Question

When σ is unknown and the sample is of size n ≥ 30, there are two...

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

Method 1: Use the Student's t distribution with d.f. = n − 1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 41, with sample mean x = 45.5 and sample standard deviation s = 5.5.

(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

90% 95% 99%
lower limit    
upper limit    


(b) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

90% 95% 99%
lower limit    
upper limit    


(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

Yes. The respective intervals based on the t distribution are shorter.No. The respective intervals based on the t distribution are shorter.     Yes. The respective intervals based on the t distribution are longer.No. The respective intervals based on the t distribution are longer.


(d) Now consider a sample size of 71. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

90% 95% 99%
lower limit    
upper limit    


(e) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

90% 95% 99%
lower limit    
upper limit    

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ. Method 1: Use the Student's t distribution with d.f. = n − 1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ. Method 1: Use the Student's t distribution with d.f. = n − 1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the...
When σ is unknown and the sample is of size n ≥ 30, there are two...
When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ. Method 1: Use the Student's t distribution with d.f. = n − 1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the...
When ? is unknown and the sample is of size n?30, there are two methods for...
When ? is unknown and the sample is of size n?30, there are two methods for computing confidence intervals for ?. Method 1: Use the Student's t distribution with d.f.= n?1. This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When n ? 30, use the sample standard deviation sas an estimate for ?, and then use the standard normal distribution. This method is...
a.) Given a normal distribution with σ = 0.380. Find the required sample size for a...
a.) Given a normal distribution with σ = 0.380. Find the required sample size for a 95% confidence level (estimating the mean), given a margin-of-error of 6%. b.) Given the sample results taken from a normal population distribution: mean = 4.65, σ = 0.32, and n = 17. For a 99% confidence interval, find the margin-of-error for the population mean. (use 2 decimal places) c.) Given the sample results taken from a normal population distribution: mean = 1.25, σ =...
A sample of n = 16 is to be taken from a distribution that can reasonably...
A sample of n = 16 is to be taken from a distribution that can reasonably be assumed to be Normal with a standard deviation σ of 100. The sample mean comes out to be 110. 1. The standard error of the mean, that is, the standard deviation of the sample mean, is σx¯ = σ/√ n. What is its numerical value? 2. The 97.5 percentile, 1.96, of the standard Normal distribution is used for a 95% confi- dence interval....
Use the sample information x¯ = 34, σ = 4, n = 10 to calculate the...
Use the sample information x¯ = 34, σ = 4, n = 10 to calculate the following confidence intervals for μ assuming the sample is from a normal population. (a) 90 percent confidence. (Round your answers to 4 decimal places.)    The 90% confidence interval is from to (b) 95 percent confidence. (Round your answers to 4 decimal places.)    The 95% confidence interval is from to (c) 99 percent confidence. (Round your answers to 4 decimal places.)    The...
Use the sample information x¯x¯ = 36, σ = 7, n = 20 to calculate the...
Use the sample information x¯x¯ = 36, σ = 7, n = 20 to calculate the following confidence intervals for μ assuming the sample is from a normal population. (a) 90 percent confidence. (Round your answers to 4 decimal places.)    The 90% confidence interval is from  to (b) 95 percent confidence. (Round your answers to 4 decimal places.)    The 95% confidence interval is from  to (c) 99 percent confidence. (Round your answers to 4 decimal places.)    The 99% confidence...
A simple random sample of size n is drawn from a population that is normally distributed....
A simple random sample of size n is drawn from a population that is normally distributed. The sample​ mean, x, is found to be 113, and the sample standard deviation, s, is found to be 10. ​(a) Construct a 90​% confidence interval about μ if the sample​ size, n, is 22. ​(b) Construct a 90​% confidence interval about μ if the sample​ size, n, is 15. ​(c) Construct an 80​% confidence interval about μ if the sample​ size, n, is...
Use the sample information x¯ = 35, σ = 7, n = 16 to calculate the...
Use the sample information x¯ = 35, σ = 7, n = 16 to calculate the following confidence intervals for μ assuming the sample is from a normal population. (a) 90 percent confidence. (Round your answers to 4 decimal places.) The 90% confidence interval is from to (b) 95 percent confidence. (Round your answers to 4 decimal places.) The 95% confidence interval is from to (c) 99 percent confidence. (Round your answers to 4 decimal places.) The 99% confidence interval...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT