I am not sure how to even get started on this question could someone help please.
A college advisor wants to estimate the mean annual income of all college students who graduated last year. It is believed that the incomes follow a normal distribution with population standard deviation of $2000. Based on a sample of 25 college graduates, the advisor obtains X=$22500. Construct a 95% confidence interval for the population mean u.
95% = mean ± 1.96 SEm
SEm = SD/√n
To construct a confidence interval for the population mean, we need to follow these steps:
Step 1: Determine the level of confidence.
In this case, it is given that we need to construct a 95% confidence interval.
Step 2: Find the critical value.
The critical value is determined based on the level of confidence and the distribution. Since we know that the population follows a normal distribution, we can use the Z-distribution. For a 95% confidence interval, the critical value (Z) is 1.96. This value can be found in a Z-table or using statistical software.
Step 3: Calculate the standard error.
The standard error is the standard deviation of the sample means, which is equal to the population standard deviation divided by the square root of the sample size. In this case, the population standard deviation is $2000, and the sample size is 25. Therefore, the standard error (SE) is $2000 / √25 = $400.
Step 4: Calculate the margin of error.
The margin of error is determined using the standard error and the critical value. It is calculated by multiplying the standard error by the critical value. In this case, the margin of error (ME) is 1.96 * $400 = $784.
Step 5: Calculate the confidence interval.
To calculate the confidence interval, we add and subtract the margin of error from the sample mean. In this case, the sample mean (X) is given as $22500. Therefore, the confidence interval can be calculated as follows:
Lower Limit = X - ME = $22500 - $784 = $21716
Upper Limit = X + ME = $22500 + $784 = $23284
So the 95% confidence interval for the population mean u is $21716 to $23284.
It is important to note that the interpretation of the confidence interval is that we are 95% confident that the true population mean falls within this range.