This is the question:
To estimate the average speed of cars on a specific highway, an investigator
collected speed data from a random sample of 75 cars driving on the
highway. The sample mean and sample standard deviation are 58 miles per
hour and 15 miles per hour, respectively.
Construct a 90% confidence interval for the mean speed.
I have the answer for it, and this is the answer:
But, I don't understand why Z0.05 =1.645. On the Standard Normal Distribution Table, P(Z < -1.645) = 0.05. Therefore, Z0.05 should be -1.645 instead of 1.645
Best Answer
Since the normal distribution is symmetric, the sign of $Z_{\alpha /2}$ is not as important as the fact that 5% of the area under the curve is in each tail of the bell curve. Since your confidence interval is constructed by using $$\bar{x} \pm Z_{\alpha /2} \frac{s}{\sqrt{n}}$$ you will be using both the positive a negative of this Z value.