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.


Light Bulbs A light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 750 hours. A random sample of 25 light bulbs has a mean life of 745 hours. Assume the population is normally distributed and the population standard deviation is 60 hours. At alpha= 0.02, 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 and alternative hypotheses. The claim is that the mean life of the light bulbs is at least 750 hours. This translates to the null hypothesis H₀: μ ≥ 750. The alternative hypothesis Ha is the opposite of the claim, so Ha: μ < 750. This is a left-tailed test because the alternative hypothesis involves a 'less than' inequality.
2. Step 2: Determine the critical value(s) and rejection region(s). Since the significance level (α) is 0.02 and this is a left-tailed test, use a z-table to find the z-critical value corresponding to a cumulative probability of 0.02. The rejection region will be z < z_critical, where z_critical is the critical value obtained from the z-table.
3. Step 3: Calculate the standardized test statistic z. Use the formula z = (x̄ - μ₀) / (σ / √n), where x̄ is the sample mean (745), μ₀ is the hypothesized population mean (750), σ is the population standard deviation (60), and n is the sample size (25). Substitute the given values into the formula to compute z.
4. Step 4: Compare the calculated z value to the critical value. If the calculated z value falls in the rejection region (z < z_critical), reject the null hypothesis H₀. Otherwise, fail to reject H₀.
5. Step 5: Interpret the decision in the context of the original claim. If you reject H₀, it means there is enough evidence to conclude that the mean life of the light bulbs is less than 750 hours, contradicting the manufacturer's claim. If you fail to reject H₀, it means there is not enough evidence to refute the manufacturer's claim that the mean life is at least 750 hours.

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 the status quo or a claim to be tested, and the alternative hypothesis (Ha), which represents a new claim. The goal is to determine whether there is enough evidence in the sample to reject H0 in favor of Ha.

Rejection Region

The rejection region is a range of values for the test statistic that leads to the rejection of the null hypothesis. It is determined by the significance level (alpha), which indicates the probability of making a Type I error (rejecting H0 when it is true). In this case, with alpha set at 0.02, the rejection region will be defined based on the critical value(s) corresponding to this significance level.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations a sample mean is from the population mean under the null hypothesis. It is calculated using the formula z = (X̄ - μ) / (σ/√n), where X̄ is the sample mean, μ is the population mean under H0, σ is the population standard deviation, and n is the sample size. This statistic helps determine whether the observed sample mean is significantly different from the hypothesized population mean.