Hypothesis Testing Using a P-Value In Exercises 33–38,
         
a. identify the claim and state and .
b. find the standardized test statistic z.
c. find the corresponding P-value.
d. decide whether to reject or fail to reject the null hypothesis.
e. interpret the decision in the context of the original claim.


Sprinkler Systems A manufacturer of sprinkler systems designed for fire protection claims that the average activating temperature is at least 135°F. To test this claim, you randomly select a sample of 32 systems and find the mean activation temperature to be 133°F. Assume the population standard deviation is 3.3°F. At alpha=0.10, 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 the alternative hypothesis (H₁). The claim is that the average activating temperature is at least 135°F. This translates to H₀: μ ≥ 135°F and H₁: μ < 135°F. Note that this is a left-tailed test because the alternative hypothesis is testing for a value less than 135°F.
2. Step 2: Calculate the standardized test statistic z. Use the formula: $z=\frac{\mathrm{x̄}-\mu }{\frac{\sigma }{\sqrt{n}}}$, where x̄ is the sample mean (133°F), μ is the claimed population mean (135°F), σ is the population standard deviation (3.3°F), and n is the sample size (32). Plug in the values to compute z.
3. Step 3: Find the corresponding P-value. Using the z-value calculated in Step 2, refer to a standard normal distribution table or use statistical software to find the P-value. Since this is a left-tailed test, the P-value is the area to the left of the calculated z-value.
4. Step 4: Compare the P-value to the significance level α (0.10). If the P-value is less than α, 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 average activating temperature is less than 135°F, 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 average activating temperature is at least 135°F.

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 (H1), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.

P-Value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, and if it is less than the predetermined significance level (alpha), we reject the null hypothesis.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, is a value that indicates how many standard deviations an element is from the mean. In hypothesis testing, it is calculated using the sample mean, population mean under the null hypothesis, population standard deviation, and sample size. This statistic is crucial for determining the P-value and making decisions about the null hypothesis.