Hypothesis Testing Using Rejection Region(s) In Exercises 39–44, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.


[APPLET] Fluorescent Lamps A compact fluorescent lamp (CFL) bulb manufacturer guarantees that the mean life of a CFL bulb is at least 10,000 hours. You want to test this guarantee. To do so, you record the lives of a random sample of 32 CFL bulbs. The results (in hours) are listed. Assume the population standard deviation is 1850 hours. At alpha=0.11, do you have enough evidence to reject the manufacturer’s claim?

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (H0) and alternative hypothesis (Ha). The claim is that the mean life of a CFL bulb is at least 10,000 hours. This translates to H0: μ ≥ 10,000 and Ha: μ < 10,000, where μ represents the population mean life of CFL bulbs.
2. Step 2: Determine the critical value and rejection region. Since the significance level (alpha) is 0.11 and the test is one-tailed (left-tailed), use a z-table to find the critical value corresponding to alpha = 0.11. The rejection region is z < critical value.
3. Step 3: Calculate the standardized test statistic z. First, compute the sample mean (x̄) using the provided data. Then, use the formula z = (x̄ - μ) / (σ / √n), where μ = 10,000 (claimed mean), σ = 1850 (population standard deviation), and n = 32 (sample size).
4. Step 4: Compare the calculated z-value to the critical value. If the z-value falls within the rejection region (z < critical value), reject the null hypothesis; otherwise, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is enough evidence to reject the manufacturer's claim that the mean life of CFL bulbs is at least 10,000 hours. If the null hypothesis is not rejected, conclude that there is not enough evidence to reject the manufacturer's claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject H0 in favor of Ha.

Rejection Region

The rejection region is a set of values for the test statistic that leads to the rejection of the null hypothesis. It is determined based on the significance level (alpha), which defines the probability of making a Type I error (rejecting H0 when it is true). In hypothesis testing, if the calculated test statistic falls within this region, the null hypothesis is rejected, indicating that the sample provides sufficient evidence against H0.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, is a measure that indicates how many standard deviations an element is from the mean. In the context of hypothesis testing, it is calculated using the sample mean, the population mean under the null hypothesis, and the standard deviation of the population. This statistic is crucial for determining whether the observed sample mean is significantly different from the hypothesized population mean, thus aiding in the decision to reject or fail to reject H0.