Question

Acid rain-rain with a pH value less than \(5.7,\) caused by the reaction of certain air pollutants with rainwater-is a growing problem in the United States. Suppose water samples from 40 rainfalls are analyzed for \(\mathrm{pH}\), and \(\bar{x}\) and \(s\) are equal to 3.7 and .5 , respectively. Find a \(99 \%\) confidence interval for the mean \(\mathrm{pH}\) in rainfall and interpret the interval. What assumption must be made for the confidence interval to be valid?

Short answer

Explain any assumptions you made. Answer: The 99% confidence interval for the mean pH in rainfall is (3.486, 3.914). We assume that the sampled rainfall pH values are normally distributed, or the sample size is large enough for the central limit theorem to apply, making the distribution of the sample mean approximately normal.

Step-by-step solution

Identify the formula for the confidence interval

The formula for the confidence interval using the t-distribution is: \(CI = \bar{x} \pm t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(t_{\frac{\alpha}{2}}\) is the t-distribution critical value, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Calculate the t-distribution critical value

To find the t-distribution critical value, we need to find the degrees of freedom and the significance level \(\alpha\). Degrees of freedom (\(df\)) = sample size - 1 = 40 - 1 = 39. Since the confidence level is \(99\%\), we have \(\alpha = 1 - 0.99 = 0.01\). Therefore, we need to find \(t_{0.005}\) (since it is a two-tailed test, we divide \(\frac{\alpha}{2} = 0.005\)) with 39 degrees of freedom. Using a t-distribution table or a calculator, we find that \(t_{0.005} = 2.707\).

Calculate the margin of error

Using the calculated t-distribution critical value, we can now find the margin of error. Margin of error = \(t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}} = 2.707\frac{0.5}{\sqrt{40}} = 0.214\)

Compute the confidence interval

Using the margin of error, we can calculate the confidence interval: \(CI = \bar{x} \pm \text{Margin of error} = 3.7 \pm 0.214 = (3.486, 3.914)\)

Interpret the confidence interval

We can interpret the confidence interval as follows: We are \(99\%\) confident that the true mean pH of rainfall in the population is between \(3.486\) and \(3.914\).

Identify the necessary assumption

The assumption that must be made for the confidence interval to be valid is that the sampled rainfall pH values are normally distributed (or the sample size is large enough for the central limit theorem to apply, which makes the distribution of the sample mean approximately normal).

Key concepts

T-Distribution

Understanding the t-distribution is crucial when working with smaller sample sizes or when the population standard deviation is unknown. It is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but with heavier tails. This means it is more prone to producing values that fall far from the mean.

As the sample size increases, the t-distribution approaches the normal distribution. The degrees of freedom (df), which is calculated as the sample size minus one, determines the shape of the t-distribution. In our example, with 39 degrees of freedom, the t-distribution will have heavier tails than the standard normal distribution.

For confidence intervals, the t-distribution is preferred over the normal distribution unless the sample size is large enough (typically over 30), in which case the two distributions are similar enough that the normal distribution can be used instead. The critical value from the t-distribution accounts for the added uncertainty of estimating the population standard deviation from the sample.

Sample Size

Sample size, often denoted as 'n', is the number of observations or data points that are selected from a population to be included in a sample. The sample size plays a significant role in statistical analysis.

In the context of confidence intervals, the sample size affects the width of the interval. Generally, a larger sample size will yield a smaller margin of error, which produces a more precise confidence interval. This is because larger samples tend to more accurately represent the population, reducing the variability and uncertainty of the estimate.

In our example, the sample size is 40, which falls just above the commonly used threshold of 30 for applying the central limit theorem. This number allows us to use the t-distribution with a reasonable degree of confidence that our interval will be an accurate representation of the true population parameter.

Margin of Error

The margin of error reflects the range of uncertainty in estimating a population parameter. In other words, it indicates how much we can expect the sample estimate to differ from the true population parameter.

The margin of error is influenced by several factors: the level of confidence we wish to have in our interval estimate, the variability of the data, and the size of our sample. In the solution to our exercise, the margin of error is calculated using the critical value from the t-distribution and the sample's standard deviation, revealing how precise our estimate of the mean pH value is.

With a 99% confidence interval, we choose a high level of confidence, which results in a wider margin of error. This means we can be more certain that our interval captures the true mean, but at the expense of a less precise interval. The calculated margin of error for the mean pH in rainfall was 0.214.

Normal Distribution Assumption

The normal distribution assumption is a foundational concept in creating confidence intervals and in statistical hypothesis testing. This assumption posits that the data are distributed in a bell-shaped pattern that is symmetric about the mean.

This assumption justifies the use of specific statistical techniques, such as the t-distribution for calculating confidence intervals. If the sample size is large enough, the central limit theorem comes into play, which states that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the distribution of the population.

In this exercise, we must assume that the rainfall pH values are normally distributed. If the sample size were smaller, this assumption would be more critical to validate, possibly with a normality test. Since the sample we are working with has 40 values, we are at the borderline where the central limit theorem would allow us to relax this assumption slightly, but for high accuracy, especially with small standard deviations or outliers, checking normality may still be advisable.