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 (c) greater than 5900 MMT CO2 eq. Compare your answers with those in Exercise 33.

검증된 단계별 안내

Identify the key information from the problem: the population mean (μ), the population standard deviation (σ), the sample size (n), and the value of interest (5900 MMT CO2 eq). These values should be referenced from Exercise 33.

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 error (SE) is calculated as SE = σ / √n, where σ is the population standard deviation and n is the sample size.

Standardize the value of interest (5900 MMT CO2 eq) to a z-score using the formula: z = (X̄ - μ) / SE, where X̄ is the sample mean (5900), μ is the population mean, and SE is the standard error.

Use the z-score to find the probability that the sample mean is greater than 5900. This can be done by looking up the z-score in a standard normal distribution table or using statistical software to find the area to the right of the z-score.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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-Score

A Z-score measures how many standard deviations an element is from the mean of a distribution. In the context of probability, Z-scores are used to determine the probability of a sample mean being above or below a certain value by converting the sample mean into a standard normal variable, facilitating the use of standard normal distribution tables.