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.


MCAT Scores A random sample of 100 medical school applicants at a university has a mean total score of 505 on the MCAT. According to a report, the mean total score for the school’s applicants is more than 503. Assume the population standard deviation is 10.6. At alpha=0.01, is there enough evidence to support the report’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 mean total score for the school’s applicants is more than 503. This is a one-tailed test. H₀: μ ≤ 503 (the mean is less than or equal to 503), and H₁: μ > 503 (the mean is greater than 503).
2. Step 2: Calculate the standardized test statistic z. Use the formula: $z=\frac{\mathrm{x̄}-\mu }{\frac{\sigma }{\surd }}$, where x̄ is the sample mean (505), μ is the population mean under H₀ (503), σ is the population standard deviation (10.6), and n is the sample size (100).
3. Step 3: Find the corresponding P-value. Use the z-value obtained in Step 2 and refer to the standard normal distribution table (or use statistical software) to find the P-value for the one-tailed test.
4. Step 4: Compare the P-value to the significance level α = 0.01. 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 H₀ is rejected, there is enough evidence to support the claim that the mean total score for the school’s applicants is more than 503. If H₀ is not rejected, there is not enough evidence to support the 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 (H1), which indicates the presence of an effect or difference. 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 researchers typically compare the P-value to a predetermined significance level (alpha) to decide whether to reject H0.

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, population standard deviation, and sample size. The z-score helps to determine the position of the sample mean in relation to the null hypothesis and is essential for finding the corresponding P-value.