Final Answer:
Reject the null hypothesis if the calculated value of the test statistic is less than 108.51 or greater than 133.49; otherwise, do not reject the null hypothesis.
Step-by-step explanation:
In hypothesis testing, the critical value(s) help determine when to reject the null hypothesis. The decision rule is established based on the significance level (alpha), which is 0.1 in this case, indicating a two-tailed test. Since alpha is divided equally between the two tails (0.1/2 = 0.05), we find the critical values corresponding to a cumulative probability of 0.05 in both tails of the distribution.
Using a standard normal distribution table or a statistical software, we find the critical z-values associated with a cumulative probability of 0.025 in each tail, resulting in critical values of approximately -1.65 and 1.65.
To translate these critical values into the context of the test statistic, we use the formula: Critical Value = Mean of the Null Hypothesis (μ) ± (Z-Score for alpha/2 * Standard Deviation of the Population). Plugging in the given values (μ = 121, Z = 1.65, σ = 27), we calculate critical values of 133.49 and 108.51.
Therefore, the decision rule is to reject the null hypothesis if the calculated test statistic falls outside this range, indicating that the sample mean is significantly different from the hypothesized population mean. If the test statistic falls within this range, we do not have sufficient evidence to reject the null hypothesis.