Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problem 7.2.2

Chapter 7, Problem 7.2.2

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?

Verified step by step guidance

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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Key Concepts

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

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.

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Textbook Question

Describe the difference between calculating the standardized test statistic, Z^2, for a chi-square test for variance and a chi-square test for standard deviation.

Textbook Question

Can a critical value for the chi-square test be negative? Explain.

Textbook Question

Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.

P= 0.2802

Textbook Question

Identifying a Test In Exercises 21–24, determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.

Ha: σ^2 = 142

H0: σ ≠ 142

Textbook Question

Writing You are testing a claim and incorrectly use the standard normal sampling distribution instead of the t-sampling distribution, mistaking the sample standard deviation for the population standard deviation. Does this make it more or less likely to reject the null hypothesis? Is this result the same no matter whether the test is left-tailed, right-tailed, or two-tailed? Explain your reasoning.

Textbook Question

Stating the Null and Alternative Hypotheses In Exercises 25–30, write the claim as a mathematical statement. State the null and alternative hypotheses, and identify which represents the claim.

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