(d) Find the Two Critical Values

For a two-tailed test at the 0.05 level of significance, the critical values correspond to the z-scores that cut off the lower and upper 2.5% of the standard normal distribution.

The critical z-values can be found using standard z-tables or statistical software.

For a 0.05 significance level (α = 0.05) in a two-tailed test:

• Each tail has an area of 0.025.
• The critical z-values correspond to the 0.025 and 0.975 percentiles of the standard normal distribution.

These critical z-values are:

$z_{\text{critical}} = \pm 1.96$

(e) Can We Conclude That the Population Mean Adult Sodium Level Differs From That Given in the Textbook?

To make the decision, compare the calculated z-test statistic (2.828) with the critical z-values ($\pm 1.96$):

• If the z-test statistic falls outside the range defined by the critical values ($-1.96$ and $1.96$), we reject the null hypothesis.
• If it falls within the range, we fail to reject the null hypothesis.

In this case, the calculated z-value (2.828) is outside the range of the critical values. Therefore, we reject the null hypothesis.

Conclusion:

Yes, we can conclude that the population mean adult sodium level differs from that given in the textbook at the 0.05 level of significance.

Do you have any further questions, or would you like more details on any part of the solution?

Here are some additional questions for further exploration:

1. How would the result change if the sample mean were 143 mEq/L instead of 147 mEq/L?
2. What is the power of the test, and how could it be calculated in this scenario?
3. How would the conclusion change if the sample size were reduced to 16?
4. What assumptions are made about the population distribution when performing this test?
5. How does the choice of significance level (e.g., 0.01 vs. 0.05) affect the test outcome?
6. What role does the standard deviation play in the hypothesis test?
7. How would you interpret the results of this test in a real-world medical context?
8. What steps should be taken if the population distribution is not normally distributed?

Tip: Always make sure to compare the test statistic to the correct critical values based on the significance level and the direction (one-tailed or two-tailed) of your test.