Ch. 9 - Inferences from Two Samples

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 9 - Inferences from Two Samples

Problem 9.R.3

Chapter 9, Problem 9.R.3

Forecast and Actual Temperatures Listed below are actual temperatures (°F) along with the temperatures that were forecast five days earlier (data collected by the author). Use a 0.05 significance level to test the claim that differences between actual temperatures and temperatures forecast five days earlier are from a population with a mean of 0°F.

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State the null hypothesis (H₀) and the alternative hypothesis (H₁): H₀: μ_d = 0 (the mean difference between actual and forecast temperatures is 0°F), H₁: μ_d ≠ 0 (the mean difference is not 0°F).

Calculate the differences (d) between the actual temperatures and the forecast temperatures for each data pair. Then, compute the mean of these differences (d̄) and the standard deviation of the differences (s_d).

Determine the test statistic using the formula: t = (d̄ - μ_d) / (s_d / √n), where μ_d is the hypothesized mean difference (0°F), n is the number of data pairs, and s_d is the standard deviation of the differences.

Find the critical t-value(s) for a two-tailed test at the 0.05 significance level using the degrees of freedom (df = n - 1). Compare the calculated t-value to the critical t-value(s).

Make a decision: If the calculated t-value falls within the critical region (beyond the critical t-values), reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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). In this context, the null hypothesis states that the mean difference between actual and forecast temperatures is 0°F, while the alternative suggests it is not. The process includes calculating a test statistic and comparing it to a critical value to determine if the null hypothesis can be rejected.

Significance Level

The significance level, denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. A common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In this scenario, using a 0.05 significance level means that if the p-value obtained from the test is less than 0.05, the null hypothesis can be rejected, suggesting that the differences in temperatures are statistically significant.

Mean Difference

The mean difference refers to the average of the differences between paired observations—in this case, the actual temperatures and the forecasted temperatures. Calculating the mean difference helps to assess whether the forecast is accurate. If the mean difference is significantly different from 0°F, it indicates that the forecasts are systematically off, which is the primary claim being tested in this analysis.

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Difference in Means: Confidence Intervals

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