A sample of 12 observations is selected from a normal population for which the population standard deviation is known to be 4. The sample mean is 19. (Round your answers to 3 decimal places.)

(a) The standard error of the mean is
The 90 percent confidence interval for the population mean is between and

To calculate the standard error of the mean, you can use the formula:

Standard Error = Population Standard Deviation / Square Root of Sample Size

In this case, the population standard deviation is known to be 4, and the sample size is 12. Therefore, we can plug these values into the formula to find the standard error:

Standard Error = 4 / √12

Calculating this value gives us:

Standard Error = 4 / 3.464

The result is approximately 1.155 (rounded to 3 decimal places).

Now let's move on to calculating the 90 percent confidence interval for the population mean.

To calculate the confidence interval, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value * Standard Error)

In this case, the sample mean is given as 19, and we need to find the critical value corresponding to a 90 percent confidence level. The critical value can be found using a z-table or a statistical software.

For a 90 percent confidence level, with a sample size of 12, the critical value is approximately 1.753 (rounded to 3 decimal places).

Now, we can plug in the values into the formula to find the confidence interval:

Confidence Interval = 19 ± (1.753 * 1.155)

Calculating this gives us:

Confidence Interval = 19 ± 2.021

Therefore, the 90 percent confidence interval for the population mean is between 16.979 and 21.021 (rounded to 3 decimal places).