The mean of a random sample of 18 test scores is x_bar. The standard deviation of the population of all test scores is sigma= 6. Under what condition can you use a z-test to decide whether to reject a claim that the population mean is mu=88?

검증된 단계별 안내

Step 1: Understand the z-test conditions. A z-test is appropriate when the population standard deviation (σ) is known, the sample size is sufficiently large (typically n ≥ 30), or the population is normally distributed regardless of sample size.

Step 2: Check if the population standard deviation (σ) is provided. In this problem, σ = 6 is given, which satisfies one of the conditions for using a z-test.

Step 3: Evaluate the sample size (n). The sample size is n = 18, which is less than 30. Therefore, the z-test can only be used if the population of test scores is normally distributed.

Step 4: Confirm the normality of the population distribution. If the population of test scores is known to be normally distributed, then the z-test can be used even with a sample size of n = 18.

Step 5: Conclude the condition for using the z-test. You can use the z-test to decide whether to reject the claim that the population mean is μ = 88 if the population of test scores is normally distributed and the population standard deviation (σ = 6) is known.

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Z-Test

A Z-test is a statistical method used to determine if there is a significant difference between the means of two groups, or between a sample mean and a known population mean. It is applicable when the population standard deviation is known, and the sample size is sufficiently large (typically n > 30) or when the population is normally distributed, allowing for the use of the standard normal distribution.

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is large enough (usually n ≥ 30). This theorem justifies the use of the Z-test for sample means, as it allows for the assumption of normality in the sampling distribution.

Population Standard Deviation

The population standard deviation (σ) is a measure of the dispersion or spread of a set of values in a population. In hypothesis testing, knowing the population standard deviation is crucial for calculating the Z-score, which helps determine how far the sample mean is from the population mean under the null hypothesis. In this case, σ = 6 is provided, allowing for the application of the Z-test.