8.3 - Estimating a Population Mean, mu

Constructing Confidence Intervals for mu

Assumptions: In order to construct a confidence interval the following conditions must be satisfied

  • The sample is a simple random sample
  • Either or both of the following are satisfied:

i) The population is normally distributed, or

ii) n ≥ 30 (The sample has 30 or more values)

Procedure for Constructing a Confidence Interval for μ (with Known σ)

1)Verify that the required assumptions are satisfied.

2)Decide if you will use a z- or a t-score (z-table or t-table)

  1. Use z when given the standard deviation of the population (σ)
  2. Use t when given the standard deviation of a sample OR if the problem contains data, use 1-Var-Stats to find x-bar and s. (ignore the sigma you see in the calculator)

3)Find the margin of error E (E = ) or (E = )

4)Construct the interval: or

Properties of the t-distribution:

  1. The t-distribution is different for different sample sizes
  2. The mean is t = 0 and the standard deviation is greater than one
  3. The t-distribution is bell shaped like the SND but extends further out because it has a larger variability
  4. The area under the curve is 1
  5. As the sample size n gets larger, the t-distribution gets closer to the standard normal distribution

Using the TI-83 to Construct Confidence Intervals for μ:

STAT>TESTS select 7:ZInterval or 8:TInterval

If you are given DATA, use the Data option, otherwise use the Stats option

8.4 - SAMPLE SIZE FORMULAS for estimating population means

  • ,

where E is the margin of error (Note: our book use the symbol m for the margin of error)

If sigma is not available, use a sample standard deviation from a similar study or use another educated guess. Read example 10, page 383 wheresigma is estimated using a reasonable value for the range since we can consider that Range ~ 6

1) The health of the bear population in Yellowstone National Park is monitored by periodic measurements taken from anesthetized bears. A sample of 54 bears has a mean weight of 182.9 lb. Assuming that sigma is known to be 121.8 lbs., find a 95% confidence interval estimate of the mean of the population of all such bear weights.

a)What is the point estimate for mu?

b)Verify that the requirements for constructing a confidence interval about x-bar are satisfied.

c)Construct a 95% confidence interval estimate for the mean weight of all such beaars. (Are you using z or t? Why?)

d)Now check with the calculator feature

e)The statement “95% confident” means that, if 100 samples of size ______were taken, about _____ intervals will contain the parameter μ and about ____ will not.

f)Complete the following: We are _____% confident that the mean weight of all such bears is between ______and ______

g)With ______% confidence we can say that the mean weights of the bears is ______with a margin of error of ______

h)For ______% of intervals constructed with this method, the sample mean would not differ from the actual population mean by more than ______

i)How can you produce a more precise confidence interval?

j)Section 8.4 - If we want an estimate which is within 25 lbs. of the actual population mean mu, what should be the sample size selected?

2) In order to correctly diagnose the disorder of hydrocephalus, a pediatrician investigates head circumferences of two month old babies. 100 two-month old babies are selected at random and the sample mean observed is 40.573 cm with a sample standard deviation of 1.649.

a)What is the point estimate?

b)Verify that the requirements for constructing a confidence interval about x-bar are satisfied.

c)Construct a 99% confidence interval estimate for the head circumference of all two months old babies. (Are you using z or t? Why?)

d)The statement “99% confident” means that, if 100 samples of size _____ were taken, about _____ intervals will contain the parameter μ and about ____ will not.

e)Complete the following: We are _____% confident that the mean head circumference of all two months old babies is between ______and ______

f)With ______% confidence we can say that the mean head circumference of all two months old babies is ______with a margin of error of ______

g)For ______% of intervals constructed with this method, the sample mean would not differ from the actual population mean by more than ______

h)How can you produce a more precise confidence interval?

i)Section 8.4 - How large of a sample should be selected in order to be 99% confident that the point estimate x-bar will be within 0.2 cm of the true population mean? (We will use the standard deviation of this study, s = ______as an estimate for sigma)

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