Confidence intervals to estimate the mean  of a population.

Discuss with your partner (2min) * About properties of sampling distributions of sample mean and ** Come up with the formula for the Confidence intervals to estimate the mean  of a population.

Confidence intervals for  when  is known Conditions to be met so as to find a formula for a confidence interval for a population mean  1) x is the sample mean from a random sample, 2) Either the population is normal or the sample size n is large (n > 30), and 3) , the population standard deviation, is known.

Confidence intervals for  when  is known Is this typically known? Point estimate Standard deviation of the statistic Bound on error of estimation

Example: Cosmic radiation levels rise with increasing altitude, promoting researchers to consider how pilots and flight crews might be affected by increased exposure to cosmic radiation. A study reported a mean annual cosmic radiation dose of 219 mrems (The millirem (abbreviated "mrem") is a unit used to measure the effect of radiation on the human body.) for a sample of flight personnel of Ethiopian Airlines. Suppose this mean is based on a random sample of 100 flight crew members. Let  = 35 mrems. Calculate and interpret a 95% confidence interval for the actual mean annual cosmic radiation exposure for Ethiopian flight crew members. 1)Data is from a random sample of crew members 2)Sample size n is large (n > 30) 3)  is known First, verify that the conditions are met.

Cosmic Radiation Continued... Let x = 219 mrems n = 100 flight crew members  = 35 mremsCalculate and interpret a 95% confidence interval for the actual mean annual cosmic radiation exposure for Ethiopian flight crew members. What would happen to the width of this interval if the confidence level was 90% instead of 95%?

What does this mean in context? We are 95% confident that the actual mean annual cosmic radiation exposure for Ethiopian flight crew members is between mrems and mrems.

Find a 90% confidence interval estimate for the true mean fills of catsup from this machine. Try Me A certain filling machine has a true population standard deviation  = ounces when used to fill catsup bottles. A random sample of 36 bottles of catsup was selected from the output from this machine and the sample mean was ounces.

How can we compute the minimum sample size to estimate population mean?

The bound on error of estimation associated with a 95% confidence interval is Solve this for n: Choosing a Sample Size We can use this to find the necessary sample size for a particular bound on error of estimation. This requires  to be known – which is rarely the case! When  is unknown, a preliminary study can be performed to estimate  OR make an educated guess of the value of . A rough estimate for  (used with distributions that are not too skewed) is the range divided by 4. When  is unknown, a preliminary study can be performed to estimate  OR make an educated guess of the value of . A rough estimate for  (used with distributions that are not too skewed) is the range divided by 4.

Example:The financial aid office wishes to estimate the mean cost of textbooks per quarter for students at a particular university. For the estimate to be useful, it should be within $20 of the true population mean. How large a sample should be used to be 95% confident of achieving this level of accuracy? The financial aid office believes that the amount spent on books varies with most values between $150 to $550. To estimate  :

The financial aid office wishes to estimate the mean cost of textbooks per quarter for students at a particular university. For the estimate to be useful, it should be within $20 of the true population mean. How large a sample should be used to be 95% confident of achieving this level of accuracy? Always round sample size up to the next whole number !

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Class work-1 Estimated the mean families at a 99% confidence level. It is known that the standard deviation for the size of all the families is 0.6. How large a sample should be if the estimate is going to be within 0.1 of the population?

Class work-2 The doctor wanted to estimate the mean cholesterol level for all adult males. He took a sample of 25 adult males and found that the mean cholesterol level for this sample is 186 with a s.d of 12. Assume that the cholesterol levels for all adult males are approximately normally distributed. Construct a 95% confidence interval for the population mean (Mu).