Final answer:
To test the hypothesis at the 0.05 significance level in a right-tailed test, we reject the null hypothesis if the test statistic is greater than 1.645.
Step-by-step explanation:
The decision rule for testing the hypothesis that the population mean is greater than 590 at a 0.05 significance level is found by rejecting the null hypothesis (H0) when the test statistic is greater than the critical value from the normal distribution. Since this is a right-tailed test (Ha: μ > 590), we use the positive critical z-value associated with the significance level α = 0.05. As provided in the information, the critical value for α = 0.05 in the normal distribution is a z-score of -1.645. However, since our test is right-tailed, we take the positive value of this critical z-score, which is 1.645.
Therefore, the correct decision rule is: Reject H0 when the test statistic is greater than 1.645. This decision rule corresponds to answer choice a, where the critical value would be rounded to three decimal places as 1.761 if the tail probability were exactly 0.05. But since we are given a critical value of -1.645, which corresponds to a slightly lower tail probability, rounding to three decimal places gives us 1.645, not 1.761.
Conclusion:
We should reject the null hypothesis if our computed test statistic exceeds 1.645, indicating that there is enough evidence at the 5% level of significance to support the claim that the population mean is greater than 590.