lead in the bloodstream. These values are typically measured in micrograms per liter ( ext{µg/L}). The study in question focused on police officers, revealing they had an average blood lead level significantly higher than those in a suburb with no known exposure.
Understanding blood lead levels is crucial to assessing environmental and occupational hazards. Elevated levels, like those found in urban officers, suggest extra sources of exposure, possibly due to factors like proximity to vehicular exhaust.

• Averages and standard deviations were used to summarize and infer differences between groups.
• High blood lead levels can lead to significant health concerns and potential regulatory intervention.
• The results demonstrated in this study highlight the need for regular blood tests and preventive measures for individuals in high-risk areas.