An old medical textbook states that the mean sodium level for healthy adults is 140 mEq  per liter of blood. A medical researcher believes that, because of modern dietary habits, the mean sodium level for healthy adults, , now differs from that given in the textbook. A random sample of 32 healthy adults is evaluated. The mean sodium level for the sample is  147 mEq per liter of blood. It is known that the population standard deviation of adult sodium levels is 14 mEq. Assume that the population is normally distributed. Can we conclude, at the 0.05 level of significance, that the population mean adult sodium level differs from that given in the textbook?
Perform a two-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places, and round your responses as specified below. (a)	State the null hypothesis H0 and the alternative hypothesis H1. (b)	Determine the type of test statistic to use. (c)Find the value of the test statistic. (Round to three or more decimal places.)

To determine if we can conclude that the population mean adult sodium level differs from that given in the textbook, we will perform a two-tailed hypothesis test. Here’s how we proceed step by step:

### (a) State the Null Hypothesis \( H_0 \) and the Alternative Hypothesis \( H_1 \).

- **Null Hypothesis (\( H_0 \))**: The population mean sodium level is equal to the textbook value.
  \[
  H_0: \mu = 140 \text{ mEq/L}
  \]
  
- **Alternative Hypothesis (\( H_1 \))**: The population mean sodium level is different from the textbook value.
  \[
  H_1: \mu \neq 140 \text{ mEq/L}
  \]

### (b) Determine the Type of Test Statistic to Use

Given that the population standard deviation (\( \sigma \)) is known and the sample size (\( n = 32 \)) is greater than 30, we will use the **z-test** statistic.

### (c) Find the Value of the Test Statistic

The formula for the z-test statistic is:
\[
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]
where:
- \( \bar{x} = 147 \) mEq/L (sample mean)
- \( \mu_0 = 140 \) mEq/L (population mean under the null hypothesis)
- \( \sigma = 14 \) mEq/L (population standard deviation)
- \( n = 32 \) (sample size)

Substituting the given values:
\[
z = \frac{147 - 140}{14 / \sqrt{32}}
\]

Let me calculate this value.

The value of the test statistic is approximately \( z = 2.828 \).

Would you like to continue with determining the critical value and the conclusion of the test? If you have any questions or need further details, feel free to ask.

Here are some related questions you might consider:

1. What are the critical z-values for a two-tailed test at the 0.05 level of significance?
2. How do you determine the p-value for this z-test statistic?
3. What does the p-value tell you about the null hypothesis?
4. What would the test conclusion be if the test were one-tailed?
5. How does sample size affect the z-test statistic?
6. What are the implications of using a z-test versus a t-test in hypothesis testing?
7. Can the results of this test be generalized to all populations? Why or why not?
8. How would the test conclusion change if the level of significance were 0.01 instead of 0.05?

**Tip:** Always check if the sample size is large enough to justify the use of the z-test when the population standard deviation is known.

Learn how to perform a hypothesis test to determine if the population mean sodium level differs from the textbook value of 140 mEq/L. Includes step-by-step calculations using the z-test with provided sample mean, population standard deviation, and sample size.

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