You have an SRS of 15 observations from a Normally distributed population. What critical value would you use to obtain a 98% confidence interval for the mean μ of the population?
I know the answer is 2.602...but I do not understand why
Tell me if I'm wrong:
n=15 therefore the df = 14 (n-1)
Area = .98
So I plugged this into the inverse T:
area: .98
df: 14
Inverse = 2.26 (but this is not the right critical value)
Can someone guide me through the steps please...and i know there is a chart you can use but i want to figure it out on my Ti-89 because that's what we use for the exam
n=15 therefore the df=14 (n-1)
Area=0.01 1-0.98=0.02 --> 0.02/2=0.01
invT(0.01,14)= -2.62
To obtain the critical value for the 98% confidence interval, you need to use the t-distribution because the sample size is small (n = 15) and the population standard deviation is unknown.
Here are the steps to find the critical value using your Ti-89 calculator:
Step 1: Press the "2nd" button and then the "DISTR" button to access the distribution menu.
Step 2: Move the cursor to highlight "t" (the t-distribution) and press ENTER.
Step 3: Enter the desired confidence level, in this case, 0.98 since you want a 98% confidence interval. So enter 0.98 in the "Area" field and press ENTER.
Step 4: Enter the degrees of freedom (df) in the appropriate field. In this case, since your sample size is 15, the degrees of freedom would be 14 (n - 1). Enter 14 and press ENTER.
Step 5: The calculator will display the critical value for the desired confidence level. In this case, the critical value is approximately 2.602.
It seems you made a mistake in step 3. The correct critical value is 2.602, not 2.26.
Remember that the critical value represents the number of standard deviations away from the mean of the t-distribution corresponding to the desired confidence level. In a 98% confidence interval, you want to capture 98% of the area under the curve, leaving 1% in each tail. The critical value reflects how far you need to go in each tail to capture that 1% in total.
Using the t-distribution and finding the critical value ensures that the confidence interval accounts for the uncertainty associated with estimating the population mean from a small sample.
To find the critical value for a 98% confidence interval for the mean of a population, you need to use the t-distribution. Here are the steps to find the correct critical value:
1. Determine the sample size (n) from the given information. In this case, n = 15.
2. Find the degrees of freedom (df) for the t-distribution. For a sample size of n = 15, the degrees of freedom is n - 1, which is 15 - 1 = 14.
3. Determine the confidence level (C). In this case, C = 98%, which is equivalent to 0.98.
4. Find the critical value using the inverse t-distribution function. This can be done using a table or a calculator.
If you're using your TI-89 calculator, follow these steps:
- Press the "2ND" key, then press the "DISTR" key.
- Scroll down or up to select "invT(".
- Enter the confidence level (C) as the first argument. In this case, enter 0.98.
- Enter the degrees of freedom (df) as the second argument. In this case, enter 14.
- Press the closing parenthesis ")" and then press "ENTER".
The calculator will give you the critical value, which should be approximately 2.602 for a 98% confidence interval.
So, the correct critical value for a 98% confidence interval for the mean of a population with a sample size of 15 is 2.602.