Questions: Question 20 of 20 (1 point) Question Attempt: 1 of Unlimited What are you drinking? Environmental Protection Agency standards require that the amount of lead in drinking water be less than 15 micrograms per liter. Ten samples of water from a particular source have the following concentrations, in units of micrograms per liter. Assume the population standard deviation is σ=4. If appropriate, perform a hypothesis test to determine whether you can conclude that the mean concentration of lead meets the EPA standards. Use the α=0.01 level of significance and the P′-value method with the 11-84 Plus calculator. 11.7 14.3 11.6 14.7 15.8 9.6 12.6 8.6 11.5 17.5 Part 1 of 7 (a) Explain why it is necessary to check that the population is approximately normal before performing a hypothesis test.

Question 20 of 20 (1 point) Question Attempt: 1 of Unlimited

What are you drinking? Environmental Protection Agency standards require that the amount of lead in drinking water be less than 15 micrograms per liter. Ten samples of water from a particular source have the following concentrations, in units of micrograms per liter. Assume the population standard deviation is σ=4. If appropriate, perform a hypothesis test to determine whether you can conclude that the mean concentration of lead meets the EPA standards. Use the α=0.01 level of significance and the P′-value method with the 11-84 Plus calculator.

11.7 14.3 11.6 14.7 15.8 9.6
12.6 8.6 11.5 17.5

Part 1 of 7
(a) Explain why it is necessary to check that the population is approximately normal before performing a hypothesis test.

Transcript text: Question 20 of 20 (1 point) | Question Attempt: 1 of Unlimited What are you drinking? Environmental Protection Agency standards require that the amount of lead in drinking water be less than 15 micrograms per liter. Ten samples of water from a particular source have the following concentrations, in units of micrograms per liter. Assume the population standard deviation is $\sigma=4$. If appropriate, perform a hypothesis test to determine whether you can conclude that the mean concentration of lead meets the EPA standards. Use the $\alpha=0.01$ level of significance and the $P^{\prime}$-value method with the $11-84$ Plus calculator. \begin{tabular}{cccccc} \hline 11.7 & 14.3 & 11.6 & 14.7 & 15.8 & 9.6 \\ 12.6 & 8.6 & 11.5 & 17.5 & & \\ \hline \end{tabular} Part 1 of 7 (a) Explain why it is necessary to check that the population is approximately normal before performing a hypothesis test.

Solution

Solution Steps

Step 1: Calculate the Sample Mean

The sample mean ($\bar{x}$) is calculated as follows:

\[ \bar{x} = \frac{\sum x_i}{n} = \frac{11.7 + 14.3 + 11.6 + 14.7 + 15.8 + 9.6 + 12.6 + 8.6 + 11.5 + 17.5}{10} = 12.79 \]

Step 2: Calculate the Standard Error

The standard error ($SE$) is calculated using the population standard deviation ($\sigma$) and the sample size ($n$):

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{4}{\sqrt{10}} \approx 1.2649 \]

Step 3: Calculate the Test Statistic

The Z-test statistic is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{12.79 - 15}{1.2649} \approx -1.7472 \]

Step 4: Calculate the P-value

For a left-tailed test, the P-value is determined as follows:

\[ P = T(z) \approx 0.0403 \]

Step 5: Conclusion

We compare the P-value to the significance level ($\alpha = 0.01$):

Since $P \approx 0.0403 > 0.01$, we fail to reject the null hypothesis. This indicates that there is not enough evidence to conclude that the mean concentration of lead is less than 15 micrograms per liter.

Final Answer

$\boxed{\text{Fail to reject the null hypothesis: There is not enough evidence to conclude that the mean concentration of lead is less than 15 micrograms per liter.}}$

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