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


Refer to Exercise 33. A random sample of 2 years is selected. Find the probability that the mean amount of greenhouse gases for the sample is (a) less than 5500 MMT CO2 eq. Compare your answers with those in Exercise 33.

Verified step by step guidance

1. Identify the key information from the problem: the population mean (μ), population standard deviation (σ), sample size (n), and the value for which the probability is to be calculated (5500 MMT CO2 eq). These values should be referenced from Exercise 33.
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 value 5500 MMT CO2 eq to a z-score using the formula: z = (X̄ - μ) / (σ/√n), where X̄ is the sample mean (5500 in this case), μ is the population mean, and σ/√n is the standard error.
4. Use the z-score obtained in the previous step to find the cumulative probability from the standard normal distribution table or a statistical software. This cumulative probability represents the probability that the sample mean is less than 5500 MMT CO2 eq.
5. Compare the calculated probability with the results from Exercise 33 to interpret how the probability changes when considering the sample mean instead of individual values.

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.

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for a sufficiently large sample size, the distribution of the sample mean will be approximately normally distributed, regardless of the population's distribution. This theorem allows statisticians to make inferences about population parameters using sample statistics, particularly when calculating probabilities.

Z-Scores and Normal Distribution

A Z-score measures how many standard deviations an element is from the mean of a distribution. In the context of the normal distribution, Z-scores are used to find probabilities associated with specific values of the sample mean, allowing for the comparison of the sample mean to the population mean in terms of standard deviations.