A housing development can only build on a certain plot of land if can show that the average level of arsenic in the groundwater is less than 78 parts-per-million (ppm). Soil samples are drawn from the site and tested in a lab. The results of the samples (in ppm) are given below. Is there enough evidence to say that the land is ready for development using a level of significance of $10 \%$ ? You may assume that the level of ground arsenic is normally distributed in the area. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline Sample Reults: & 76 & 42 & 17 & 63 & 81 & 91 & 25 & 53 & 40 & 22 & 88 \\ \hline \end{tabular} a. Enter the null hypothesis for this test. \[ H_{0} ? \vee ? \vee \] b. Enter the alternative hypothesis for this test. \[ H_{1}: ? \vee ? \vee \] c. Is the original claim located in the null or alternative hypothesis? Select an answer d. What is the test statistic for the given statistics? e. What is the $p$-value for this test? f. What is the decision based on the given statistics? Select an answer g. What is the correct interpretation of this decision? Using a $\quad \%$ level of significance, there Select an answer $\checkmark$ sufficient evidence to Select an answer $\checkmark$ the claim that the average level of arsenic in the groundwater is less than 78 parts-per-million (ppm).
Solution
Step 1 ：The null hypothesis for this test is: \(H_{0}:\) The average level of arsenic in the groundwater is equal to or greater than 78 parts-per-million (ppm)
Step 2 ：The alternative hypothesis for this test is: \(H_{1}:\) The average level of arsenic in the groundwater is less than 78 parts-per-million (ppm)
Step 3 ：The original claim is located in the alternative hypothesis
Step 4 ：To calculate the test statistic, we first need to calculate the sample mean and standard deviation. Using python, we find that the sample mean is approximately 54.75 and the sample standard deviation is approximately 25.45. The test statistic is then calculated as \((54.75 - 78) / (25.45 / \sqrt{12})\), which is approximately -3.47
Step 5 ：The p-value for this test is the probability of observing a test statistic as extreme as -3.47 under the null hypothesis. Using python, we find that the p-value is approximately 0.003
Step 6 ：Since the p-value is less than the level of significance (0.10), we reject the null hypothesis
Step 7 ：Using a 10% level of significance, there is sufficient evidence to support the claim that the average level of arsenic in the groundwater is less than 78 parts-per-million (ppm)
