[APPLET] A researcher claims that the mean age of the residents of a small town is more than 38 years. The ages (in years) of a random sample of 30 residents are listed below. At α=0.10, is there enough evidence to support the researcher’s claim? Assume the population standard deviation is 9 years.

검증된 단계별 안내

Step 1: Formulate the null and alternative hypotheses. The null hypothesis (H₀) is that the mean age of the residents is 38 years (μ = 38). The alternative hypothesis (H₁) is that the mean age of the residents is greater than 38 years (μ > 38).

Step 2: Calculate the sample mean (x̄). Add all the ages provided in the sample and divide by the total number of residents (n = 30). Use the formula:

x̄ = \frac{\sum x}{n}

Step 3: Compute the test statistic using the z-test formula for a population mean. The formula is:

z = \frac{x̄ - μ}{\frac{σ}{\sqrt{n}}}

, where μ = 38, σ = 9, and n = 30.

Step 4: Determine the critical value for α = 0.10 in a one-tailed z-test. Look up the z-value corresponding to a significance level of 0.10 in a z-table. This critical value will help decide whether to reject or fail to reject the null hypothesis.

Step 5: Compare the calculated z-test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis and conclude that there is enough evidence to support the researcher’s claim. Otherwise, fail to reject the null hypothesis.

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주요 개념

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Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1). In this case, the null hypothesis states that the mean age is 38 years or less, while the alternative hypothesis posits that it is greater than 38 years. The goal is to determine if there is enough evidence to reject the null hypothesis at a specified significance level.

Significance Level (α)

The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.10, meaning there is a 10% risk of concluding that the mean age is greater than 38 years when it is not. This threshold helps researchers decide how strong the evidence must be to support the alternative hypothesis.

Z-Test for Means

A Z-test for means is a statistical test used to determine if there is a significant difference between the sample mean and a known population mean when the population standard deviation is known. In this case, the sample of 30 residents' ages will be analyzed using the Z-test to compare the sample mean against the hypothesized mean of 38 years, utilizing the provided population standard deviation of 9 years to calculate the Z-score.