Question

a local weather enthusiast wants to see if the last five summers have been warmer than normal for eloy, arizona. from the national oceanic and atmospheric administration (noaa), he knows the 30-year average for july was 90° with a standard deviation of 2.1°. the last five summers recorded have an average of 93.5°. if he uses α = 5%, which of the following would be valid based on the data he has gathered?
○ he can conclude that the summers have been warmer than the 30-year average.
○ he can conclude that the summers will continue to be warmer for the next 10 years.
○ he does not have enough information to make a decision.
○ the average for the last five years fits with the 30-year average.

Explanation:

Step1: Identify the test type

This is a hypothesis test for the mean. We know the population mean ($\mu = 90^\circ$), population standard deviation ($\sigma = 2.1^\circ$), sample size ($n = 5$), and sample mean ($\bar{x} = 93.5^\circ$). We can use a z - test (since $\sigma$ is known) to test if the sample mean is significantly different from the population mean.

Step2: Calculate the z - score

The formula for the z - score in a one - sample z - test is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.
Substitute the values: $\bar{x}=93.5$, $\mu = 90$, $\sigma=2.1$, and $n = 5$.
First, calculate the standard error: $\frac{\sigma}{\sqrt{n}}=\frac{2.1}{\sqrt{5}}\approx\frac{2.1}{2.236}\approx0.939$.
Then, calculate the z - score: $z=\frac{93.5 - 90}{0.939}=\frac{3.5}{0.939}\approx3.73$.

Step3: Determine the critical value

For a one - tailed test with $\alpha = 0.05$, the critical z - value (from the standard normal table) is $z_{\alpha}=1.645$.

Step4: Compare z - score and critical value

Since our calculated z - score ($z\approx3.73$) is greater than the critical z - value ($1.645$), we reject the null hypothesis that the sample mean is equal to the population mean. This means that the sample mean (average of the last five summers) is significantly greater than the 30 - year average.

Now let's analyze each option:

• Option 1: "He can conclude that the summers have been warmer than the 30 - year average." Since we rejected the null hypothesis (that the mean is equal to the 30 - year average) in favor of the alternative hypothesis (that the mean is greater than the 30 - year average), this option is correct.
• Option 2: "He can conclude that the summers will continue to be warmer for the next 10 years." The data we have is about the past five summers and the 30 - year average. We cannot make predictions about the next 10 years based on this hypothesis test. This option is incorrect.
• Option 3: "He does not have enough information to make a decision." We have enough information (population mean, population standard deviation, sample size, sample mean) to perform a hypothesis test and make a decision. This option is incorrect.
• Option 4: "The average for the last five years fits with the 30 - year average." Since we rejected the null hypothesis that they are equal, this option is incorrect.

Answer:

He can conclude that the summers have been warmer than the 30 - year average.