In Exercises 55–60, find the indicated probabilities and interpret the results.


Refer to Exercise 34. A random sample of six days is selected. Find the probability that the mean surface concentration of carbonyl sulfide for the sample is (c) more than 11.1 picomoles per liter. Compare your answers with those in Exercise 34.

Verified step by step guidance

1. Identify the key components of the problem: the sample size (n = 6), the population mean (μ), the population standard deviation (σ), and the threshold value (11.1 picomoles per liter). These values should be referenced from Exercise 34.
2. Determine the sampling distribution of the sample mean. The mean of the sampling distribution is the same as the population mean (μ), and the standard deviation of the sampling distribution (standard error) is calculated as σ/√n, where n is the sample size.
3. Standardize the threshold value (11.1) to a z-score using the formula: z = (X̄ - μ) / (σ/√n), where X̄ is the threshold value, μ is the population mean, and σ/√n is the standard error.
4. Use the standard normal distribution table (or a statistical software) to find the probability corresponding to the calculated z-score. Since the problem asks for the probability that the mean is more than 11.1, calculate the area to the right of the z-score (1 - cumulative probability).
5. Compare the result with the probabilities calculated in Exercise 34 to interpret how the sample size affects the probability. Larger sample sizes typically result in smaller standard errors, which can influence the probabilities.

Key Concepts

Sampling Distribution

The sampling distribution is the probability distribution of a statistic (like the sample mean) obtained from a large number of samples drawn from a specific population. It describes how the sample mean varies from sample to sample and is crucial for understanding how to calculate probabilities related to sample means, especially when the sample size is small.

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for a sufficiently large sample size, the distribution of the sample mean will approximate a normal distribution, regardless of the population's distribution. This theorem is essential for calculating probabilities related to sample means, particularly when the sample size is small, as it allows us to use normal distribution properties.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. In this context, it involves determining whether the sample mean of carbonyl sulfide concentrations exceeds a specified value (11.1 picomoles per liter) and interpreting the significance of this result in relation to the null hypothesis.