Deciding on a Distribution In Exercises 31 and 32, decide whether you should use the standard normal sampling distribution or a t-sampling distribution to perform the hypothesis test. Justify your decision. Then use the distribution to test the claim. Write a short paragraph about the results of the test and what you can conclude about the claim.


Tuition and Fees An education publication claims that the mean in-state tuition and fees at public four-year institutions by state is more than \$10,500 per year. A random sample of 30 states has a mean in-state tuition and fees at public four-year institutions of \$10,931 per year. Assume the population standard deviation is \$2380. At α=0.01, test the publication’s claim.

Verified step by step guidance

1. Step 1: Determine the appropriate distribution to use. Since the population standard deviation (σ) is provided and the sample size (n = 30) is greater than 30, the standard normal (Z) distribution should be used for this hypothesis test. The t-distribution is typically used when the population standard deviation is unknown or the sample size is small (n < 30).
2. Step 2: State the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: μ ≤ 10,500, and the alternative hypothesis is Hₐ: μ > 10,500. This is a one-tailed test because the claim is that the mean is more than \$10,500.
3. Step 3: Calculate the test statistic using the formula Z = (x̄ - μ₀) / (σ / √n), where x̄ is the sample mean, μ₀ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size. Substitute the given values: x̄ = 10,931, μ₀ = 10,500, σ = 2380, and n = 30.
4. Step 4: Determine the critical value for the Z-test at a significance level of α = 0.01. Use a Z-table or standard normal distribution table to find the critical value corresponding to a one-tailed test with α = 0.01.
5. Step 5: Compare the calculated test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Based on this decision, write a short paragraph summarizing whether the claim that the mean in-state tuition and fees is more than \$10,500 is supported by the data.

Key Concepts

Standard Normal Distribution vs. t-Distribution

The standard normal distribution is used when the population standard deviation is known and the sample size is large (typically n > 30). In contrast, the t-distribution is applied when the population standard deviation is unknown or the sample size is small (n ≤ 30). The t-distribution has heavier tails, which accounts for the increased uncertainty in estimating the population mean from a small sample.

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), then using sample data to determine whether to reject H0 in favor of H1. The significance level (α) indicates the probability of making a Type I error, which is rejecting a true null hypothesis.

P-Value and Conclusion

The p-value is the probability of observing the sample data, or something more extreme, assuming the null hypothesis is true. In hypothesis testing, if the p-value is less than the significance level (α), we reject the null hypothesis. The conclusion drawn from the test indicates whether there is sufficient evidence to support the alternative hypothesis, in this case, that the mean tuition exceeds $10,500.