Historically, the mean daily high temperature in Evansville in July has been 84 degrees with a standard deviation of 4. However, over the most recent 46 days, the mean daily high temperature has been 86.54 degrees. At the 10% significance level, can we conclude that the mean daily temperature has increased from 84 degrees based on this recent, limited sample?

State the null hypothesis and the alternate hypothesis.

A. H₀: μ > 84; H₁: μ = 84

B. H₀: μ ≥ 84; H₁: μ < 84

C. H₀: μ = 84; H₁: μ ≠ 84

D. H₀: μ ≤ 84; H₁: μ > 84

The null hypothesis H₀ for the hypothesis test would assert that there is no change in the mean daily high temperature, and the alternative hypothesis H₁ would suggest an increase. The correct hypotheses in this case are H₀: μ ≤ 84 and H₁: μ > 84, indicating we are testing for an increase in the mean temperature from the historical mean.

Explanation:

The question is about determining whether the mean daily high temperature in Evansville in July has statistically increased from the historical mean of 84 degrees based on a recent sample. When performing a hypothesis test, the null hypothesis (H₀) is a statement of no effect or no difference, and the alternative hypothesis (H₁) is what we are trying to find evidence for. In this case, the correct hypotheses would be:

H₀: μ = 84 (The mean daily high temperature has not changed from the historical mean.)
H₁: μ > 84 (The mean daily high temperature has increased from the historical mean.)

Therefore, the correct answer from the given options would be:

D. H₀:μ ≤ 84 ;H₁: μ > 84

Statistics like the mean temperature, standard deviation, and sample size are used to conduct hypothesis testing to determine if observed differences are statistically significant or if they could be due to random variation.