Question

In: Statistics and Probability

The mean and standard deviation of a random sample of n measurements are equal to 33.5...

The mean and standard deviation of a random sample of n measurements are equal to 33.5 and 3.5, respectively.

a. Find a 90% confidence interval for m if n = 144.

b. Find a 90% confidence interval for m if n = 576.

c. Find the widths of the confidence intervals found in parts a and b.

What is the effect on the width of a confidence interval of quadrupling the sample size while holding the confidence coefficient fixed?

Solutions

Expert Solution

a. Find a 90% confidence interval for m if n = 144.

We need to construct the 90% confidence interval for the population mean . The following information is provided:

Sample Mean   33.5
Population Standard Deviation 3.5
Sample Size 144

The critical value for α = 0.1 is = 1.645 . The corresponding confidence interval is computed as shown below:

CI = (33.02, 33.98)  

b. Find a 90% confidence interval for m if n = 576

We need to construct the 90% confidence interval for the population mean . The following information is provided:

Sample Mean   33.5
Population Standard Deviation 3.5
Sample Size 576

The critical value for α = 0.1 is = 1.645 . The corresponding confidence interval is computed as shown below:

CI = (33.26, 33.74)  

c. Find the widths of the confidence intervals found in parts a and b.

Width when n = 144

33.98 - 33.02 = 0.96

Width when n = 576

33.74 - 33.26 = 0.48

What is the effect on the width of a confidence interval of quadrupling the sample size while holding the confidence coefficient fixed?

As the sample size is quadrupeled the width is reduced to half


Related Solutions

The mean and standard deviation of a random sample of n measurements are equal to 33.2...
The mean and standard deviation of a random sample of n measurements are equal to 33.2 and 3.6?, respectively. a. Find a 90?% confidence interval for mu if nequals81. b. Find a 90?% confidence interval for mu if nequals324. c. Find the widths of the confidence intervals found in parts a and b. What is the effect on the width of a confidence interval of quadrupling the sample size while holding the confidence coefficient? fixed?
The mean and standard deviation of a random sample of n measurements are equal to 33.6...
The mean and standard deviation of a random sample of n measurements are equal to 33.6 and 3.5, respectively. A) The 99% confidence interval for π if n = 49 is approximately ( ___ , ___ ) B) The 99% confidence interval π if n = 196 is approximately ( ___ , ___ )
A random sample of ?n measurements was selected from a population with standard deviation 13.913.9 and...
A random sample of ?n measurements was selected from a population with standard deviation 13.913.9 and unknown mean ?μ. Calculate a 9999 % confidence interval for ?μ for each of the following situations: (a) ?=35, ?⎯⎯⎯=104.9n=35, x¯=104.9 (b) ?=60, ?⎯⎯⎯=104.9n=60, x¯=104.9 (c) ?=85, ?⎯⎯⎯=104.9n=85, x¯=104.9 (d) In general, we can say that for the same confidence level, increasing the sample size  the margin of error (width) of the confidence interval. (Enter: ''DECREASES'', ''DOES NOT CHANGE'' or ''INCREASES'', without the quotes.)
A random sample of ?n measurements was selected from a population with standard deviation ?=14.6 and...
A random sample of ?n measurements was selected from a population with standard deviation ?=14.6 and unknown mean ?. Calculate a 99% confidence interval for ? for each of the following situations: (a) ?=45, ?⎯⎯⎯=76.3 ____ ≤?≤ _______ (b)  ?=65, ?⎯⎯⎯=76.3 ____  ≤?≤ ______ (c)  ?=85, ?⎯⎯⎯=76.3 ______ ≤?≤ _______
A random sample of ?n measurements was selected from a population with standard deviation ?=11.7 and...
A random sample of ?n measurements was selected from a population with standard deviation ?=11.7 and unknown mean ?. Calculate a 95% confidence interval for ? for each of the following situations: (a) ?=60, ?=85 ≤?≤ (b)  ?=75, ?=85 ≤?≤ (c)  ?=100, ?=85 ≤?≤
A random sample of n measurements was selected from a population with standard deviation σ=19.8 and...
A random sample of n measurements was selected from a population with standard deviation σ=19.8 and unknown mean μ. Calculate a 99 % confidence interval for μ for each of the following situations (I did the third one right but can't get these two to work so figured I'd ask before emailing my instructor): (a) n=65, x¯=96.7 90.36380962, 103.0361904 = wrong (b) n=80, x¯=96.7 90.989, 102.411 = wrong Am I wrong or is it the system lol?
A random sample of n measurements was selected from a population with standard deviation σ=13.6and unknown...
A random sample of n measurements was selected from a population with standard deviation σ=13.6and unknown mean μ. Calculate a 90 % confidence interval for μ for each of the following situations: (a) n=35, x=78.5 (b) n=50, x¯=78.5 (c)n=70, x¯=78.5
A simple random sample with n = 56 provided a sample mean of 26.5 and a sample standard deviation of 4.4.
  A simple random sample with n = 56 provided a sample mean of 26.5 and a sample standard deviation of 4.4. (Round your answers to one decimal place.) (a) Develop a 90% confidence interval for the population mean. to (b) Develop a 95% confidence interval for the population mean. to (c) Develop a 99% confidence interval for the population mean. to (d) What happens to the margin of error and the confidence interval as the confidence level is increased?...
A random sample of ? measurements was selected from a population with standard deviation ?=13.7 and...
A random sample of ? measurements was selected from a population with standard deviation ?=13.7 and unknown mean ?. Calculate a 99 % confidence interval for ? for each of the following situations: (a) n=55, x¯=100.4 ≤?≤ (b)  n=75, x¯=100.4 ≤?≤ (c)  n=105, x¯=100.4 ≤?≤
A random sample of 50 measurements resulted in a sample mean of 62 with a sample standard deviation 8. It is claimed that the true population mean is at least 64.
A random sample of 50 measurements resulted in a sample mean of 62 with a sample standard deviation 8. It is claimed that the true population mean is at least 64.(a) Is there sufficient evidence to refute the claim at the 2% level of significance?(b) What is the p-value?(c) What is the smallest value of α for which the claim will be rejected?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT