Question

Lead Levels in Blood A study reported in the American Journal of Public Health (Science News) followed blood lead levels in handgun hobbyists using indoor firing ranges. \({ }^{6}\) Lead exposure measurements were made on 17 members of a law enforcement trainee class before, during, and after a 3 -month period of firearm instruction at an indoor firing range. No trainee had elevated blood lead levels before the training, but 15 of the 17 ended their training with blood lead levels deemed "elevated." If use of the indoor firing range causes an increase in a person's blood lead levels, then \(p\), the probability that a person's lead level increases will be greater than .5. Use the sign test to determine whether using an indoor firing range has the effect of increasing a person's blood lead level with \(\alpha=.05 .\) (HINT: The normal approximation to binomial probabilities is fairly accurate for \(n=17 .)\)

Short answer

Answer: Yes, using an indoor firing range significantly increases a person's blood lead level.

Step-by-step solution

State the hypotheses

Write the null and alternative hypotheses: $$ H_0: p \leq 0.5 $$ $$ H_a: p > 0.5 $$ where \(p\) is the probability that a person's blood lead level increases.

Calculate the test statistic

In this case, the test statistic is the number of successes (trainees with increased blood lead levels): \(X = 15\) out of \(n = 17\) trainees.

Determine the critical region

We are using the normal approximation to the binomial distribution. The mean and the standard deviation of a binomial distribution with parameters \(n = 17\) and \(p = 0.5\) can be calculated as follows: Mean (\(\mu\)): \(\mu = np = 17 \times 0.5 = 8.5\) Standard deviation (\(\sigma\)): \(\sigma = \sqrt{np(1-p)} = \sqrt{17 \times 0.5 \times 0.5} \approx 2.07\) Now, find the corresponding \(z\)-score for the significance level \(\alpha = 0.05\) (one-tailed test) from the standard normal distribution table: \(z_\alpha = 1.645\) Calculate the critical value for the number of successes (\(X_c\)) as follows: $$ X_c = \mu + z_\alpha \times \sigma \approx 8.5 + 1.645 \times 2.07 \approx 12.00 $$ Since we are dealing with whole numbers, round up to the nearest whole number: \(X_c = 13\) If the number of successes, \(X\), is greater than or equal to the critical value, \(X_c\), we will reject the null hypothesis.

Make the decision

Compare the test statistic, \(X = 15\), with the critical value, \(X_c = 13\). Since \(X > X_c\), we reject the null hypothesis \(H_0\).

State the conclusion

We have significant evidence at the \(\alpha = 0.05\) level to reject the null hypothesis and conclude that using an indoor firing range has the effect of increasing a person's blood lead level.

Key concepts

Understanding Probability in Binomial Distributions

Probability is a measure of the likelihood of a particular event occurring. In the context of binomial distributions, which involve scenarios with two possible outcomes (success or failure), probability plays a central role. For example, the chance of a trainee at a firing range having increased blood lead levels due to exposure is represented by a probability, denoted as \(p\).

When we focus on such binary events and perform many trials, we can compile the results into a binomial distribution -- a type of distribution that reflects the number of successes in a fixed number of independent trials, given the same probability of success in each trial. To apply this to real-world scenarios, like the firing range case study involving lead exposure, students must ask questions such as 'What is the probability of a success?', and consider the implications when this rate is unexpectedly high or low. Did an external factor, like increasing lead in the environment due to the firing range, influence these probabilities? This is where statistical methods, such as hypothesis testing, can provide some answers.

Decoding Hypothesis Testing with the Sign Test

Hypothesis testing is a systematic method used to decide if the data at hand sufficiently supports a particular hypothesis. The sign test is one of the simplest forms of hypothesis tests which can be used when dealing with a binomial distribution.

At its core, this test involves formulating two hypotheses: the null hypothesis (\(H_0\)), which typically represents the status quo or a baseline belief, and the alternative hypothesis (\(H_a\)), which suggests that there is an effect or a difference. In our example, the null hypothesis posits that the indoor firing range does not increase blood lead levels, while the alternative hypothesis claims the opposite.

To conduct the test, we calculate a test statistic that reflects our observed data. In the lead exposure study, the test statistic is the number of trainees with increased blood lead levels. The decision about which hypothesis is more likely true comes by comparing this test statistic to a critical value derived from probability theory. If our observed statistic is extreme enough (in the sense it falls into the critical region), we then have grounds to reject the null hypothesis, favoring the idea that the firing range indeed affects blood lead levels.

Normal Approximation to Binomial – When and How

The normal approximation to binomial distributions is a helpful tool when dealing with a large number of trials, n, or when calculation of exact binomial probabilities is complex. Under certain conditions, this approximation allows us to use the properties of the normal distribution to estimate probabilities for a binomial distribution.

For the normal approximation to be appropriate, we typically look for two things: the sample size should be large enough (a common rule is that both \(np\) and \(n(1-p)\) should be greater than 5), and the probability of success should be reasonably balanced (not too close to 0 or 1). In the exercise regarding blood lead levels, the sample size of 17 meets the condition for using the normal approximation.

Applying the normal approximation involves calculating the mean (\(\text{expected number of successes}\)) and standard deviation of the binomial distribution, then converting our binomial question into a z-score question in the context of the normal distribution. In our exercise, this is crucial for determining whether the observed success count is high enough to be considered statistically significant, or if it could be attributed to chance.