True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

If the sample size is at least 30, then you can use z-scores to determine the probability that a sample mean falls in a given interval of the sampling distribution.

Verified step by step guidance

Step 1: Understand the context of the problem. The statement is about using z-scores to determine probabilities for a sample mean when the sample size is at least 30. This relates to the Central Limit Theorem (CLT), which states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's original distribution.

Step 2: Recall the conditions for using z-scores. Z-scores are used when the population standard deviation (σ) is known, and the sampling distribution of the sample mean is approximately normal. For a sample size of at least 30, the CLT ensures that the sampling distribution is approximately normal, even if the population distribution is not normal.

Step 3: Evaluate the statement. The statement is true if the population standard deviation (σ) is known. If σ is unknown, you would typically use a t-distribution instead of z-scores, especially for smaller sample sizes. However, for large sample sizes (n ≥ 30), the t-distribution and z-distribution become very similar.

Step 4: If the statement is false, rewrite it as a true statement. A true version of the statement would be: 'If the sample size is at least 30 and the population standard deviation is known, then you can use z-scores to determine the probability that a sample mean falls in a given interval of the sampling distribution.'

Step 5: Conclude by emphasizing the importance of verifying whether the population standard deviation is known before deciding to use z-scores. If it is unknown, consider using the t-distribution instead.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the population distribution, the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, typically becoming approximately normal when the sample size is 30 or more. This theorem is fundamental in statistics as it allows for the use of normal probability methods for inference about population means.

Z-scores

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores are used in hypothesis testing and confidence intervals to determine how many standard deviations an element is from the mean, which is particularly useful when the sample size is large.

Sampling Distribution

A 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 provides a framework for understanding how sample statistics vary and is crucial for making inferences about the population based on sample data. The shape of the sampling distribution is influenced by the sample size and the population distribution.

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